# Supplemental Modules (Linear Algebra) - Mathematics

• Matrices
An m by n matrix is an array of numbers with m rows and n columns.
• 2: Determinants and Inverses
• 3: Linear Systems
We know that for two by two linear systems of equation, the geometry is that of two lines that either intersect, are parallel, or are the same line. If they intersect then there is exactly one solution, if they are parallel then there are no solutions, and if they are the same line, then there are infinitely many solutions. For three by three systems, the situation is different. The solution set is either the empty set, a point, a line, or a whole plane.
• 4: Linear Programming

## MATH-MATHEMATICS

This course will develop students’ ability to work with and interpret numerical data, to apply logical and symbolic analysis to a variety of problems, and/or to model phenomena with mathematical or logical reasoning. Topics include financial mathematics used in everyday life situations, statistics, and optional topics from a wide array of authentic contexts. Prerequsisite(s): adequate scoring on the Mathematics Placement Exam, or any ACT/SAT and GPA combination that is considered equivalent, or a C- or better in CCDM 113 N or CCDM 114 N or A S 103 or higher

MATH 1134. Fundamentals of Elementary Mathematics I

3 Credits (3)

Numbers and the four operations of arithmetic. Understanding and comparing multiple representations of numbers and operations, in particular how these representations build from whole numbers to integers to fractions and decimals. Applying properties of numbers and operations in contextual situations. Reasoning, communicating, and problem solving with numbers and operations. Applications to ratio, and connections with algebra. Taught primarily through student activities and investigations. Restricted to: EDUC,EPAR,E ED,ECED majors. Prerequsisite(s): adequate scoring on the Mathematics Placement Exam, or any ACT/SAT and GPA combination that is considered equivalent, or a C- or better in MATH 1215 or higher

MATH 1215. Intermediate Algebra

3 Credits (3)

A study of linear and quadratic functions, and an introduction to polynomial, absolute value, rational, radical, exponential, and logarithmic functions. A development of strategies for solving single-variable equations and contextual problems. Prerequsisite(s): adequate scoring on the Mathematics Placement Exam, or any ACT/SAT and GPA combination that is considered equivalent, or a C- or better in CCDM 113 N or CCDM 114 N or A S 103 or higher

MATH 1217. General Supplemental Instruction I

1 Credit (2P)

Collaborative workshop for students enrolled in Intermediate Algebra. Graded: S/U Grading (S/U, Audit). Corequisite(s): MATH 1215

MATH 1220G. College Algebra

3 Credits (3)

The study of equations, functions and graphs, reviewing linear and quadratic functions, and concentrating on polynomial, rational, exponential and logarithmic functions. Emphasizes algebraic problem solving skills and graphical representation of functions. Prerequsisite(s): adequate scoring on the Mathematics Placement Exam, or any ACT/SAT and GPA combination that is considered equivalent, or a C- or better in MATH 1215 or higher

MATH 1221. General Supplemental Instruction II

1 Credit (1+2P)

Collaborative workshop for students enrolled in College Algebra. Graded: S/U Grading (S/U, Audit).

Corequisite(s): MATH 1220G.

MATH 1250G. Trigonometry & Pre-Calculus

4 Credits (3+2P)

Trigonometry & Pre-Calculus includes the study of functions in general with emphasis on the elementary functions: algebraic, exponential, logarithmic, trigonometric and inverse trigonometric functions. Topics include rates of change, limits, systems of equations, conic sections, sequences and series, trigonometric equations and identities, complex number, vectors, and applications.Prerequsisite(s): adequate scoring on the Mathematics Placement Exam, or any ACT/SAT and GPA combination that is considered equivalent, or a C- or better in MATH 1220G or higher

MATH 1350G. Introduction to Statistics

3 Credits (3)

This course discusses the fundamentals of descriptive and inferential statistics. Students will gain introductions to topics such as descriptive statistics, probability and basic probability models used in statistics, sampling and statistical inference, and techniques for the visual presentation of numerical data. These concepts will be illustrated by examples from a variety of fields. Prerequsisite(s): adequate scoring on the Mathematics Placement Exam, or any ACT/SAT and GPA combination that is considered equivalent, or a C- or better in MATH 1215 or higher

MATH 1430G. Applications of Calculus I

3 Credits (2+2P)

An algebraic and graphical study of derivatives and integrals, with an emphasis on applications to business, social science, economics and the sciences. Prerequsisite(s): adequate scoring on the Mathematics Placement Exam, or any ACT/SAT and GPA combination that is considered equivalent, or a C- or better in MATH 1220G or higher

MATH 1435. Applications of Calculus I

3 Credits (3)

Intuitive differential calculus with applications to engineering.

Prerequisite(s): C- or better in MATH 1250G.

MATH 1440. Applications of Calculus II

3 Credits (3)

Topics in this second course of Applications of Calculus include functions of several variables, techniques of integration, an introduction to basic differential equations, and other applications.

Prerequisites: C or better in MATH 1430G or in MATH 1521G, or in MATH 1521H.

MATH 1511G. Calculus and Analytic Geometry I

4 Credits (4)

Limits and continuity, theory and computation of derivatives, applications of derivatives, extreme values, critical points, derivative tests, L'Hopital's Rule. Prerequsisite(s): adequate scoring on the Mathematics Placement Exam, or any ACT/SAT and GPA combination that is considered equivalent, or a C- or better in MATH 1250G or higher

MATH 1521G. Calculus and Analytic Geometry II

4 Credits (4)

Riemann sums, the definite integral, antiderivatives, fundamental theorems, techniques of integration, applications of integrals, improper integrals, Taylor polynomials, sequences and series, power series and Taylor series.

Prerequisite(s): C or better in MATH 1511G.

MATH 1521H. Calculus and Analytic Geometry II Honors

4 Credits (3+1P)

A more advanced treatment of the material of MATH 1521G with additional topics. Consent of Instructor required. Restricted to Las Cruces campus only. Consent of Department.

MATH 1531. Introduction to Higher Mathematics

3 Credits (3)

Logic sets, relations, and functions introduction to mathematical proofs.

Prerequisite(s): C- or better in MATH 1521G or MATH 1521H.

MATH 1996. Topics in Mathematics

Topics to be announced in the Schedule of Classes. Maximum of 3 credits per semester. Total credit not to exceed 6 credits. Community Colleges only.

Prerequisite: consent of instructor.

MATH 2134G. Fundamentals of Elementary Math II

3 Credits (3)

Geometry and measurement. Multiple approaches to solving problems and understanding concepts in geometry. Analyzing and constructing two- and three-dimensional shapes. Measurable attributes, including angle, length, area, and volume. Understanding and applying units and unit conversions. Transformations, congruence, and symmetry. Scale factor and similarity. Coordinate geometry and connections with algebra. Reasoning and communicating about geometric concepts. Taught primarily through student activities and investigations.

Prerequisite(s): C or better in MATH 1134.

MATH 2234. Fundamentals of Elementary Mathematics III

3 Credits (3)

Probability, statistics, ratios, and proportional relationships. Experimental and theoretical probability. Collecting, analyzing, and displaying data, including measurement data. Multiple approaches to solving problems involving proportional relationships, with connections to number and operation, geometry and measurement, and algebra. Understanding data in professional contexts of teaching. Taught primarily through student activities and investigations.

Prerequisite(s): C or better in MATH 2134G.

MATH 2350G. Statistical Methods

3 Credits (3)

Exploratory data analysis. Introduction to probability, random variables and probability distributions. Concepts of Central Limit Theorem and Sampling Distributions such as sample mean and sample proportion. Estimation and hypothesis testing single population parameter for means and proportions and difference of two population parameters for means and proportions. Analysis categorical data for goodness of fit. Fitting simple linear regression model and inference for regression parameters. Analysis of variance for several population means. Techniques in data analysis using statistical packages. Prerequsisite(s): adequate scoring on the Mathematics Placement Exam, or any ACT/SAT and GPA combination that is considered equivalent, or a C- or better in MATH 1215 or higher

MATH 2415. Introduction to Linear Algebra

3 Credits (3)

Systems of equations, matrices, vector spaces and linear transformations. Applications to computer science.

Prerequisite(s): Grade of C- or better in MATH 1521G or MATH 1521H.

MATH 2530G. Calculus III

3 Credits (3)

The purpose of this course, which is a continuation of Calculcus II, is to study the methods of calculus in more detail. The course will cover the material in the textbook from Chapters 10-14.Vectors in the plane and 3-space, vector calculus in two-dimensions, partial differentiation, multiple integration, topics in vector calculus, and complex numbers and functions.

Prerequisite(s): Grade of C- or better in MATH 1521G or MATH 1521H.

MATH 2992. Directed Study

May be repeated for a maximum of 6 credits. Graded S/U.

Prerequisite: consent of the instructor.

A selection of readings and reports in the mathematical sciences, the breadth and depth of which is deemed to fit the needs of the student. Graded S/U.

Prerequisite: consent of instructor.

MATH 313. Fundamentals of Algebra and Geometry I

3 Credits (3+1P)

Covers algebra combined with geometry based on measurements of distance (metric geometry). Secondary mathematics education majors may take course as a math elective. MATH 313 does not substitute for other required math courses. Does not fulfill requirements for major in mathematics.

Prerequisites: MATH 1134 and MATH 2134G.

MATH 316. Calculus with Hands-on Applications

3 Credits (3)

This course, primarily for prospective teachers, is taught in an interactive laboratory format. Students design and construct physical objects for which the planning stage requires calculus techniques. All numerical computations are carried out on graphing calculators. Meets simultaneously with MATH 516, primarily for practicing teachers. Secondary math education majors may take course as a math elective. MATH 316 does not fulfill requirements for majors in mathematics. Consent of instructor required.

MATH 331. Introduction to Modern Algebra

3 Credits (3)

Elements of abstract algebra, including groups, rings and fields.

Prerequisite: C or better in MATH 1531 and MATH 2415.

MATH 332. Introduction to Analysis

3 Credits (3)

Development of the real numbers, a rigorous treatment of sequences, limits, continuity, differentiation, and integration.

Prerequisite: C or better in MATH 1521G or MATH 1521H and MATH 1531.

MATH 377. Introduction to Numerical Methods

3 Credits (3)

Basic numerical methods for interpolation, approximation, locating zeros of functions, integration, and solution of linear equations. Computer-oriented methods will be emphasized.

Prerequisites: grade of C or better in MATH 1521G or MATH 1521H and some programming experience.

MATH 391. Vector Analysis

3 Credits (3)

Calculus of vector valued functions, Green's and Stokes' theorems and applications.

Prerequisite: grade of C or better in MATH 2530G.

MATH 392. Introduction to Ordinary Differential Equations

3 Credits (3)

Introduction to differential equations and dynamical systems with emphasis on modeling and applications. Basic analytic, qualitative and numerical methods. Equilibria and bifurcations. Linear systems with matrix methods, real and complex solutions.

Prerequisite: C or better in MATH 1521G or MATH 1521H or B or better in MATH 1440.

May be repeated for a maximum of 6 credits. Graded S/U.

Prerequisite: consent of faculty member.

MATH 401. Special Topics

1-3 Credits (1-3)

Specific subjects to be announced in the Schedule of Classes. May be used to fulfill a course requirement for the mathematics major. Consent of Instructor required.

MATH 411V. Great Theorems: The Art of Mathematics

3 Credits (3)

Adopts the view of mathematics as art, using original sources displaying the creation of mathematical masterpieces from antiquity to the modern era. Original sources are supplemented by cultural, biographical, and mathematical history placing mathematics in a broad human context.

Prerequisites: Grades of B or better in MATH 1521G or MATH 1521H and any upper division MATH course, with overall GPA of 3.2 or better, or consent of instructor.

MATH 450. Introduction to Topology

3 Credits (3)

Topological spaces: general spaces and specific examples such as metric spaces, Hausdorff spaces and/or normed vector spaces separation axioms continuity, compactness, connectedness related theorems. Crosslisted with: MATH 520.

Prerequisite(s): MATH 332.

MATH 451. Introduction to Differential Geometry

3 Credits (3)

Applies calculus to curves and surfaces in three dimensional Euclidean space.

Prerequisite(s): C- or better in each of MATH 2415 and MATH 391, or consent of instructor.

MATH 452. Foundations of Geometry

3 Credits (3)

Topics in projective, axiomatic Euclidean or non-Euclidean geometries. Restricted to: Main campus only.

Prerequisite(s): C or better in MATH 331 or MATH 332.

MATH 454. Logic and Set Theory

3 Credits (3)

Propositional and first order logic axioms, proofs, models. Semantic and syntactic consequence. Soundness, completeness, compactness, and Loewenheim –Skolem theorems. The Zermelo-Fraenkel axioms for set theory. Well orderings, ordinals, cardinals, the axiom of choice, and the von Neumann hierarchy. Crosslisted with: MATH 524.

Prerequisite(s): C- or better in MATH 331 or MATH 332, or consent of instructor.

MATH 455. Elementary Number Theory

3 Credits (3)

Covers primes, congruences and related topics.

Prerequisite: grade of C or better in MATH 331 or consent of instructor.

MATH 456. Abstract Algebra I: Groups and Rings

3 Credits (3)

Group theory, including cyclic groups, homomorphisms, cosets, quotient groups and Lagrange's theorem. Introduction to rings: ring homomorphisms, ideals, quotient rings, polynomial rings, and principal ideal domains. Crosslisted with: MATH 526.

Prerequisite(s): MATH 331 or consent of instructor.

MATH 459. Survey of Geometry

3 Credits (3)

Basic concepts of Euclidean geometry, ruler and compass constructions. May include topics in non-Euclidean geometry. For non-math majors. Restricted to: Main campus only.

Prerequisite(s): C or better in MATH 331 or MATH 332.

MATH 471. Complex Variables

3 Credits (3)

A first course in complex function theory, with emphasis on applications.

