## Courses

» Please see the Schedule of Classes for the current semester’s offerings.

Except as noted, all courses are 3 credits. Except in the case of Topics Courses and Problem Seminars, the first semester of any two-semester course will include a survey of the important topics in the course. The second semester, which will be given only upon sufficient student demand, will consist of a deeper treatment of certain topics covered in the first semester.

**MAT 5115 Elementary Theory of Numbers**

Divisibility, congruence, quadratic reciprocity, elementary results in quadratic forms, diophantine equations, and rational approximation to irrationals. Applications to cryptology and data security.

**MAT 5117 Linear Algebra**

Vector spaces, basis, dimension, direct sums, factor spaces; linear transformations, functionals, dual spaces, matrices, determinants; systems of linear equations; diagonalization, normal and canonical forms, elementary divisors; bilinear and quadratic forms; inner products, euclidean and unitary space, orthogonal and symmetric matrices; tensors and exterior algebra. Applications to Markov chains and linear regression.

**MAT 5118 Introduction to Analysis**

A survey of analytic methods which are of practical significance for applications, as well as the mathematical foundations, contexts, and limitations of those methods.

**MAT 5127 Functions of a Complex Variable**

Integration and differentiation in the complex domain. Cauchy's Theorem, Cauchy Integral Formula, Laurent expansion, residues. Elements of conformal mapping, special functions, series and product representations.

**MAT 5200, 5201 Problem Seminar**

Students are trained in applying their knowledge in various areas to the solution of specific problems arising in industrial and technological applications of mathematics: operations management, risk theory, shock wave theory, atomic force microscopy, materials science.

**MAT 5209 Ordinary Differential Equations**

Differential and integral equations in the real domain; existence and stability theory, Sturm-Liouville problem for linear equations, techniques of solution for special classes. Differential and integral equations in the complex domain; equations of Fuchsian type and special functions; transform methods. Transition to chaos.

**MAT 5210, 5211, 5215 Partial Differential Equations I, II, III**

Introduction to the theory of partial differential equations of second order. Problem of Cauchy, boundary value problems of potential theory, variational principles. Equations of mixed type. Applications to finance, geometry, plasma, and gas dynamics.

**MAT 5230, 5231 Functional Analysis I, II**

Banach and Hilbert spaces, linear functionals, Hahn-Banach theorem, dual spaces, linear operators, closed graph theorem, Riesz theory for compact operators, spectral theory.

**MAT 5253 Algebra**

Sets, Boolean algebra, cardinal numbers. Groups, rings and ideals, integral domains, fields, algebraic number fields, Galois theory, combinatorics. Diverse applications to computer science.

**MAT 5256, 5257 Topics in Probability Theory I, II**

Probability and risk measures. Applications to financial decision-making, operations management, and the theory of gambling. Random walks, Brownian motion, fractional Brownian motion , white noise processes, Markov chains, Markov processes, time series, convexity methods. Nonperiodic cycles and processes with memory.

**MAT 5258 Topics in Geometry**

Synthetic geometry. Projective spaces. Elements of algebraic geometry. Applications are chosen from computer graphics, finite-element analysis, geometrical optics, and the theory of caustics.

**MAT 5259, 5260 Differential and Riemannian Geometry I, II**

Classical differential geometry of curves and surfaces in space. Intrinsic geometry on a surface. Tensor calculus with applications to geometry in *n* dimensions. Elements of geometric analysis (harmonic maps). Applications to special and general relativity.

**MAT 5261 Topics in Modern Differential Geometry**

Definition and elementary properties of Lie groups and Lie algebras; vector bundles and connections. Morse theory. Elements of Hodge theory. Applications to high-energy physics and gauge-field theory.

**MAT 5262, 5263 Topology I, II**

A rigorous introductory treatment of point-set topology, differential topology, homotopy and homology.

**MAT 5265 Functions of a Real Variable**

Fundamentals of real analysis and applications. Lebesgue measure and integral. Integrals on sigma algebras. Probability measures. Introduction to Hilbert spaces and *LP*-spaces. Applications to Fourier series and to Fourier and more general transforms.

**MAT 5266 Mathematical Statistics**

Development of statistical models as corollaries of theorems in probability, and a rigorous presentation of topics related to the practice of statistics and data analysis.

**MAT 5267 Convex Optimization**

Convex analysis in finite dimensions; linear programming; convex optimization with constraints; vector (multi-criteria) optimization problems from theoretical and computational perspectives. Applications to finance and economics, including convex risk measures, portfolio optimization and utility maximization problems.

** MAT 5270** **Data
Science: Fundamentals and Applications**

Statistical and computational fundamentals that form the
basis for contemporary data science applications in biomedical science, finance
and other cognate 'big data' disciplines are introduced. Core components
include data exploration, data modeling, the use of data mining technologies
and application examples. Course material will be complemented by hands-on
programming experience, using the iPython programming environment, to allow the
class to gain a hands-on experience of data science analytics.

** MAT 5272 Applied Data Science: Contexts and Methodologies**

Examination of exemplar data science publications from the domains of biomedical science, quantitative finance, geoscience and the astronomical sciences.

**MAT 5301, 5302 Topics and Problems in Analysis**

Techniques of problem-solving and estimation, and related concepts in real and complex analysis. An introduction to working analysis in distinction to theoretical analysis. Financial and engineering applications are emphasized.

**MAT 5310, 5311 Topics in Partial Differential Equations I, II**

General theory of linear partial differential equations. Semilinear, quasilinear, and fully nonlinear equations. Variational theory. Cauchy and boundary value problems, Estimates and regularity of solutions. Topics will be chosen from contemporary application to geometry, finance, continuum mechanics, plasma physics, and engineering.

**MAT 5312 Mathamatical Logic and Computability Theory**

Boolean logics, truth functions, quantification theory, Turing machines. Horn algebras, lattices, quasivarities. Applications to computer science and, in particular, artificial intelligence.

**MAT 5315 Readings in Mathematical Logic**

Topics to be arranged, depending on the interests and backgrounds of the students.

Given only by arrangement with the instructor.

**MAT 5317 Readings in Linear/Modern Algebra**

Topics to be arranged, depending on the interests and backgrounds of the students.

Given only by arrangement with the instructor.

**MAT 5320 Complex Systems**

Nonlinear and fractal time series; computational methods; applications include econophysics, fractal statistics, and neural physics.

**MAT 5330, 5331 Topics in Functional Analysis**

Hilbert and Banach spaces, operator theory. Applications.

**MAT 5365 Fourier Analysis**

Fourier integrals. Applications to signal processing, imaging science, time series analysis, and the theory of waves.

**MAT 5931, 5932 Graduate Seminar**

Faculty-supervised reading/research on a topic in contemporary mathematics.

**MAT 6401, 6402 Readings in Analysis**

Reading course for doctoral students. Variable credit.

**MAT 6431, 6432 Readings in Algebra/Geometry**

Reading course for doctoral students. Variable credit.

**MAT 6451, 6452 Readings in Mathematics**

Reading course for doctoral students. Variable credit.

**MAT 7705 Thesis Preparation**

*Prerequisite*: a grade of Pass on the Advanced Qualifying Examination. Variable credit.