Prerequisite(s): C- or better in MATH 391 or C- or better in both MATH 392 and MATH 2530G.

MATH 472. Fourier Series and Boundary Value Problems

3 Credits (3)

Fourier series and methods of solution of the boundary value problems of applied mathematics.

Prerequisite(s): C- or better in MATH 392.

MATH 473. Calculus of Variations and Optimal Control

3 Credits (3)

Euler's equations, conditions for extrema, direct methods, dynamic programming, and the Pontryagin maximal principle.

Prerequisite(s): C- or better in MATH 392.

MATH 480. Matrix Theory and Applied Linear Algebra

3 Credits (3)

An application driven course, whose topics include rectangular systems, matrix algebra, vector spaces and linear transformations, inner products, and eigenvalues and eigenvectors. Applications may include LU factorization, least squares, data compression, QR factorization, singular value decomposition, and search engines.

Prerequisite(s): C or better in any 300-level course with a MATH prefix.

3 Credits (3)

Rigorous treatment of vector spaces and linear transformations including canonical forms, spectral theory, inner product spaces and related topics.

Prerequisite: grade of C or better in MATH 331.

MATH 491. Introduction to Real Analysis I

3 Credits (3)

Rigorous discussion of the topics introduced in calculus. Sequences, series, limits, continuity, differentiation.

Prerequisite: grade of C or better in MATH 332 or consent of instructor.

MATH 492. Introduction to Real Analysis II

3 Credits (3)

Continuation of MATH 491. Integration, metric spaces and selected topics.

Prerequisite(s): C- or better in MATH 491 or consent of instructor.

May be repeated for a maximum of 6 credits. Graded S/U.

MATH 499. Complex Analysis

3 Credits (3)

Rigorous treatment of complex differentiation and integration, properties of analytic functions, series and Cauchy's integral representations. Crosslisted with: MATH 529.

Prerequisite(s): MATH 332.

MATH 501. Introduction to Differential Geometry

3 Credits (3)

MATH 502. Foundations of Geometry

3 Credits (3)

MATH 505. Elementary Number Theory

3 Credits (3)

MATH 511. Fundamentals of Elementary Mathematics I

3 Credits (3+1P)

Topics from real numbers, geometry, measurement, and algorithms, incorporating calculator technology. Intended for K-8 teachers. As part of course students mentor MATH 1134 undergraduates. Does not fulfill degree requirements for M.S. in mathematics.

MATH 512. Fundamentals of Elementary Mathematics II

3 Credits (3+1P)

Real numbers, geometry, and statistics, incorporating calculator technology. Intended for K-8 teachers. Students serve as mentors to MATH 2134G undergraduates. Does not fulfill degree requirements for M.S. in mathematics.

MATH 513. Fundamentals of Algebra and Geometry I

3 Credits (3+1P)

Algebra and metric geometry, incorporating appropriate calculator technology. Intended for K-8 teachers. Students serve as mentors to MATH 313 undergraduates. Does not fulfill degree requirements for M.S. in mathematics.

MATH 516. Calculus with Hands-on Application

3 Credits (3)

This course, primarily for in-service teachers, is taught in an interactive laboratory format. Students design and construct physical objects for which the planning stage requires calculus techniques. All numerical computations are carried out on graphing calculators. Meets simultaneously with Math 316, primarily for prospective teachers. Does not fulfill degree requirements for M.S. in Mathematics.

Prerequisite(s): MATH 511 and MATH 512 or consent of instructor.

MATH 517. Complex Variables

3 Credits (3)

MATH 518. Fourier Series and Boundary Value Problems

3 Credits (3)

MATH 519. Calculus of Variations and Optimal Control

3 Credits (3)

MATH 520. Introduction to Topology

3 Credits (3)

Same as MATH 450 with additional work for graduate students. Crosslisted with: MATH 450.

MATH 524. Logic and Set Theory

3 Credits (3)

Same as MATH 454 with additional assignments for graduate students. Crosslisted with: MATH 454.

Prerequisite(s): consent of instructor.

3 Credits (3)

Same as MATH 481 with additional work for graduate students. May be repeated up to 3 credits.

MATH 526. Abstract Algebra I: Groups and Rings

3 Credits (3)

Same as MATH 456 with additional work for graduate students. Crosslisted with: MATH 456.

Prerequisite(s): MATH 525 or consent of instructor.

MATH 527. Introduction to Real Analysis I

3 Credits (3)

MATH 528. Introduction to Real Analysis II

3 Credits (3)

MATH 529. Complex Analysis

3 Credits (3)

Same as Math 499 with additional work for graduate students. Crosslisted with: MATH 499.

Prerequisite(s): MATH 528.

MATH 530. Special Topics

Specific subjects to be announced in the Schedule of Classes. May be for unlimited credit with approval of the department.

MATH 531. Ordinary Differential Equations

3 Credits (3)

Linear algebra and linear ordinary differential equations, existence and uniqueness of solution, smooth dependence on initial conditions, flows, introduction to smooth dynamical systems. May be repeated up to 3 credits.

Prerequisite(s): MATH 527, or consent of instructor.

MATH 532. Nonlinear Dynamics

3 Credits (3)

Introduction to nonlinear dynamics and deterministic chaos. Core topics include stability and bifurcations chaos in one dimensional maps universality and re-normalization group. Further topics include symbolic dynamics, fractals, sensitive dependence on initial data, self-organization and complexity and cellular automata. Knowledge of differential equations and linear algebra is desired.

May be repeated for a maximum of 6 credits. Consent of instructor required. Graded: S/U.

MATH 541. Topology I

3 Credits (3)

Connectedness and compactness of topological spaces, introduction to the quotient topology, elementary homotopy theory, the fundamental group, the Seifert-van Kampen theorem

Prerequisite(s): MATH 525 and MATH 528, or consent of instructor.

MATH 542. Topology II

3 Credits (3)

Covering spaces and their classification, singular homology, degree theory, Brouwer's fixed point theorem, CW-complexes and cellular homology, and other applications.

Prerequisite(s): MATH 541 or consent of instructor.

MATH 551. Mathematical Structures in Logic

3 Credits (3)

Lattices, distributive lattices, Boolean algebras, Heyting algebras. Lindenbaum-Tarski algebras of classical and intuitionistic logics. Representation theorems. Modal logics and their algebraic counterparts. Kripke semantics. Goedel translation.

Prerequisite(s): MATH 524.

MATH 552. Universal Algebra and Model Theory

3 Credits (3)

Universal algebra, homomorphisms, subalgebras, products, congruences. Varieties and class operators. Free algebras and Birkhoff's theorem. Ultraproducts and Los's theorem. Congruence distributive varieties and Jonsson's theorem. Universal classes and quasi-varieties.

Prerequisite(s): MATH 524.

MATH 555. Differentiable Manifolds

3 Credits (3)

Differentiable structures, tangent bundles, vector fields and differential equations. Additional topics may include differential forms, De Rham cohomology, Riemannian geometry, and topics chosen by the instructor. May be repeated for a maximum of 9 credits. Consent of instructor required.

Prerequisite(s): MATH 525 and MATH 528, or consent of instructor.

MATH 562. History and Theories of Mathematics Education

3 Credits (3)

A study of the history of the mathematics taught in American schools, including an examination of authentic original textbooks and the changes in their content and the approach to the subject over time, together with writings of people who have influenced the development and changes of mathematics education. Theories of learning mathematics, and current issues in mathematics education.

MATH 563. Algebra with Connections

3 Credits (3)

Connections between Algebra and other K-12 curriculum strands, especially Geometry and Probability / Data Analysis. Apply algebraic modeling and reasoning to a variety of mathematical problem solving situations. Does not fulfill requirements for degrees in mathematics. Consent of instructor required.

Prerequisite(s): Admittance into the MC2-LIFT program.

MATH 564. From Number to Algebra

3 Credits (3)

The progression from Number to Algebra in the K-12 curriculum as a concrete-to-abstract progression. Key concepts considered across the grade levels include the different uses of variables, equivalence in different contexts, patterns, and ratios. Does not fulfill requirements for degrees in mathematics. Consent of instructor required.

Prerequisite(s): Admittance into the MC2-LIFT program.

MATH 566. Data Analysis with Applications

3 Credits (3)

Statistical concepts and terminology in professional uses of data by teachers, such as standardized test score reports and educational research visual displays of data measures of variation and central tendency consideration of how K-12 topics in Data Analysis are developed from one grade level to the next. Does not fulfill requirements for degrees in mathematics. Consent of instructor required.

Prerequisite(s): Admittance into the MC2-LIFT program.

MATH 567. From Measurement to Geometry

3 Credits (3)

The progression from Measurement to Geometry in the K-12 curriculum as a concrete-to abstract progression. Important concepts such as angle, length, and area progress from concrete, measurable situations to more abstract problems which require reasoning and proof. Does not fulfill requirements for degrees in mathematics. Consent of instructor required.

Prerequisite(s): Admittance into the MC2-LIFT program.

MATH 568. Using Number Throughout the Curriculum

3 Credits (3)

Understand number concepts more deeply by seeing many examples of those concepts applied in other content strands. Develop mathematical knowledge and understanding to build a repertoire of ways for students to practice and review basic number skills and concepts as part of later, more advanced courses. Does not fulfill requirements for degrees in mathematics. Consent of instructor required.

Prerequisite(s): Admittance into the MC2-LIFT program.

MATH 569. Geometry with Connections

3 Credits (3)

Connections between Geometry and other K-12 curriculum strands, especially Algebra and Probability / Data Analysis. Address key attributes of geometric concepts by considering their connections within and across grade levels. Does not fulfill requirements for degrees in mathematics. Consent of instructor required.

Prerequisite(s): Admittance into the MC2-LIFT program.

MATH 571. Partial Differential Equations I

3 Credits (3)

The basic equations of mathematical physics. Laplace, Heat and Wave Equations. The method of characteristics, introduction to conservation laws, special solutions.

Prerequisite(s): MATH 518 and MATH 528 or consent of instructor.

MATH 572. Partial Differential Equations II

3 Credits (3)

Sobolev spaces theory: basic definitions and properties, embedding theorems, weak solutions of boundary value problems and variational methods for partial differential equations.

Prerequisite(s): MATH 593 or consent of instructor.

MATH 581. Abstract Algebra II: Fields, Rings and Modules

3 Credits (3)

Topics covered include field extensions algebraic closure polynomials rings irreducibility criteria Noetherian rings algebraic sets Nullstellensatz modules applications to linear algebra.

Prerequisite(s): MATH 526 or consent of instructor.

MATH 582. Module Theory and Homological Algebra

3 Credits (3)

Introductory concepts of homological algebra, including projective, injective and flat modules projective and injective resolutions exactness of functors homology of chain complexes derived functors.

Prerequisite(s): MATH 581 or consent of instructor.

MATH 583. Introduction to Commutative Algebra and Algebraic Geometry

3 Credits (3)

Introduction to the basic notions and techniques of modern algebraic geometry, including the necessary commutative algebra foundation. Topics likely to include algebraic and projective varieties, Nullstellensatz, morphisms, rational and regular functions, local properties. Other topics may include Noether normalization, dimension theory, singularities, sheaves, schemes, Grobner bases.

Prerequisite(s): MATH 581 or consent of instructor.

MATH 593. Measure and Integration

3 Credits (3)

Measure spaces, measurable functions, extension and decomposition theorems for measures, integration on measure spaces, absolute continuity, iterated integrals.

Prerequisite: MATH 528 or consent of instructor.

MATH 594. Real Analysis

3 Credits (3)

Differentiation, Lp spaces, Banach spaces, measure and topology, other selected topics.

Prerequisite: MATH 593.

MATH 595. Introduction to Functional Analysis

3 Credits (3)

Banach spaces. The three basic principles: uniform boundedness principle, closed graph/open mapping theorems, Hahn-Banach theorem.

Prerequisite(s): MATH 594, or consent of instructor.

MATH 599. Master's Thesis

1-15 Credits

MATH 600. Doctoral Research

1-15 Credits

MATH 698. Selected Topics

1-15 Credits

MATH 700. Doctoral Dissertation

1-15 Credits

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## Supplemental Modules (Linear Algebra) - Mathematics

CALCULUS (SINGLE VARIBALE)

MODULE 1A: CALCULUS: (12 LECTURES)

INTERVALS, CONVERGENCE OF SEQUENCES AND SERIES OF REAL NUMBERS, LIMIT AND CONTINUITY OF FUNCTIONS, DIFFERENTIABILITY OF FUNCTIONS, ROLLE’S THEOREM, MEAN VALUE THEOREMS, TAYLOR’S AND MACLAURIN THEOREMS WITH REMAINDERS INDETERMINATE FORMS AND L'HOSPITAL'S RULE MAXIMA AND MINIMA, RIEMANN INTEGRATION, FUNDAMENTAL THEOREM OF CALCULUS.

MODULE 1B: CALCULUS: (8 LECTURES)

EVOLUTES AND INVOLUTES EVALUATION OF DEFINITE AND IMPROPER INTEGRALS BETA AND GAMMA FUNCTIONS AND THEIR PROPERTIES APPLICATIONS OF DEFINITE INTEGRALS TO EVALUATE SURFACE AREAS AND VOLUMES OF REVOLUTIONS.

MODULE 1C: SERIES: (PREREQUISITE 2B) (8 LECTURES)

POWER SERIES, TAYLOR'S SERIES. SERIES FOR EXPONENTIAL, TRIGONOMETRIC AND LOGARITHMIC FUNCTIONS FOURIER SERIES: HALF RANGE SINE AND COSINE SERIES, PARSEVAL’S THEOREM]

TEXTBOOKS/REFERENCES:

• B. THOMAS AND R.L. FINNEY, CALCULUS AND ANALYTIC GEOMETRY, 9TH EDITION, PEARSON, REPRINT, 2002.
• VEERARAJAN T., ENGINEERING MATHEMATICS FOR FIRST YEAR, TATA MCGRAWHILL, NEW DELHI, 2008. &
• RAMANA B.V., HIGHER ENGINEERING MATHEMATICS, TATA MCGRAW HILL NEW DELHI, 11TH REPRINT, 2010. &
• P. BALI AND MANISH GOYAL, A TEXT BOOK OF ENGINEERING MATHEMATICS, LAXMI PUBLICATIONS,REPRINT, 2010.
• S. GREWAL, HIGHER ENGINEERING MATHEMATICS, KHANNA PUBLISHERS, 35TH EDITION, 2000.

MATRICES AND LINEAR ALGEBRA

MODULE 2A: MATRICES (IN CASE VECTOR SPACES IS NOT TO BE TAUGHT) (14 LECTURES)

ALGEBRA OF MATRICES, INVERSE AND RANK OF A MATRIX, RANK-NULLITY THEOREM SYSTEM OF LINEAR EQUATIONS SYMMETRIC, SKEW-SYMMETRIC AND ORTHOGONAL MATRICES DETERMINANTS EIGENVALUES AND EIGENVECTORS DIAGONALIZATION OF MATRICES CAYLEY-HAMILTON THEOREM, ORTHOGONAL TRANSFORMATION AND QUADRATIC TO CANONICAL FORMS.

MODULE 2B: MATRICES (IN CASE VECTOR SPACES IS TO BE TAUGHT) (8 LECTURES)

MATRICES, VECTORS: ADDITION AND SCALAR MULTIPLICATION, MATRIX MULTIPLICATION LINEAR SYSTEMS OF EQUATIONS, LINEAR INDEPENDENCE, RANK OF A MATRIX, DETERMINANTS, CRAMER’S RULE, INVERSE OF A MATRIX, GAUSS ELIMINATION AND GAUSS-JORDAN ELIMINATION.

MODULE 2C: VECTOR SPACES (PREREQUISITE 4B) (10 LECTURES)

VECTOR SPACE, LINEAR DEPENDENCE OF VECTORS, BASIS, DIMENSION LINEAR TRANSFORMATIONS (MAPS), RANGE AND KERNEL OF A LINEAR MAP, RANK AND NULLITY, INVERSE OF A LINEAR TRANSFORMATION, RANK- NULLITY THEOREM, COMPOSITION OF LINEAR MAPS, MATRIX ASSOCIATED WITH A LINEAR MAP.

MODULE 2D: VECTOR SPACES (PREREQUISITE 4B-C) (10 LECTURES)

EIGENVALUES, EIGENVECTORS, SYMMETRIC, SKEW-SYMMETRIC AND ORTHOGONAL MATRICES, EIGENBASES. DIAGONALIZATION INNER PRODUCT SPACES, GRAM-SCHMIDT ORTHOGONALIZATION.

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This programme is accredited.

### Module details

Due to the impact of COVID-19 we're changing how the course is delivered.

#### Year One Compulsory Modules

##### Calculus I (MATH101)

1. To introduce the basic ideas of differential and integral calculus, to develop the basic skills required to work with them and to apply these skills to a range of problems.

2. To introduce some of the fundamental concepts and techniques of real analysis, including limits and continuity.

3. To introduce the notions of sequences and series and of their convergence.

(LO1) Understand the key definitions that underpin real analysis and interpret these in terms of straightforward examples.

(LO2) Apply the methods of calculus and real analysis to solve previously unseen problems (of a similar style to those covered in the course).

(LO3) Understand in interpret proofs in the context of real analysis and apply the theorems developed in the course to straightforward examples.

(LO4) Independently construct proofs of previously unseen mathematical results in real analysis (of a similar style to those demonstrated in the course).

(LO5) Differentiate and integrate a wide range of functions

(LO6) Sketch graphs and solve problems involving optimisation and mensuration

(LO7) Understand the notions of sequence and series and apply a range of tests to determine if a series is convergent

##### Calculus II (MATH102)

To discuss local behaviour of functions using Taylor’s theorem.
To introduce multivariable calculus including partial differentiation, gradient, extremum values and double integrals.

(LO1) Use Taylor series to obtain local approximations to functions

(LO2) Obtain partial derivatives and use them in several applications such as, error analysis, stationary points change of variables.

(LO3) Evaluate double integrals using Cartesian and Polar Co-ordinates.

##### Introduction to Linear Algebra (MATH103)

• To develop techniques of complex numbers and linear algebra, including equation solving, matrix arithmetic and the computation of eigenvalues and eigenvectors.
• To develop geometrical intuition in 2 and 3 dimensions.
• To introduce students to the concept of subspace in a concrete situation.
• To provide a foundation for the study of linear problems both within mathematics and in other subjects

(LO1) Manipulate complex numbers and solve simple equations involving them, solve arbitrary systems of linear equations.

(LO2) Understand and use matrix arithmetic, including the computation of matrix inverses.

(LO3) Compute and use determinants.

(LO4) Understand and use vector methods in the geometry of 2 and 3 dimensions.

(LO5) Calculate eigenvalues and eigenvectors.

##### Introduction to Statistics Using R (MATH163)

1. Use software R to display and analyse data, perform tests and demonstrate basic statistical concepts.

2. Describe statistical data and display it using variety of plots and diagrams.

3. Understand basic laws of probability: law of total probability, independence, Bayes’ rule.

4. Be able to estimate mean and variance.

5. Be familiar with properties of some probability distributions and relations between them: Binomial, Poisson, Normal, t, Chi-squared.

6. To perform simple statistical tests: goodness-of-fit test, z-test, t-test.

7. To understand and be able to interpret p-values.

8. To be able to report finding of statistical outcomes to non-specialist audience.

9. Group work will help students to develop transferable skills such as communication, the ability to coordinate and prioritise tasks, time management and teamwork.

(LO1) An ability to apply statistical concepts and methods covered in the module's syllabus to well defined contexts and interpret results.

(LO2) An ability to understand, communicate, and solve straightforward problems related to the theory and derivation of statistical methods covered in the module's syllabus.

(LO3) An ability to understand, communicate, and solve straightforward theoretical and applied problems related to probability theory covered in the syllabus.

(LO4) Use the R programming language fluently in well-defined contexts. Students should be able to understand and correct given code select appropriate code to solve given problems select appropriate packages to solve given problems and independently write small amounts of code.

##### Mathematical It Skills (MATH111)

•To acquire key mathematics-specific computer skills.
•To reinforce mathematics as a practical discipline by active experience and experimentation, using the computer as a tool.
•To illustrate and amplify mathematical concepts and techniques.
•To initiate and develop problem solving, group work and report writing skills.
•To initiate and develop modelling skills.
•To develop team work skills.

(LO1) After completing the module, students should be able to tackle project work, including writing up of reports detailing their solutions to problems.

(LO2) After completing the module, students should be able to use computers to create documents containing formulae, tables, plots and references.

(LO3) After completing the module, students should be able to use mathematical software packages such as Maple and Matlab to manipulate mathematical expressions and to solve simple problems.

(LO4) After completing the module, students should be able to better understand the mathematical topics covered, through direct experimentation with the computer.

(S1) Problem solving skills

(S8) Mathematical modelling skills

##### Introduction to Study and Research in Mathematics (MATH107)

This module addresses what it means to be a mathematician, as an undergraduate and beyond that into academia or industry, and prepares students to succeed as such. It aims to:
- bridge the gap in language and philosophy between A-level and (more rigorous) University mathematics
- equip students with the basic tools they need for their mathematical careers
- enable students to take responsibility for their learning and become active learners
- familiarise students with mathematics research as conducted within the department
- build students' confidence in handling various forms of mathematical communication.

(LO1) Foundational knowledge of objects, processes, logic and reasoning required for university level mathematics.

(LO2) Awareness of the nature of mathematics at University and beyond, and the implications of this for themselves.

(LO3) Demonstrate proactive engagement in the student's own learning.

(LO4) Development of skills for mathematical communication (including mathematics proofs).

##### Newtonian Mechanics (MATH122)

To provide a basic understanding of the principles of Classical Mechanics and their application to simple dynamical systems. Learning Outcomes: After completing the module students should be able to analyse real world problems involving: - the motions of bodies under simple force systems - conservation laws for momentum and energy - rigid body dynamics using centre of mass, angular momentum and moments of inertia

(LO1) the motions of bodies under simple force systems

(LO2) conservation laws for momentum and energy

(LO3) rigid body dynamics using centre of mass, angular momentum and moments

(LO4) oscillation, vibration, resonance

(S1) Representing physical problems in a mathematical way

(S2) Problem Solving Skills

##### Numbers, Groups and Codes (MATH142)

- To provide an introduction to rigorous reasoning in axiomatic systems exemplified by the framework of group theory.

- To give an appreciation of the utility and power of group theory as the study of symmetries.

- To introduce public-key cryptosystems as used in the transmission of confidential data, and also error-correcting codes used to ensure that transmission of data is accurate. Both of these ideas are illustrations of the application of algebraic techniques.

(LO1) Be able to apply the Euclidean algorithm to find the greatest common divisor of a pair of positive integers, and use this procedure to find the inverse of an integer modulo a given integer.

(LO2) Be able to solve linear congruences and apply appropriate techniques to solve systems of such congruences.

(LO3) Be able to perform a range of calculations and manipulations with permutations.

(LO4) Recall the definition of a group and a subgroup and be able to identify these in explicit examples.

(LO5) Be able to prove that a given mapping between groups is a homomorphism and identify isomorphic groups.

(LO6) To be able to apply group theoretic ideas to applications with error correcting codes.

(LO7) Engage in group project work to investigate applications of the theoretical material covered in the module.

### Programme Year Two

In Year Two, you will choose some compulsory and some optional modules from the list below. Please note that we regularly review our teaching so the choice of modules may change.

#### Year Two Compulsory Modules

##### Differential Equations (MATH221)

•To familiarize students with basic ideas and fundamental techniques to solve ordinary differential equations.

•To illustrate the breadth of applications of ODEs and fundamental importance of related concepts.

(LO1) To understand the basic properties of ODE, including main features of initial value problems and boundary value problems, such as existence and uniqueness of solutions.

(LO2) To know the elementary techniques for the solution of ODEs.

(LO3) To understand the idea of reducing a complex ODE to a simpler one.

(LO4) To be able to solve linear ODE systems (homogeneous and non-homogeneous) with constant coefficients matrix.

(LO5) To understand a range of applications of ODE.

(S1) Problem solving skills

##### Vector Calculus With Applications in Fluid Mechanics (MATH225)

To provide an understanding of the various vector integrals, the operators div, grad and curl and the relations between them. To give an appreciation of the many applications of vector calculus to physical situations. To provide an introduction to the subjects of fluid mechanics and electromagnetism.

(LO1) After completing the module students should be able to: - Work confidently with different coordinate systems. - Evaluate line, surface and volume integrals. - Appreciate the need for the operators div, grad and curl together with the associated theorems of Gauss and Stokes. - Recognise the many physical situations that involve the use of vector calculus. - Apply mathematical modelling methodology to formulate and solve simple problems in electromagnetism and inviscid fluid flow. All learning outcomes are assessed by both examination and course work.

##### Linear Algebra and Geometry (MATH244)

To introduce general concepts of linear algebra and its applications in geometry and other areas of mathematics.

(LO1) To understand the geometric meaning of linear algebraic ideas.

(LO2) To know the concept of an abstract vector space and how it is used in different mathematical situations.

(LO3) To be able to apply a change of coordinates to simplify a linear map.

(LO4) To be able to work with matrix groups, in particular GL(n), O(n) and SO(n),.

(LO5) To understand bilinear forms from a geometric point of view.

(S1) Problem solving skills

##### Statistics and Probability I (MATH253)

Use the R programming language fluently to analyse data, perform tests, ANOVA and SLR, and check assumptions.

Develop confidence to understand and use statistical methods to analyse and interpret data check assumptions of these methods.

Develop an awareness of ethical issues related to the design of
studies.

(LO1) An ability to apply advanced statistical concepts and methods covered in the module's syllabus to well defined contexts and interpret results.

(LO2) Use the R programming language fluently for a broad selection of statistical tests, in well-defined contexts.

(S1) Problem solving skills

##### Complex Functions (MATH243)

To introduce the student to a surprising, very beautiful theory having intimate connections with other areas of mathematics and physical sciences, for instance ordinary and partial differential equations and potential theory.

(LO1) To understand the central role of complex numbers in mathematics.

(LO2) To develop the knowledge and understanding of all the classical holomorphic functions.

(LO3) To be able to compute Taylor and Laurent series of standard holomorphic functions.

(LO4) To understand various Cauchy formulae and theorems and their applications.

(LO5) To be able to reduce a real definite integral to a contour integral.

(LO6) To be competent at computing contour integrals.

(S1) Problem solving skills

#### Year Two Optional Modules

##### Classical Mechanics (MATH228)

To provide an understanding of the principles of Classical Mechanics and their application to dynamical systems.

(LO1) To understand the variational principles, Lagrangian mechanics, Hamiltonian mechanics.

(LO2) To be able to use Newtonian gravity and Kepler's laws to perform the calculations of the orbits of satellites, comets and planetary motions.

(LO3) To understand the motion relative to a rotating frame, Coriolis and centripetal forces, motion under gravity over the Earth's surface.

(LO4) To understand the connection between symmetry and conservation laws.

(LO5) To be able to work with inertial and non-inertial frames.

(S1) Applying mathematics to physical problems

(S2) Problem solving skills

##### Metric Spaces and Calculus (MATH242)

To introduce the basic elements of the theory of metric spaces and calculus of several variables.

(LO1) After completing the module students should: Be familiar with a range of examples of metric spaces. Have developed their understanding of the notions of convergence and continuity.

(LO2) Understand the contraction mapping theorem and appreciate some of its applications.

(LO3) Be familiar with the concept of the derivative of a vector valued function of several variables as a linear map.

(LO4) Understand the inverse function and implicit function theorems and appreciate their importance.

(LO5) Have developed their appreciation of the role of proof and rigour in mathematics.

(S1) problem solving skills

##### Commutative Algebra (MATH247)

To give an introduction to abstract commutative algebra and show how it both arises naturally, and is a useful tool, in number theory.

(LO1) After completing the module students should be able to: • Work confidently with the basic tools of algebra (sets, maps, binary operations and equivalence relations). • Recognise abelian groups, different kinds of rings (integral, Euclidean, principal ideal and unique factorisation domains) and fields. • Find greatest common divisors using the Euclidean algorithm in Euclidean domains. • Apply commutative algebra to solve simple number-theoretic problems.

##### Statistics and Probability II (MATH254)

To introduce statistical distribution theory which forms the basis for all applications of statistics, and for further statistical theory.

(LO1) To have an understanding of basic probability calculus.

(LO2) To have an understanding of a range of techniques for solving real life problems of probabilistic nature.

(S1) Problem solving skills

##### Financial Mathematics (MATH260)

To provide an understanding of basic theories in Financial Mathematics used in the study process of actuarial/financial interest.

To provide an introduction to financial methods and derivative pricing financial instruments in discrete time set up.

(LO1) Know how to optimise portfolios and calculating risks associated with investment.

(LO2) Demonstrate principles of markets.

(LO3) Assess risks and rewards of financial products.

(LO4) Understand mathematical principles used for describing financial markets.

##### Numerical Methods for Applied Mathematics (MATH266)

To demonstrate how these ideas can be implemented using a high-level programming language, leading to accurate, efficient mathematical algorithms.

(LO1) To strengthen students’ knowledge of scientific programming, building on the ideas introduced in MATH111.

(LO2) To provide an introduction to the foundations of numerical analysis and its relation to other branches of Mathematics.

(LO3) To introduce students to theoretical concepts that underpin numerical methods, including fixed point iteration, interpolation, orthogonal polynomials and error estimates based on Taylor series.

(LO4) To demonstrate how analysis can be combined with sound programming techniques to produce accurate, efficient programs for solving practical mathematical problems.

(S2) Problem solving skills

##### Operational Research (MATH269)

The aims of the module are to develop an understanding of how mathematical modelling and operational research techniques are applied to real-world problems and to gain an understanding of linear and convex programming, multi-objective problems, inventory control and sensitivity analysis.

(LO1) To understand the operational research approach.

(LO2) To be able to apply standard methods of operational research to a wide range of real-world problems as well as to problems in other areas of mathematics.

(LO3) To understand the advantages and disadvantages of particular operational research methods.

(LO4) To be able to derive methods and modify them to model real-world problems.

(LO5) To understand and be able to derive and apply the methods of sensitivity analysis.

(LO6) To understand the importance of sensitivity analysis.

(S2) Problem solving skills

(S4) Self-management readiness to accept responsibility (i.e. leadership), flexibility, resilience, self-starting, initiative, integrity, willingness to take risks, appropriate assertiveness, time management, readiness to improve own performance based on feedback/reflective learning

##### Mathematics Education and Communication (MATH291)

1.Improving communication skills.

2.Exposing students to current pedagogical practice and issues related to child protection

3.Encouraging students to reflect on mathematics with which they are familiar in a teaching context.

(LO1) Confidence in planning and presenting mathematics to school-age children.

(LO2) Knowledge of current best pedagogical practice and child protection issues.

(LO3) Ability to work in a team.

(LO4) Understanding the role of outreach in mathematics education.

(S1) Improving own learning/performance - Reflective practice

(S2) Communication (oral, written and visual) - Presentation skills – oral

(S3) Communication (oral, written and visual) - Presentation skills - written

(S4) Communication (oral, written and visual) - Presentation skills - visual

(S5) Communication (oral, written and visual) - Report writing

(S6) Working in groups and teams - Group action planning

### Programme Year Three

Choose 75 credits of FHEQ Level 6 Modules (MATH3XX) from the list below Choose 45 credits of FHEQ Level 7 modules (MATH4XX) from the list below MATH490 cannot be taken in Year 3.
If MATH499 is taken in Year 3 then MATH490 must be taken in Year 4.

#### Year Three Optional Modules

##### Further Methods of Applied Mathematics (MATH323)

•To give an insight into some specific methods for solving important types of ordinary differential equations.

•To provide a basic understanding of the Calculus of Variations and to illustrate the techniques using simple examples in a variety of areas in mathematics and physics.

•To build on the students'' existing knowledge of partial differential equations of first and second order.

(LO1) After completing the module students should be able to:
- use the method of "Variation of Arbitrary Parameters" to find the solutions of some inhomogeneous ordinary differential equations.

- solve simple integral extremal problems including cases with constraints

- classify a system of simultaneous 1st-order linear partial differential equations, and to find the Riemann invariants and general or specific solutions in appropriate cases

- classify 2nd-order linear partial differential equations and, in appropriate cases, find general or specific solutions.

[This might involve a practical understanding of a variety of mathematics tools e.g. conformal mapping and Fourier transforms.]

##### Cartesian Tensors and Mathematical Models of Solids and VIscous Fluids (MATH324)

To provide an introduction to the mathematical theory of viscous fluid flows and solid elastic materials. Cartesian tensors are first introduced. This is followed by modelling of the mechanics of continuous media. The module includes particular examples of the flow of a viscous fluid as well as a variety of problems of linear elasticity.

(LO1) To understand and actively use the basic concepts of continuum mechanics such as stress, deformation and constitutive relations.

(LO2) To apply mathematical methods for analysis of problems involving the flow of viscous fluid or behaviour of solid elastic materials.

(S1) Problem solving skills

##### Quantum Mechanics (MATH325)

The aim of the module is to lead the student to an understanding of the way that relatively simple mathematics (in modern terms) led Bohr, Einstein, Heisenberg and others to a radical change and improvement in our understanding of the microscopic world.

(LO1) To be able to solve Schrodinger's equation for simple systems.

(LO2) To have an understanding of the significance of quantum mechanics for both elementary systems and the behaviour of matter.

(S1) Problem solving skills

##### Relativity (MATH326)

(i) To introduce the physical principles behind Special and General Relativity and their main consequences

(ii) To develop the competence in the mathematical framework of the subjects - Lorentz transformation and Minkowski space-time, semi-Riemannian geometry and curved space-time, symmetries and conservation laws, Variational principles.

(iii) To develop the understanding of the dynamics of particles and of the Maxwell field in Minkowski space-time, and of particles in curved space-time

(iv) To develop the knowledge of tests of General Relativity, including the classical tests (perihelion shift, gravitational deflection of light)

(v) To understand the basic concepts of black holes and (time permitting) relativistic cosmology and gravitational waves.

(LO1) To be proficient at calculations involving Lorentz transformations, the kinematical and dynamical quantities associated to particles in Minkowski space-times, and the application of the conservation law for the four-momentum to scattering processes.

(LO2) To know the relativistically covariant form of the Maxwell equations .

(LO3) To know the action principles for relativistic particles, the Maxwell field and the gravitational field.

(LO4) To be proficient at calculations in semi-Riemannian geometry as far as needed for General Relativity, including calculations involving general coordinate transformations, tensor fields, covariant derivatives, parallel transport, geodesics and curvature.

(LO5) To understand the arguments leading to the Einstein's field equations and how Newton's law of gravity arises as a limiting

(LO6) To be able to calculate the trajectories of bodies in a Schwarzschild space-time.

(S1) problem solving skills

##### Number Theory (MATH342)

To give an account of elementary number theory with use of certain algebraic methods and to apply the concepts to problem solving.

(LO1) To understand and solve a wide range of problems about integers numbers.

(LO2) To have a better understanding of the properties of prime numbers.

(S1) Problem solving skills

##### Group Theory (MATH343)

To introduce the basic techniques of finite group theory with the objective of explaining the ideas needed to solve classification results.

(LO1) Understanding of abstract algebraic systems (groups) by concrete, explicit realisations (permutations, matrices, Mobius transformations).

(LO2) The ability to understand and explain classification results to users of group theory.

(LO3) The understanding of connections of the subject with other areas of Mathematics.

(LO4) To have a general understanding of the origins and history of the subject.

(S1) Problem solving skills

##### Differential Geometry (MATH349)

This module is designed to provide an introduction to the methods of differential geometry, applied in concrete situations to the study of curves and surfaces in euclidean 3-space. While forming a self-contained whole, it will also provide a basis for further study of differential geometry, including Riemannian geometry and applications to science and engineering.

(LO1) 1a. Knowledge and understanding: Students will have a reasonable understanding of invariants used to describe the shape of explicitly given curves and surfaces.

(LO2) 1b. Knowledge and understanding: Students will have a reasonable understanding of special curves on surfaces.

(LO3) 1c. Knowledge and understanding: Students will have a reasonable understanding of the difference between extrinsically defined properties and those which depend only on the surface metric.

(LO4) 1d. Knowledge and understanding: Students will have a reasonable understanding of the passage from local to global properties exemplified by the Gauss-Bonnet Theorem.

(LO5) 2a. Intellectual abilities: Students will be able to use differential calculus to discover geometric properties of explicitly given curves and surfaces.

(LO6) 2b. Intellectual abilities: Students will be able to understand the role played by special curves on surfaces.

(LO7) 3a. Subject-based practical skills: Students will learn to compute invariants of curves and surfaces.

(LO8) 3b. Subject-based practical skills: Students will learn to interpret the invariants of curves and surfaces as indicators of their geometrical properties.

(LO9) 4a. General transferable skills: Students will improve their ability to think logically about abstract concepts,

(LO10) 4b. General transferable skills: Students will improve their ability to combine theory with examples in a meaningful way.

(S1) Problem solving skills

##### Applied Probability (MATH362)

To give examples of empirical phenomena for which stochastic processes provide suitable mathematical models. To provide an introduction to the methods of probabilistic model building for &lsquo&lsquodynamic" events occurring over time. To familiarise students with an important area of probability modelling.

(LO1) 1. Knowledge and Understanding After the module, students should have a basic understanding of:
(a) some basic models in discrete and continuous time Markov chains such as random walk and Poisson processes
(b) important subjects like transition matrix, equilibrium distribution, limiting behaviour etc. of Markov chain
(c) special properties of the simple finite state discrete time Markov chain and Poisson processes, and perform calculations using these.
2. Intellectual Abilities After the module, students should be able to:
(a) formulate appropriate situations as probability models: random processes
(b) demonstrate knowledge of standard models (c) demonstrate understanding of the theory underpinning simple dynamical systems
3. General Transferable Skills
(a) numeracy through manipulation and interpretation of datasets
(b) communication through presentation of written work and preparation of diagrams
(c) problem solving through tasks set in tutorials
(d) time management in the completion of practicals and the submission of assessed work
(e) choosing, applying and interpreting results of probability techniques for a range of different problems.

##### Linear Statistical Models (MATH363)

- To understand how regression methods for continuous data extend to include multiple continuous and categorical predictors, and categorical response variables.

- To provide an understanding of how this class of models forms the basis for the analysis of experimental and also observational studies.

- To understand generalized linear models.

- To develop skills in using an appropriate statistical software package.

(LO1) Be able to understand the rationale and assumptions of linear regression and analysis of variance.

(LO2) Be able to understand the rationale and assumptions of generalized linear models.

(LO3) Be able to recognise the correct analysis for a given experiment.

(LO4) Be able to carry out and interpret linear regressions and analyses of variance, and derive appropriate theoretical results.

(LO5) Be able to carry out and interpret analyses involving generalised linear models and derive appropriate theoretical results.

(LO6) Be able to perform linear regression, analysis of variance and generalised linear model analysis using an appropriate statistical software package.

##### Game Theory (MATH331)

To explore, from a game-theoretic point of view, models which have been used to understand phenomena in which conflict and cooperation occur. To see the relevance of the theory not only to parlour games but also to situations involving human relationships, economic bargaining (between trade union and employer, etc), threats, formation of coalitions, war, etc. To treat fully a number of specific games including the famous examples of "The Prisoners' Dilemma" and "The Battle of the Sexes". To treat in detail two-person zero-sum and non-zero-sum games. To give a brief review of n-person games. In microeconomics, to look at exchanges in the absence of money, i.e. bartering, in which two individuals or two groups are involved.To see how the Prisoner's Dilemma arises in the context of public goods.

(LO1) To extend the appreciation of the role of mathematics in modelling in Economics and the Social Sciences.

(LO2) To be able to formulate, in game-theoretic terms, situations of conflict and cooperation.

(LO3) To be able to solve mathematically a variety of standard problems in the theory of games and to understand the relevance of such solutions in real situations.

##### Numerical Methods for Ordinary and Partial Differential Equations (MATH336)

Many real-world systems in mathematics, physics and engineering can be described by differential equations. In rare cases these can be solved exactly by purely analytical methods, but much more often we can only solve the equations numerically, by reducing the problem to an iterative scheme that requires hundreds of steps. We will learn efficient methods for solving ODEs and PDEs on a computer.

(LO1) Demonstrate an advanced knowledge of the analysis of ODEs and PDEs underpinning the scientific programming within our context.

(LO2) Demonstrate an extended understanding of scientific programming and its application to numerical analysis and to other branches of Mathematics.

(LO3) Continuous engagement with putting practical problems into mathematical language.

(S2) Problem solving skills

##### Combinatorics (MATH344)

To provide an introduction to the problems and methods of Combinatorics, particularly to those areas of the subject with the widest applications such as pairings problems, the inclusion-exclusion principle, recurrence relations, partitions and the elementary theory of symmetric functions.

(LO1) After completing the module students should be able to: understand of the type of problem to which the methods of Combinatorics apply, and model these problems solve counting and arrangement problems solve general recurrence relations using the generating function method appreciate the elementary theory of partitions and its application to the study of symmetric functions.

##### The Magic of Complex Numbers: Complex Dynamics, Chaos and the Mandelbrot Set (MATH345)

1. To introduce students to the theory of the iteration of functions of one complex variable, and its fundamental objects

2. To introduce students to some topics of current and recent research in the field

3. To study various advanced results from complex analysis, and show how to apply these in a dynamical setting

4. To illustrate that many results in complex analysis are "magic", in that there is no reason to expect them in a real-variable context, and the implications of this in complex dynamics

5. To explain how complex-variable methods have been instrumental in questions purely about real-valued one-dimensional dynamical systems, such as the logistic family.

6. To deepen students' appreciations for formal reasoning and proof. After completing the module, students should be able to:
1. understand the compactification of the complex plane to the Riemann sphere, and use spherical distances and derivatives.
2. use Möbius transformations to transform the Riemann sphere and to normalise complex dynamical systems.
3. state and apply the definitions of Julia and Fatou sets of polynomials, and understand their basic properties.
4. determine whether points with simple orbits, such as certain periodic points, belong to the Julia set or the Fatou set.
5. apply advanced results from complex analysis in the setting of complex dynamics.
6. determine whether certain types of quadratic polynomials belong to the Mandelbrot set or not.

(LO1) To understand the compactification of the complex plane to the Riemann sphere, and be able to use spherical distances and derivatives.

(LO2) To be able to use Möbius transformations to transform the Riemann sphere and to normalise complex dynamical systems.

(LO3) To be able to state and apply the definitions of Julia and Fatou sets of polynomials, and understand their basic properties.

(LO4) To be able to determine whether points with simple orbits, such as certain periodic points, belong to the Julia set or the Fatou set.

(LO5) To know how to apply advanced results from complex analysis in a dynamical setting.

(LO6) To be able to determine whether certain types of quadratic polynomials belong to the Mandelbrot set or not.

(S1) Problem solving/ critical thinking/ creativity analysing facts and situations and applying creative thinking to develop appropriate solutions.

(S2) Problem solving skills

##### Topology (MATH346)

1. To introduce students to the mathematical notions of space and continuity.
2. To develop students’ ability to reason in an axiomatic framework.
3. To provide students with a foundation for further study in the area of topology and geometry, both within their degree and subsequently.
4. To introduce students to some basic constructions in topological data analysis.
5. To enhance students’ understanding of mathematics met elsewhere within their degree (in particular real and complex analysis, partial orders, groups) by placing it within a broader context.
6. To deepen students’ understanding of mathematical objects commonly discussed in popular and recreational mathematics (e.g. Cantor sets, space-filling curves, real surfaces).

(BH1) An understanding of the ubiquity of topological spaces within mathematics.

(BH2) Knowledge of a wide range of examples of topological spaces, and of their basic properties.

(BH3) The ability to construct proofs of, or counter-examples to, simple statements about topological spaces and continuous maps.

(BH4) The ability to decide if a (simple) space is connected and/or compact.

(BH5) The ability to construct the Cech and Vietoris-Rips complexes of a point set in Euclidean spac. e

(BH6) The ability to compute the fundamental group of a (simple) space, and to use it to distinguish spaces.

##### Applied Stochastic Models (MATH360)

To give examples of empirical phenomena for which stochastic processes provide suitable mathematical models. To provide an introduction to the methods of stochastic model building for 'dynamic' events occurring over time or space. To enable further study of the theory of stochastic processes by using this course as a base.

(LO1) To understand the theory of continuous-time Markov chains.

(LO2) To understand the theory of diffusion processes.

(LO3) To be able to solve problems arising in epidemiology, mathematical biology, financial mathematics, etc. using the theory of continuous-time Markov chains and diffusion processes.

(LO4) To acquire an understanding of the standard concepts and methods of stochastic modelling.

(S1) Problem solving skills

##### Theory of Statistical Inference (MATH361)

To introduce some of the concepts and principles which provide theoretical underpinning for the various statistical methods, and, thus, to consolidate the theory behind the other second year and third year statistics options.

(LO1) To acquire a good understanding of the classical approach to, and especially the likelihood methods for, statistical inference.

(LO2) To acquire an understanding of the blossoming area of Bayesian approach to inference.

(S1) Problem solving skills

##### Medical Statistics (MATH364)

The aims of this module are to: demonstrate the purpose of medical statistics and the role it plays in the control of disease and promotion of health explore different epidemiological concepts and study designs apply statistical methods learnt in other programmes, and some new concepts, to medical problems and practical epidemiological research enable further study of the theory of medical statistics by using this module as a base.

(LO1) identify the types of problems encountered in medical statistics

(LO3) apply appropriate statistical methods to problems arising in epidemiology and interpret results

(LO4) explain and apply statistical techniques used in survival analysis

(LO5) critically evaluate statistical issues in the design and analysis of clinical trials

(LO6) discuss statistical issues related to systematic review and apply appropriate methods of meta-analysis

(LO7) apply Bayesian methods to simple medical problems.

(S1) Problem solving skills

##### Measure Theory and Probability (MATH365)

The main aim is to provide a sufficiently deepintroduction to measure theory and to the Lebesgue theory of integration. Inparticular, this module aims to provide a solid background for the modernprobability theory, which is essential for Financial Mathematics.

(LO1) After completing the module students should be ableto:

(LO2) master the basic results about measures and measurable functions

(LO3) master the basic results about Lebesgue integrals and their properties

(LO4) to understand deeply the rigorous foundations ofprobability theory

(LO5) to know certain applications of measure theoryto probability, random processes, and financial mathematics.

(S1) Problem solving skills

##### Mathematical Risk Theory (MATH366)

•To provide an understanding of the mathematical risk theory used in the study process of actuarial interest

• To provide an introduction to mathematical methods for managing the risk in insurance and finance (calculation of risk measures/quantities)

• To develop skills of calculating the ruin probability and the total claim amount distribution in some non‐life actuarial risk models with applications to insurance industry

• To prepare the students adequately and to develop their skills in order to be ready to sit for the exams of CT6 subject of the Institute of Actuaries (MATH366 covers 50% of CT6 in much more depth).

(LO1) After completing the module students should be able to:
(a) Define the loss/risk function and explain intuitively the meaning of it, describe and determine optimal strategies of game theory, apply the decision criteria's, be able to decide a model due to certain model selection criterion, describe and perform calculations with Minimax and Bayes rules.
(b) Understand the concept (and the mathematical assumptions) of the sums of independent random variables, derive the distribution function and the moment generating function of finite sums of independent random variables.
(c) Define and explain the compound Poisson risk model, the compound binomial risk model, the compound geometric risk model and be able to derive the distribution function, the probability function, the mean, the variance, the moment generating function and the probability generating function for exponential/mixture of exponential severities and gamma (Erlang) severities, be able to calculate the distribution of sums of independent compound Poisson random variables.
(d) Understand the use of convolutions and compute the distribution function and the probability function of the compound risk model for aggregate claims using convolutions and recursion relationships.
(e) Define the stop‐loss reinsurance and calculate the (mean) stop‐loss premium for exponential and mixtures of exponential severities, be able to compare the original premium and the stoploss premium in numerical examples.
(f) Understand and be able to use Panjer's equation when the number of claims belongs to theR(a, b, 0) class of distributions, use the Panjer's recursion in order to derive/evaluate the probability function for the total aggregate claims.
(g) Explain intuitively the individual risk model, be able to calculate the expected losses (as well as the variance) of group life/non‐life insurance policies when the benefits of the each person of the group are assumed to have deterministic variables.
(h) Derive a compound Poisson approximations for a group of insurance policies (individual risk model as approximation),
(i) Understand/describe the classical surplus process ruin model and calculate probabilities of the number of the risks appearing in a specific time period, under the assumption of the Poisson process.
(j) Derive the moment generating function of the classical compound Poisson surplus process, calculate and explain the importance of the adjustment coefficient, also be able to make use of Lundberg's inequality for exponential and mixtures of exponential claim severities.
(k) Derive the analytic solutions for the probability of ruin, psi(u), by solving the corresponding integro‐differential equation for exponential and mixtures of exponential claim amount severities,
(l) Define the discrete time surplus process and be able to calculate the infinite ruin probability, psi(u,t) in numerical examples (using convolutions).
(m) Derive Lundberg's equation and explain the importance of the adjustment coefficient under the consideration of reinsurance schemes.
(n) Understand the concept of delayed claims and the need for reserving, present claim data as a triangle (most commonly used method), be able to fill in the lower triangle by comparing present data with past (experience) data.
(o) Explain the difference and adjust the chain ladder method, when inflation is considered.
(p) Describe the average cost per claim method and project ultimate claims, calculate the required reserve (by using the claims of the data table).
(q) Use loss ratios to estimate the eventual loss and hence outstanding claims.
(r) Describe the Bornjuetter‐Ferguson method (be able to understand the combination of the estimated loss ratios with a projection method). Use the aforementioned method to calculate the revised ultimate losses (by making use of the credibility factor).

##### Networks in Theory and Practice (MATH367)

•To develop an appreciation of network models for real world problems.

•To describe optimisation methods to solve them.

•To study a range of classical problems and techniques related to network models.

(LO1) After completing the module students should be able to model problems in terms of networks and be able to apply effectively a range of exact and heuristic optimisation techniques.

##### Stochastic Theory and Methods in Data Science (MATH368)

1. To develop a understanding of the foundations of stochastics normally including processes and theory.

2. To develop an understanding of the properties of simulation methods and their applications to statistical concepts.

3. To develop skills in using computer simulations such as Monte-Carlo methods

4. To gain an understanding of the learning theory and methods and of their use in the context of machine learning and statistical physics.

5. To obtain an understanding of particle filters and stochastic optimisation.

(LO1) Develop understanding of the use of probability theory.

(LO2) Understand stochastic models and the use statistical data.

(LO3) Demonstrate numerical skills for the understanding of stochastic processes.

(LO4) Understand the main machine learning techniques.

##### Statistical Physics (MATH327)

1. To develop an understanding of the foundations of Statistical Physics normally including statistical ensembles and related extensive and intrinsic quantities.
2. To develop an understanding of the properties of classical and quantum gases and an appreciation of their applications to concepts such as the classical equation of state or the statistical
theory of photons.
3. To obtain a reasonable level of skill in using computer simulations for describing diffusion and transport in terms of stochastic processes.
4. To know the laws of thermodynamics and thermodynamical cycles.
5. To obtain a reasonable understanding of interacting statistical systems and related phenomenons such as phase transitions.

(LO1) Demonstrate understanding of the microcanonical, canonical and grand canonical ensembles, their relation and the derived concepts of entropy, temperature and particle number
density.

(LO2) Understand the derivation of the equation-of-state for non-interacting classical or quantum gases.

(LO3) Demonstrate numerical skills to understand diffusion from an underlying stochastic process.

(LO4) Know the laws of thermodynamics and demonstrate their application to thermodynamic cycles.

(LO5) Be aware of the effect of interactions including an understanding of the origin of phase transitions.

(S1) Problem solving skills

##### Professional Projects and Employability in Mathematics (MATH390)

The first aim of the module is to further develop students' problem solving abilities and ability to select techniques and apply mathematical knowledge to authentic work-style situations. Specifically, within this aim, the module aims to:

1) develop students' ability to solve a problem in depth over an extended period and produce reports

2) develop students' ability to communicate mathematical results to audiences of differing technical ability, including other mathematicians, business clients and the general public

3) develop an appreciation of how groups operate, different roles in group work, and the different skills required to successfully operate as a team.

The second aim of the module is to develop students' employability skills in key areas such as public speaking, task management and professionalism.

(LO1) Select appropriate techniques and apply mathematical knowledge to solve problems related to real-world phenomena.

(LO2) Communicate mathematical results to audiences of differing technical ability via different methods.

(LO3) Reflect on skills development and identify areas for further development.

(LO4) Articulate employability skills.

(LO5) Produce reports based on the development of a piece of work, in depth over an extended period of time.

(S1) Problem solving skills

##### Maths Summer Industrial Research Project (MATH391)

To acquire knowledge and experience of some of the ways in which mathematics is applied, directly or indirectly, in the workplace.
To gain knowledge and experience of work in an industrial or business environment.

Improve the ability to work effectively in small groups.

Skills in writing a substantial report, with guidance but largely independently This report will have mathematical content, and may also reflect on the work experience as a whole.

Skills in giving an oral presentation to a (small) audience of staff and students.

(LO1) To have knowledge and experience of some of the ways in which mathematics is applied, directly or indirectly, in the workplace

(LO2) To have gained knowledge and experience of work on industrial or business problems.

(LO3) To acquire skills of writing, with guidance but largely independently, a research report. This report will have mathematical content.

(LO4) To acquire skills of writing a reflective log documenting their experience of project development.

(LO5) To have gained experience in giving an oral presentation to an audience of staff, students and industry representatives.

### Programme Year Four

One of the project modules MATH499 or MATH490 must be taken in Year 4. Some modules are only delivered in alternate years.

#### Year Four Optional Modules

##### Linear Differential Operators in Mathematical Physics (MATH421)

This module provides a comprehensive introduction to the theory of partial differential equations, and it provides illustrative applications and practical examples in the theory of elliptic boundary value problems, wave propagation and diffusion problems.

(LO1) To understand and actively use the basic concepts of mathematical physics, such as generalised functions, fundamental solutions and Green's functions.

(LO2) To apply powerful mathematical methods to problems of electromagnetism, elasticity, heat conduction and wave propagation.

##### Quantum Field Theory (MATH425)

To provide a broad understanding of the essentials of quantum field theory.

(LO1) After the course the students should understand the important features of the mathematical tools necessary for particle physics. In particular they should · be able to compute simple Feynman diagrams, · understand the basic principles of regularisation and renormalisation · be able to calculate elementary scattering cross-sections.

##### Variational Calculus and Its Applications (MATH430)

This module provides a comprehensive introduction to the theory of the calculus of variations, providing illuminating applications and examples along the way.

(LO1) Students will posses a solid understanding of the fundamentals of variational calculus

(LO2) Students will be confident in their ability to apply the calculus of variations to range of physical problems

(LO3) Students will also have the ability to solve a wide class of non-physical problems using variational methods

(LO4) Students will develop an understanding of Hamiltonian formalism and have the ability to apply this framework to solve physical and non-physical problems

(LO5) Students will be confident in their ability to analyse variational symmetries and generate the associated conservation laws

(S1) Problem solving skills

##### Manifolds, Homology and Morse Theory (MATH410)

To give an introduction to the topology of manifolds, emphasising the role of homology as an invariant and the role of Morse theory as a visualising and calculational tool.

(LO1) To be able to:
• give examples of manifolds, particularly in low dimensions
• compute homology groups, Euler characteristics and degrees of maps in simple cases
• determine whether an explicitly given function is Morse and to identify its critical points and their indices
• use the Morse inequalities to estimate the ranks of homology groups
• use the Morse complex to compute Euler characteristics and, in simple cases, homology.

##### Higher Arithmetic (MATH441)

This module is designed to provide an introduction to topics in Analytic Number Theory, including the worst and average case behaviour of arithmetic functions, properties of the Riemann zeta function, and the distribution of prime numbers.

(LO1) Be able to apply analytic techniques to arithmetic functions.

(LO2) Understand basic analytic properties of the Riemann zeta function.

(LO3) Understand Dirichlet characters and L-series.

(LO4) Understand the connection between Ingham's theorem and the Prime Number Theorem.

(S2) Problem solving skills

##### Representation Theory of Finite Groups (MATH442)

Representation theory is one of the standard tools used in the investigation of finite groups, especially via the character of a representation. This module will be an introduction to these ideas with emphasis of the calculation of character tables for specific groups.

(LO1) After completing this module students should be able to · use representation theory as a tool to understand finite groups

(LO2) calculate character tables of a variety of groups.

(S1) Problem solving skills

##### Probability and Analysis (MATH465)

Understanding of the modern theoretical and applicable methods and tools of the vast field of Probability Theory (Stochastics) that are at the intersection of many mathematical disciplines.

Recognition of the central part of STOCHASTICS (probability theory, statistics, stochastic processes, stochastic analysis) within almost all science fields and of its explanatory power.

Students will be encouraged to develop:

Ability to read, understand and communicate research literature.

Ability to recognise potential research opportunities and research directions.

(LO1) Detailed understanding of how determinism, in view of complexity, can be handled by random tools.

(LO2) Ability to use probabilistic tools to model, analyse and understand complex systems.

##### Singularity Theory of Differentiable Mappings (MATH455)

To give an introduction to the study of local singularities of differentiable functions and mappings.

(LO1) To know and be able to apply the technique of reducing functions to local normal forms.

(LO2) To understand the concept of stability of mappings and its applications.

(LO3) To be able to construct versal deformations of isolated function singularities.

(S1) Problem solving skills

##### Stochastic Analysis and Its Applications (MATH483)

This module aims to demonstrate the advanced mathematical techniques underlying financial markets and the practical use of financial derivative products to analyse various problems arising in financial markets. Emphases are on the stochastic techniques, probability theory, Markov processes and stochastic calculus, together with the related applications

(LO1) A critical awareness of current problems and research issues in the fields of probability and stochastic processes, stochastic analysis and financial mathematics.

(LO2) The ability to formulate stochastic cuclulus for the purpose of modelling particular financial questions.

(LO3) The ability to read, understand and communicate research literature in the fields of probability, stochastic analysis and financial mathematics.

(LO4) The ability to recognise potential research opportunities and research directions.

##### Math499 - Project for M.math. (MATH499)

To demonstrate a critical understanding and historical appreciation of some branch of mathematics by means of directed reading and preparation of a report. Students doing this project in certain areas will become effective in the use of appropriate software/coding.

(LO1) To gain a greater understanding of the chosen mathematical topic and an appreciation of the historical context.

(LO2) To learn how to understand abstract mathematical concepts and explain them.

(LO3) To have gained experience in consulting related relevant literature.

(LO4) To learn how to construct a written project report.

(LO5) To have gained experience in making an oral presentation.

(LO6) To have gained familiarity with a scientific word-processing package such as LaTeX or TeX.

##### Math490 - Project for M.math. (MATH490)

To demonstrate a critical understanding and historical appreciation of some branch of mathematics by means of directed reading and preparation of a report.

The MATH490 project should treat its subject at a more advanced level and in greater depth than the MATH399 or MATH499 projects. Working on this year-long project can provide a good base to continue mathematical studies through PhD.

(LO1) To gain a greater understanding of the chosen mathematical topic and an appreciation of the historical context.

(LO2) To learn how to understand abstract mathematical concepts and explain them

(LO3) To have gained experience in consulting related relevant literature.

(LO4) To learn how to construct a written project report.

(LO5) To have gained experience in making an oral presentation.

(LO6) To have gained familiarity with a scientific word-processing package such as LaTeX or TeX.

##### Advanced Topics in Mathematical Biology (MATH426)

To introduce some hot problems of contemporary mathematical biology, including analysis of developmental processes, networks and biological mechanics.

To further develop mathematical skills in the areas of difference equations and ordinary and partial differential equations.

To explore biological applications of fluid dynamics in the limit of low and high Reynolds number.

(LO1) To familiarise with mathematical modelling methodology used in contemporary mathematical biology.

(LO2) Be able to use techniques from difference equations and ordinary and partial differential equations in tackling problems in biology.

##### Waves, Mathematical Modelling (MATH427)

This module gives an introduction to the mathematical theory of linear and non-linear waves. Illustrative applications involve problems of acoustics, gas dynamics and examples of solitary waves.

(LO1) To understand essential modelling techniques in problems of wave propagation.

(LO2) To understand that mathematical models of the same type can be successfully used to describe different physical phenomena.

(LO3) To understand background mathematical theory in models of acoustics, gas dynamics and water waves.

(S1) Problem solving skills

##### Asymptotic Methods for Differential Equations (MATH433)

This module provides an introduction into the perturbation theory for partial differential equations. We consider singularly and regularly perturbed problems and applications in electro-magnetism, elasticity, heat conduction and propagation of waves.

(LO1) The ability to make appropriate use of asymptotic approximations.

(LO2) The ability to analyse boundary layer effects.

(LO3) The ability to use the method of compound asymptotic expansions in the analysis of singularly perturbed problems.

(S1) Problem solving skills

##### Elliptic Curves (MATH444)

To provide an introduction to the problems and methods in the theory of elliptic curves.

To investigate the geometry of ellptic curves and their arithmetic in the context of finite fields, p-adic fields and rationals.

To outline the use of elliptic curves in cryptography.

(LO1) The ability to describe and to work with the group structure on a given elliptic curve.

(LO2) Understanding and application of the Abel-Jacobi theorem.

(LO3) To estimate the number of points on an elliptic curve over a finite field.

(LO4) To use the reduction map to investigate torsion points on a curve over Q.

(LO5) To apply descent to obtain so-called Weak Mordell-Weil Theorem.

(LO6) Use heights of points on elliptic curves to investigate the group of rational points on an elliptic curve.

(LO7) Understanding and application of Mordell-Weil theorem. Encode and decode using public keys.

(S1) Problem solving skills

##### Riemann Surfaces (MATH445)

To introduce to a beautiful theory at the core of modern mathematics. Students will learn how to handle some abstract geometric notions from an elementary point of view that relies on the theory of holomorphic functions. This will provide those who aim to continue their studies in mathematics with an invaluable source of examples, and those who plan to leave the subject with the example of a modern axiomatic mathematical theory.

(LO1) Students should be familiar with the most basic examples of Riemann surfaces: the Riemann sphere, hyperelliptic Riemann surfaces, and smooth plane algebraic curves.

(LO2) Students should understand and be able to use the abstract notions used to build the theory: holomorphic maps, meromorphic differentials, residues and integrals, Euler characteristic and genus.

(LO3) Students should know different techniques to calculate the genus and the dimensions of spaces of meromorphic functions, and they should have acquired some understanding of uniformisation.

(S1) Problem solving skills

##### Geometry of Continued Fractions (MATH447)

To give an introduction to the current state of the art in geometry of continued fractions and to study how classical theorems can be visualized via modern techniques of integer geometry.

(LO1) To be able to find best approximations to realnumbers and to homogeneous decomposable forms.

(LO2) To be able to use techniques of geometric continuedfractions for quadratic irrationalities (Lagrange’s theorem, Markov spectrum).

(LO3) To be able to use lattice trigonometry in the studyof toric varieties.

(LO4) To be able to compute relative frequencies of facesin multidimensional continued fractions.

(LO5) To be able to use multidimensional continuedfractions to study properties of algebraic irrationalities of higher degree.

(S2) Problem solving skills

##### Algebraic Geometry (MATH448)

To give a detailed explanation of basic concepts and methods of algebraic geometry in terms of coordinates and polynomial algebra, supported by strong geometrical intuition.

To elaborate examples and to explain the basic constructions of algebraic geometry, such as projections, products, blowing up, intersection multiplicities, linear systems, vector bundles, etc.

To understand in detail the proofs of several fundamental results in algebraic geometry on the structure of birational maps and intersection theory.

To take the first steps in acquiring the technique of linear systems, vector bundles and differential forms.

(LO1) To know:basic concepts of smooth geometry and algebraic geometry.

(LO2) To understand: the interplay between local and global geometry, the duality between differential forms and submanifolds, between curves and divisors, between algebraic and geometric data.

(LO3) To be able to: perform elementary computations with differential forms, patch together local objects into global ones, compute elementary intersection indices.

(S1) Problem solving skills

##### Galois Theory (MATH449)

To introduce the theory of polynomial equations of one variable: Galois Theory.

To introduce criteria when a polynomial equation can be solved in radicals, when a geometric construction can be performed by a ruler and a compass.

(LO1) Know why and how a polynomial equation of degree up to 4 can be solved in radicals.

(LO2) Understand why a solution in radicals is impossible in general for the degree greater than or equal to 5.

(LO3) Understand when a polynomial can be solved in radicals.

(LO4) Know when a geometric construction can be done by a ruler and compass.

(LO5) Know what is the Galois group of a polynomial which permits the above results.

##### Introduction to String Theory (MATH423)

To provide a broad understanding of string theory, and its utilization as a theory that unifies all of the known fundamental matter and interactions.

(LO1) After completing the module the students should: - be familiar with the properties of the classical string.

(LO2) be familiar with the basic structure of modern particle physics and how it may arise from string theory.

(LO3) be familiar with the basic properties of first quantized string and the implications for space-time dimensions.

(LO4) be familiar with string toroidal compactifications and T-duality.

(S1) Problem solving skills

##### Introduction to Modern Particle Theory (MATH431)

To provide a broad understanding of the current status of elementary particle theory. To describe the structure of the Standard Model of particle physics and its embedding in Grand Unified Theories.

(LO1) To understand the Lorentz and Poincare groups and their role in classification of elementary particles.

(LO2) To understand the basics of Langrangian and Hamiltonian dynamics and the differential equations of bosonic and fermionic wave functions.

(LO3) To understand the basic elements of field quantisation.

(LO4) To understand the Feynman diagram pictorial representation of particle interactions.

(LO5) To understand the role of symmetries and conservation laws in distinguishing the strong, weak and electromagnetic interactions.

(LO6) To be able to describe the spectrum and interactions of elementary particles and their embedding into Grand Unified Theories (GUTs)

(LO7) To understand the flavour structure of the standard particle model and generation of mass through symmetry breaking.

(LO8) To understand the phenomenological aspects of Grand Unified Theories.

(S1) Problem solving skills

The programme detail and modules listed are illustrative only and subject to change.

#### Teaching and Learning

Your learning activities will consist of lectures, tutorials, practical classes, problem classes, private study and supervised project work. In Year One, lectures are supplemented by a thorough system of group tutorials and computing work is carried out in supervised practical classes. Key study skills, presentation skills and group work start in first-year tutorials and are developed later in the programme. The emphasis in most modules is on the development of problem solving skills, which are regarded very highly by employers. Project supervision is on a one-to-one basis, apart from group projects in Year Two.

#### Assessment

Most modules are assessed by a two and a half hour examination in January or May, but many have an element of coursework assessment. This might be through homework, class tests, mini-project work or key skills exercises.

### Foundation courses

International students who do not qualify for direct entry to this degree can prepare at University of Liverpool International College, where successful completion of the Foundation Certificate guarantees entry to this degree.

### Contact

Department of Mathematical Sciences
University of Liverpool
Mathematical Sciences Building
Liverpool
L69 7ZL

Got a question? Use our Ask Liverpool facility to get the answer you need.

As part of your Mathematics degree programme you may have the opportunity to study abroad. Studying abroad has huge personal and academic benefits, as well as giving you a head start in the graduate job market. Students may apply to study at universities in the US and Canada and students on a Maths with a European Language programme will be able to spend a year in a country where the relevant language is spoken. For more information, visit www.liverpool.ac.uk/goabroad.

### Year in China

The Year in China is the University of Liverpool’s exciting flagship programme enabling undergraduate students from a huge range of departments, including Mathematical Sciences, the opportunity to spend one year at our sister university Xi’an Jiaotong-Liverpool University (XJTLU), following XJTLU’s BA China Studies degree classes. See our Year in China page for more information.

## Columbia University GU4042 Introduction to Modern Algebra II, fall 2020

Textbook: Galois Theory, by Joseph Rotman, second edition (1998). You can get pdf file from Columbia Online Library (follow "SpringerLink ebooks" link on the right) as well as purchase a printed copy from Springer via MyCopy service on the same webpage as the pdf download.

• Fields and Galois Theory, by John Howie (pdf via Columbia Library). Howie covers essentially the same material as Rotman at a more leisurely pace.
• Abstract Algebra: Theory and Applications, by Thomas W. Judson. Our second semester topics start with Section 16.

This is the second semester of a 2-semester course on Modern Algebra. The first semester, which covered group theory, is the prerequisite for this course.

Syllabus: Rings and commutative rings. Rings of polynomials, residues modulo n and other examples. Matrix rings and quaternions. Integral domains and fields. Field of fractions. Homomorphisms of rings and ideals. Quotient rings and First Isomorphism Theorem for rings. Principal ideal domains and polynomial rings over fields. Prime and maximal ideals. Irreducible polynomials. Characteristic of a field. Finite fields. Linear algebra over a field. Field extensions and splitting fields. Galois group. Solvability by Radicals. Ruler and compass constructions. Independence of characters. Galois' Theorems. Applications. Fundamental Theorem of Algebra. Applications of finite fields.
If time allows: Modules over rings and representation theory. Classification of (finitely-generated) modules over PIDs. Semisimple rings. Basics of category theory.

Homework: Homework will be assigned on Wednesdays, due Wednesday the next week before class. It will be posted on this webpage. The first problem set is due September 16. The lowest homework score will be dropped. You can discuss homework problems with your fellow students, after you make a serious effort to solve each problem on your own. Homework discussion prior to submission is subject to the following rules: (1) List the name of your collaborators at the head of the problem or assignment, (2) Do not exchange written work with others, (3) Write up solutions in your own words.
Throughout the semester we'll have several 10-minute quizzes, with yes/no and multiple choice questions.

The numerical grade for the course will be the following linear combination: 5% quizzes, 20% homework, 20% each midterm, 35% final.

Supplemental resources for weeks 1-2:
Notes by Robert Friedman: Rings Polynomials Integral domains

Weeks 11-12:
Lecture 18 Notes, Monday Nov 16. Ruler-Compass constructions: Rotman Morandi (Field and Galois theory, full book available via Columbia online library).

Robert Donley (MathDoctorBob on Youtube) has an online course on Modern Algebra.

Other algebra texts: There are many that you can find online or in the library. A rather incomplete list: Michael Artin Algebra, John Fraleigh A First Course in Abstract Algebra, Joseph Gallian Contemporary Abstract Algebra, Thomas Hungerford Abstract Algebra: An Introduction, Serge Lang Undergraduate Algebra. Dummit and Foote Abstract Algebra is truly encyclopedic without losing textbook qualities, a popular graduate school textbook.

Herstein, I. N. [George Seligman, Charles Curtis, Marshall Hall, Nathan Jacobson, Arthur Mattuck, Maxwell Rosenlicht, Irving Kaplansky, Francis McNary]

Published by New York Waltham, Masachusetts Toronto : Blaisdell Publishing Company, 1964., 1964

Used - Hardcover
Condition: Good

Hardcover. Condition: Good. 1st Edition. 1st ed. viii, 342 p. : ill. 24 cm. LCCN: 63-17982 OCLC: 259785 LC: QA155 Dewey: 512.8 black cloth with gold lettering no dustjacket covers worn names on front ep Contents: Preliminary notions -- Group theory -- Ring theory -- Vector spaces and modules -- Fields -- Linear transformations -- Selected topics : Finite fields, Wedderburn's Theorem on FInite Division Rings, A theorem of Frobenius -- Integral quaternions and the four-square theorem. a few pencilings, else G. Book.

MATH 098. Intermediate Algebra. 3 Credits.

Properties of the real number system, factoring, linear and quadratic equations, functions, polynomial and rational expressions, inequalities, systems of equations, exponents, and radicals. Offered through Continuing Education. Special fee required. Does not satisfy any requirements for graduation.

MATH 103. College Algebra. 3 Credits.

Relations and functions, equations and inequalities, complex numbers polynomial, rational, exponential and logarithmic functions systems of equations, and matrices. Prereq: MATH 98 with a grade of C or higher or placement.

MATH 104. Finite Mathematics. 3 Credits.

Systems of linear equations and inequalities, matrices, linear programming, mathematics of finance, elementary probability and descriptive statistics. Prereq: MATH 98 with a grade of C or higher or placement.

MATH 105. Trigonometry. 3 Credits.

Angle measure, trigonometric and inverse trigonometric functions, trigonometric identities and equations, polar coordinates and applications. Prereq: MATH 103 or placement. Credit awarded only for MATH 105 or MATH 107, not both.

MATH 107. Precalculus. 4 Credits.

Equations and inequalities polynomial, rational, exponential, logarithmic and trigonometric functions inverse trigonometric functions algebraic and trigonometric methods commonly needed in calculus. An expedited, combined offering of MATH 103 and MATH 105. Prereq: Placement. Credit awarded only for MATH 105 or MATH 107, not both.

MATH 128. Introduction to Linear Algebra. 1 Credit.

Systems of linear equations, row operations, echelon form, matrix operations, inverses, and determinants. Prereq: MATH 105 or MATH 107. Credit awarded only for MATH 128 or MATH 129, not both.

MATH 129. Basic Linear Algebra. 3 Credits.

Systems of linear equations, matrices, determinants, vector spaces, lines and planes in space, linear transformations, eigenvalues and eigenvectors. Prereq: MATH 105 or MATH 107.

MATH 144. Mathematics for Business. 4 Credits.

Mathematics of finance, linear programming and its applications in business, limits, continuity, derivatives, implicit and logarithmic differentiation, higher order derivatives, optimization and extrema, partial differentiation, extreme values of functions of two variables. Prereq: MATH 103, MATH 107 or placement exam. Credit awarded only for MATH 144 or MATH 146, not both.

MATH 146. Applied Calculus I. 4 Credits.

Limits, derivatives, integrals, exponential and logarithmic functions and applications. Prereq: MATH 103, MATH 107, or placement. Credit awarded only for MATH 144 or MATH 146, not both.

MATH 147. Applied Calculus II. 4 Credits.

Definite integrals, trigonometry, introduction to differential equations, infinite sequences and series, probability and applications. Prereq: MATH 146.

MATH 165. Calculus I. 4 Credits.

Limits, continuity, differentiation, Mean Value Theorem, integration, Fundamental Theorem of Calculus and applications. Prereq: MATH 105, MATH 107, or placement.

MATH 166. Calculus II. 4 Credits.

Applications and techniques of integration polar equations parametric equation sequences and series, power series. Prereq: MATH 165.

MATH 194. Individual Study. 1-5 Credits.

MATH 196. Field Experience. 1-15 Credits.

MATH 199. Special Topics. 1-5 Credits.

MATH 259. Multivariate Calculus. 3 Credits.

Functions of several variables, vectors in two and three variables, partial derivatives, surfaces and gradients, tangent planes, differentials, chain rule, optimization, space curves, and multiple integrals. Prereq: MATH 166. Credit awarded only for MATH 259 or MATH 265, not both.

MATH 265. Calculus III. 4 Credits.

Multivariate and vector calculus including partial derivatives, multiple integration, applications, line and surface integrals, Green's Theorem, Stoke's Theorem, and Divergence Theorem. Prereq: MATH 166. Credit awarded only for MATH 259 or MATH 265, not both.

MATH 266. Introduction to Differential Equations. 3 Credits.

Solution of elementary differential equations by elementary techniques. Laplace transforms, systems of equations, matrix methods, numerical techniques, and applications. Prereq: MATH 259 or MATH 265. Coreq: MATH 128, MATH 129, or MATH 329.

MATH 270. Introduction to Abstract Mathematics. 3 Credits.

Sets, symbolic logic, propositions, quantifiers, methods of proof, relations and functions, equivalence relations, math induction and its equivalents, infinite sets, cardinal numbers, number systems. Prereq: MATH 166.

MATH 291. Seminar. 1-3 Credits.

MATH 294. Individual Study. 1-5 Credits.

MATH 299. Special Topics. 1-5 Credits.

MATH 329. Intermediate Linear Algebra. 3 Credits.

Vector spaces over real and complex numbers, matrices, determinants, linear transformations, eigenvalues and eigenvectors, Cayley-Hamilton Theorem, inner product spaces, selected topics and applications. Prereq: MATH 129 and MATH 165.

MATH 346. Metric Space Topology. 3 Credits.

Various metrics on Euclidean spaces, metric spaces, open and closed sets, limit points and convergence, Bolzano Weierstrass Theorem, (uniformly) continuous functions, connected spaces, compact spaces and the Heine Borel Theorem, sequence of functions. Prereq: MATH 270.

MATH 374. Special Problems In Mathematics. 1 Credit.

Diverse and challenging mathematical problems are considered with the intent of preparing the student for the Putnam Mathematics competition. May be repeated for credit. Pass/Fail only. Prereq: MATH 270.

MATH 376. Actuarial Exam Study. 1 Credit.

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Vector spaces, linear transformations, eigenvalues and eigenvectors, canonical forms, inner product spaces, and selected applications. Prereq: MATH 270. .

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Hilbert's axioms for Euclidean geometry, projective geometry, history of parallel axiom, hyperbolic geometry, elliptic geometry. Prereq: MATH 270. .

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Basic Point-Set Topology: Topological Spaces, Open/Closed Sets, Continuity, Connectedness, Compactness Surfaces: Classification, Basic Invariants Introduction to Homology Applications: Brouwer's Fix-Point Theorem, Ham and Sandwich Theorem. Prereq: MATH 346. .

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Differentiation and Riemann integration in the real numbers. Sequences and series of functions uniform convergence and power series. Prereq: MATH 346. .

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Riemann integration, spaces of continuous functions, convergence theorems, multiple integration and Fubini's Theorem and selected topics. Prereq: MATH 450..

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Cryptography and cryptanalysis of ciphers. Discrete logarithms, Diffie-Hellman key exchange, the RSA cryptosystem, elliptic curve cryptography, and selected topics. Prereq: MATH 420 or MATH 472. .

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Historical considerations emphasizing the source of mathematical ideas, growth of mathematical knowledge, and contributions of some outstanding mathematicians. Prereq: MATH 270. .

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Method of power series and method of Frobenius, oscillation theorems, special functions (Bessel functions and Legendre functions), linear systems including the exponential matrix. Sturm-Liouville and phase plane analysis as time permits. Prereq: MATH 266. .

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Discrete and continuous Fourier transforms, Fourier series, convergence and inversion theorems, mean square approximation and completeness, Poisson summation, Fast-Fourier transform. Prereq: MATH 265. .

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First and second order partial differential equations, classification, examples, solution methods for the wave, diffusion, and Laplace equations, causality and energy, boundary value problems, separation of variables, Green's identities, Green's functions. Prereq: MATH 266 and MATH 270. .

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Numerical solution of nonlinear equations, interpolation, numerical integration and differentiation, numerical solution of initial value problems for ordinary differential equations. Prereq: MATH 266. .

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Numerical solutions of linear and nonlinear systems, eigenvalue problems for matrices, boundary value problems for ordinary differential equations, selected topics. Prereq: MATH 329, MATH 488. .

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Groups, permutations, quotient groups, homomorphisms, rings, ideals, integers. .

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Division rings, integral domains, fields, field extensions, Galois Theory. Prereq: MATH 620. .

MATH 629. Linear Algebra. 3 Credits.

Vector spaces, linear transformations, eigenvalues and eigenvectors, canonical forms, inner product spaces, and selected applications. .

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Graphs and directed graphs, graph models, subgraphs, isomorphisms, paths, connectivity, trees, networks, cycles, circuits, planarity, Euler's formula, matchings, bipartite graphs, colorings, and selected advanced topics. .

MATH 636. Combinatorics. 3 Credits.

Recurrence relations, formal power series, generating functions, exponential generating functions, enumeration, binomial coefficients and identities, hypergeometric functions, Ramsey theory, Sterling and Eulerian numbers. .

MATH 639. Topics in Algebra and Discrete Mathematics. 3 Credits.

Advanced topics in algebra and discrete mathematics. Topics may vary but may include: algebraic geometry, factorization, partially ordered sets, and/or coding theory. .

MATH 640. Axiomatic Geometry. 3 Credits.

Hilbert's axioms for Euclidean geometry, projective geometry, history of parallel axiom, hyperbolic geometry, elliptic geometry. .

MATH 642. Introduction to Topology. 3 Credits.

Basic Point-Set Topology: Topological Spaces, Open/Closed Sets, Continuity, Connectedness, Compactness Surfaces: Classification, Basic Invariants Introduction to Homology Applications: Brouwer's Fix-Point Theorem, Ham and Sandwich Theorem. .

MATH 643. Differential Geometry. 3 Credits.

Local and global geometry of plane curves, local geometry of hypersurfaces, global geometry of hypersurfaces, geometry of lengths and distances. .

MATH 649. Topics in Topology and Geometry. 3 Credits.

Topics will vary and may include: Riemannian Geometry, Symplectic Topology, Dynamical Systems on Manifolds, Hamiltonian Systems, Geometric Group Theory, Descriptive Set Theory. .

MATH 650. Real Analysis I. 3 Credits.

Differentiation and Riemann integration in the real numbers. Sequences and series of functions uniform convergence and power series. .

MATH 651. Real Analysis II. 3 Credits.

Riemann integration, spaces of continuous functions, convergence theorems, multiple integration and Fubini's Theorem and selected topics. Prereq: MATH 650. .

MATH 652. Complex Analysis. 3 Credits.

Complex number systems, analytic and harmonic functions, elementary conformal mapping, integral theorems, power series, Laurent series, residue theorem, and contour integral. .

MATH 653. Introduction to Lebesgue Measure. 3 Credits.

Definition of Lebesgue measure. Measurable and Lebesgue integrable functions. Introduction to Lp spaces. .

MATH 654. Introduction to Functional Analysis. 3 Credits.

Functional analysis in sequence spaces. Standard sequence spaces and dual spaces. Hahn-Banach Theorem. Operators on sequences spaces. .

MATH 659. Topics in Analysis. 3 Credits.

Topics will vary and may include: Harmonic Analysis, Dynamical Systems, Fractals, Distribution Theory, and Approximation Theory. .

MATH 660. Intensive Mathematica. 1 Credit.

Thorough overview of the general purpose mathematical software MATHEMATICA: numerical and symbolic calculations for algebra and linear algebra, single and multivariable calculus, ordinary and partial differential equations, 2D- and 3D-graphics, animation, word processing. .

MATH 672. Number Theory. 3 Credits.

Properties of integers, number theoretic functions, quadratic residues, continued fractions, prime numbers and their distribution, primitive roots. .

MATH 673. Cryptology. 3 Credits.

Cryptography and cryptanalysis of ciphers. Discrete logarithms, Diffie-Hellman key exchange, the RSA cryptosystem, elliptic curve cryptography, and selected topics. .

MATH 678. History of Mathematics. 3 Credits.

Historical considerations emphasizing the source of mathematical ideas, growth of mathematical knowledge, and contributions of some outstanding mathematicians. .

MATH 680. Applied Differential Equations. 3 Credits.

Method of power series and method of Frobenius, oscillation theorems, special functions (Bessel functions and Legendre functions), linear systems including the exponential matrix. Sturm-Liouville and phase plane analysis as time permits. .

MATH 681. Fourier Analysis. 3 Credits.

Discrete and continuous Fourier transforms, Fourier series, convergence and inversion theorems, mean square approximation and completeness, Poisson summation, Fast-Fourier transform. .

MATH 683. Partial Differential Equations. 3 Credits.

First and second order partial differential equations, classification, examples, solution methods for the wave, diffusion, and Laplace equations, causality and energy, boundary value problems, separation of variables, Green's identities, Green's functions. .

MATH 684. Mathematical Methods of Biological Processes. 3 Credits.

This course provides an introduction to mathematical methods in biology. .

MATH 685. Topics in Applied Mathematics. 3 Credits.

Topics will vary and may include: Models in Biology and Finance, Network Theory, Calculus of Variation, Stochastic Calculus, Integral Transforms, Control Theory, and Parameter Estimation. .

MATH 688. Numerical Analysis I. 3 Credits.

Numerical solution of nonlinear equations, interpolation, numerical integration and differentiation, numerical solution of initial value problems for ordinary differential equations. .

MATH 689. Numerical Analysis II. 3 Credits.

Numerical solutions of linear and nonlinear systems, eigenvalue problems for matrices, boundary value problems for ordinary differential equations, selected topics. Prereq: MATH 688. .

MATH 690. Graduate Seminar. 1-3 Credits.

MATH 696. Special Topics. 1-5 Credits.

MATH 720. Algebra I. 3 Credits.

Graduate level survey of algebra: rings, modules, linear algebra and selected advanced topics. Prereq or Co-req: MATH 621.

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Graduate level survey of algebra: groups, fields, Galois theory, and selected advanced topics. Prereq: MATH 720.

MATH 726. Homological Algebra. 3 Credits.

An overview of the techniques of homological algebra. Topics covered will include categories and functors, exact sequences, (co)chain complexes, Mayer-Vietoris sequences, TOR and EXT. Applications to other fields will be stressed. Prereq: MATH 720.

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An introduction to the principles of bioinformatics including information relating to the determination of DNA sequencing. Prereq: STAT 661. Cross-listed with CSCI 732 and STAT 732.

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Topological spaces, convergence and continuity, separation axioms, compactness, connectedness, metrizability, fundamental group and homotopy theory. Advanced topics may include homology theory, differential topology, three-manifold theory and knot theory. Prereq: MATH 642.

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Topological spaces, convergence and continuity, separation axioms, compactness, connectedness, metrizability, fundamental group and homotopy theory. Advanced topics may include homology theory, differential topology, three-manifold theory and knot theory. Prereq: MATH 642.

MATH 750. Analysis. 3 Credits.

Lebesgue and general measure and integration theory, differentiation, product spaces, metric spaces, elements of classical Banach spaces, Hilbert spaces, and selected advanced topics. Prereq: MATH 650.

MATH 752. Complex Analysis. 3 Credits.

Analytic and harmonic functions, power series, conformal mapping, contour integration and the calculus of residues, analytic continuation, meromorphic and entire functions, and selected topics. Prereq: MATH 650.

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Normed spaces, linear maps, Hahn-Banach Theorem and other fundamental theorems, conjugate spaces and weak topology, adjoint operators, Hilbert spaces, spectral theory, and selected topics. Prereq: MATH 750.

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A survey of Harmonic analysis including: Lp spaces Fourier Series Fourier transform Hilbert transform and special selected topics. Prereq: MATH 750.

MATH 760. Ordinary Differential Equations I. 3 Credits.

Existence, uniqueness, and extensibility of solutions to initial value problems, linear systems, stability, oscillation, boundary value problems, and selected advanced topics. Prereq: MATH 650 or MATH 680.

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Review of practical mathematical methods routinely used by physicists, including applications. Focus on differential equations, variational principles, and other selected topics. Cross-listed with PHYS 752.

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Tensor analysis, matrices and group theory, special relativity, integral equations and transforms, and selected advanced topics. Prereq: MATH 629 and MATH 652. Cross-listed with PHYS 753.

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Classification in elliptic, parabolic, hyperbolic type existence and uniqueness for second order equations Green's functions, and integral representations characteristics, nonlinear phenomena. Prereq: MATH 650 or MATH 683.

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MATH 824. Topics in Commutative Algebra. 3 Credits.

Topics vary each time the course is offered and may include: dimension theory, integral dependence, factorization, regular rings, Cohen-Macaulay rings, Gorenstein rings. May be repeated for credit with change in subtopic. Prereq: MATH 720.

MATH 825. Theory Of Rings. 3 Credits.

The ideal theory of commutative rings, structure of (non-commutative) rings, and selected advanced topics. Prereq: MATH 720.

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Graduate-level survey of graph theory: paths, connectivity, trees, cycles, planarity, genus, Eulerian graphs, Hamiltonian graphs, factorizations, tournaments, embedding, isomorphism, subgraphs, colorings, Ramsey theory, girth. Prereq: MATH 630.

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Combinatorial reasoning, generating functions, inversion formulae. Topics may include design theory, finite geometry, Ramsey theory, and coding theory. Advanced topics may include cryptography, combinatorial group theory, combinatorial number theory, algebraic combinatorics, (0,1)-matrices, and finite geometry. Prereq: MATH 636.

MATH 849. Topics in Geometry & Topology. 3 Credits.

Advanced topics in Geometry and/or Topology. Topics vary but may include: differential geometry, K-theory, knot theory, or noncommutative geometry. May be repeated for credit with change in subtopic. Prereq: MATH 642, MATH 643.

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A study of basic notions of topological and symbolic dynamics. Introduction to measurable dynamics and ergodic theory. Ergodicity, mixing and entropy of dynamical systems. Prereq: MATH 750.

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Maximal monotone operators and the Hille-Yosida theorem, Sobolev spaces in dimension one and applications, Sobolev spaces in higher dimensions, extension operators, Sobolev embedding theorems, Poincare inequality, duality. May be repeated for credit with change in subtopic. Prereq: MATH 750. Co-req: MATH 751.

MATH 861. Ordinary Differential Equations II. 3 Credits.

Existence, uniqueness, and extendibility of solutions to initial value problems, linear systems, stability, oscillation, boundary value problems, difference equations, and selected advanced topics. Prereq: MATH 760.

MATH 862. Integral Equations. 3 Credits.

Existence and uniqueness of solutions of Fredholm and Volterra integral equations, Fredholm Theory, singular integral equations, and selected advanced topics. Prereq: MATH 650.

MATH 864. Calculus Of Variations. 3 Credits.

Variational techniques of optimization of functionals, conditions of Euler, Weierstrass, Legendre, Jacobi, Erdmann, Pontryagin Maximal Principle, applications, and selected advanced topics. Prereq: MATH 650.

MATH 867. Topics in Applied Mathematics. 3 Credits.

Topics will vary and may include: Optimal Control, Robust Control, Stability Analysis, Mathematics of Networks, Models in Biology, Levy Processes, Asymptotic Expansions. May be repeated for credit with change in subtopic. Prereq: MATH 650 or MATH 680.

MATH 878. Modern Probability Theory. 3 Credits.

Probability theory presented from the measure theoretic perspective. Emphasis on various types of convergence and limit theorems. Discussion of random walks, conditional expectations, and martingales. Prereq: STAT 768 or MATH 750. Cross-listed with STAT 778.

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Elements of convex analysis, constrained and unconstrained multi-dimensional linear and nonlinear optimization theory and algorithms, convergence properties and computational complexity. Prereq: CSCI 653. Cross-listed with CSCI 880.

MATH 881. Mathematical Control Theory. 3 Credits.

Standard optimal control and optimal estimation problems duality optimization in Hardy space robust control design. Prereq: MATH 650.

MATH 885. Partial Differential Equations II. 3 Credits.

Nonlinear partial differential equations, Non-variational techniques, Hamilton-Jacobi equations, Riemann invariants, Entropy/entropy-flux pairs, selected advanced topics. Prereq: MATH 784.

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Numerical solutions to partial differential and integral equations, error analysis, stability, acceleration of convergence, numerical approximation, and selected advanced topics. Prereq: MATH 688.

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## Vector Spaces, Modules, and Linear Algebra

Let’s take a little trip back in time to grade school mathematics. What is five apples plus three apples? Easy, the answer is eight apples. What about two oranges plus two oranges? The answer is four oranges. What about three apples plus two oranges? Wait, that question is literally “apples and oranges”! But we can still answer that question of course. Three apples plus two oranges is three apples and two oranges. Does that sound too easy? We ramp it up just a little bit: What is three apples and two oranges, plus one apple and five oranges? The answer is four apples and seven oranges. Even if we’re dealing with two objects we’re not supposed to mix together, we can still do mathematics with them, as long as we treat each object separately.

Such an idea can be treated with the concept of vector spaces. Another application of this concept is to quantities with magnitude and direction in physics, where the concept actually originated. Yet another application is to quantum mechanics, where things can be simultaneously on and off, or simultaneously pointing up and down, or simultaneously be in a whole bunch of different states we would never think of being capable of existing together simultaneously. But what, really, is a vector space?

We can think of vector spaces as sets of things that can be added to or subtracted from each other, or scaled up or scaled down, or combinations of all these. To make all these a little easier, we stay in the realm of what are called “finite dimensional” vector spaces, and we develop for this purpose a little notation. We go back to the example we set out at the start of this post, that of the apples and oranges. Say for example that we have three apples and two oranges. We will write this as

Now, say we want to add to this quantity, one more apple and five oranges. We write

Of course this is easy to solve, and we have already done the calculation earlier. We have

But we also said we can “scale” such a quantity. So suppose again that we have three apples and two oranges. If we were to double this quantity, what would we have? We would have six apples and four oranges. We write this operation as

We can also “scale down” such a quantity. Suppose we want to cut in half our amount of three apples and two oranges. We would have one and a half apples (or three halves of an apple) and one orange:

We can also apply what we know of negative numbers – we can for example think of a negative amount of something as being like a “debt”. With this we can now add subtraction to the operations that we can do to vector spaces. For example, let us subtract from our quantity of three apples and two oranges the quantity of one apple and five oranges. We will be left with two apples and a “debt” of three oranges. We write

Finally, we can combine all these operations:

For vector spaces, the “scaling” operation possesses a property analogous to the distributive property of multiplication over addition. So if we wanted to, we could also have performed the previous operation in another way, which gives the same answer:

We can also apply this notation to problems in physics. Suppose a rigid object acted on by a force of one Newton to the north and another force of one Newton to the east. Then adopting a convention of Cartesian coordinates with the positive x-axis oriented towards the east, we can calculate the resultant force acting on the object as follows

This is actually a force with a magnitude of around Newtons, with a direction pointing towards the northeast, but a discussion of such calculations will perhaps be best left for future posts. For now, we want to focus on the two important properties of vector spaces, its being closed under the operations of addition and multiplication by a scaling factor, or “scalar”.

In Rings, Fields, and Ideals, we discussed what it means for a set to be closed under certain operations. A vector space is therefore a set that is closed under addition among its own elements and under multiplication by a “scalar”, which is an element of a field, a concept we discussed in the same post linked to above. A set that is closed under addition among its own elements and multiplication by a scalar which is a ring instead of a field is called a module. Another concept we discussed in Rings, Fields, and Ideals and also in More on Ideals is the concept of an ideal. An ideal is a module which is also a subset of its ring of scalars.

Whenever we talk about sets, it is always important to also talk about the functions between such sets. A vector space (or a module) is just a set with special properties, namely closure under addition and scalar multiplication, therefore we want to talk about functions that are related to these properties. A linear transformation is a function between two vector spaces or modules that “respect” addition and scalar multiplication. Let and be any two elements of a vector space or a module, and let be any element of their field or ring of scalars. By the properties defining vector spaces and modules, and are also elements of the same vector space or module. A function between two vector spaces or modules is called a linear transformation if

Linear transformations are related to the equation of a line in Cartesian geometry, and they give the study of vector spaces and modules its name, linear algebra. For certain types of vector spaces or modules, linear transformations can be represented by nifty little gadgets called matrices, which are rectangular arrays of elements of the field or ring of scalars. The vectors (elements of vector spaces) which we have featured in this post can be thought of as matrices with only a single column, or sometimes called column matrices. We will not discuss matrices in this post, although perhaps in the future we will they can be found, along with many other deeper aspects of linear algebra, in most textbooks on linear algebra or abstract algebra such as Linear Algebra Done Right by Sheldon Axler or Algebra by Michael Artin.