Mathematics Course Catalog

BIOCHEM/BMI/BMOLCHEM/MATH 609: Mathematical Methods for Systems Biology

Credits unavailable

Provides a rigorous foundation for mathematical modeling of biological systems. Mathematical techniques include dynamical systems and differential equations. Applications to biological pathways, including understanding of bistability within chemical reaction systems, are emphasized.

COMPSCI/ECE/MATH 435: Introduction to Cryptography

3 credits

Cryptography is the art and science of transmitting digital information in a secure manner. Provides an introduction to its technical aspects.

COMPSCI/ISYE/MATH 425: Introduction to Combinatorial Optimization

3 credits

Focuses on optimization problems over discrete structures, such as shortest paths, spanning trees, flows, matchings, and the traveling salesman problem. We will investigate structural properties of these problems, and we will study both exact methods for their solution, and approximation algorithms.

COMPSCI/ISYE/MATH 728: Integer Optimization

Credits unavailable

Introduces optimization problems over integers, and surveys the theory behind the algorithms used in state-of-the-art methods for solving such problems. Special attention is given to the polyhedral formulations of these problems, and to their algebraic and geometric properties. Applicability of Integer Optimization is highlighted with applications in combinatorial optimization. Key topics include: formulations, relaxations, polyhedral theory, cutting planes, decomposition, enumeration. Students are strongly encouraged to have knowledge of Linear Programming (e.g.,MATH/COMP SCI/I SY E/STAT 525), including algorithms, duality and polyhedral theory.

COMPSCI/ISYE/MATH 730: Nonlinear Optimization II

Credits unavailable

Theory and algorithms for nonlinearly constrained optimization. Relevant geometric concepts, including tangent and normal cones, theorems of the alternative, and separation results. Constraint qualifications. Geometric and algebraic expression of first-order optimality conditions. Second-order optimality conditions. Duality. Nonlinear programming algorithms: merit functions and filters; interior-point, augmented Lagrangian, and sequential quadratic programming algorithms.

COMPSCI/ISYE/MATH/STAT 525: Linear Optimization

3 credits

Introduces optimization problems whose constraints are expressed by linear inequalities. Develops geometric and algebraic insights into the structure of the problem, with an emphasis on formal proofs. Presents the theory behind the simplex method, the main algorithm used to solve linear optimization problems. Explores duality theory and theorems of the alternatives.

COMPSCI/ISYE/MATH/STAT 726: Nonlinear Optimization I

3 credits

Theory and algorithms for nonlinear optimization, focusing on unconstrained optimization. Line-search and trust-region methods; quasi-Newton methods; conjugate-gradient and limited-memory methods for large-scale problems; derivative-free optimization; algorithms for least-squares problems and nonlinear equations; gradient projection algorithms for bound-constrained problems; and simple penalty methods for nonlinearly constrained optimization. Students are strongly encouraged to have knowledge of linear algebra and familiarity with basic mathematical analysis.

COMPSCI/MATH 240: Introduction to Discrete Mathematics

3 credits

Basic concepts of logic, sets, partial order and other relations, and functions. Basic concepts of mathematics (definitions, proofs, sets, functions, and relations) with a focus on discrete structures: integers, bits, strings, trees, and graphs. Propositional logic, Boolean algebra, and predicate logic. Mathematical induction and recursion. Invariants and algorithmic correctness. Recurrences and asymptotic growth analysis. Fundamentals of counting.

COMPSCI/MATH 513: Numerical Linear Algebra

Credits unavailable

Direct and iterative solution of linear and nonlinear systems and of eigenproblems. LU and symmetric LU factorization. Complexity, stability, and conditioning. Nonlinear systems. Iterative methods for linear systems. QR-factorization and least squares. Eigenproblems: local and global methods.

COMPSCI/MATH 514: Numerical Analysis

3 credits

Polynomial forms, divided differences. Polynomial interpolation. Polynomial approximation: uniform approximation and Chebyshev polynomials, least-squares approximation and orthogonal polynomials. Numerical differentiation and integration. Splines, B-splines and spline approximation. Numerical methods for solving initial and boundary value problems for ordinary differential equations.

COMPSCI/MATH 714: Methods of Computational Mathematics I

3 credits

Development of finite difference methods for hyperbolic, parabolic and elliptic partial differential equations. Analysis of accuracy and stability of difference schemes. Direct and iterative methods for solving linear systems. Introduction to finite volume methods. Applications from science and engineering. Students are strongly encouraged to have programming skills (e.g.COMP SCI 200) and some undergraduate numerical analysis (e.g.MATH/COMP SCI 514orCOMP SCI 412), analysis and differential equations (e.g.MATH 322andMATH 521) and linear algebra (e.g.MATH 341).

COMPSCI/MATH 715: Methods of Computational Mathematics II

Credits unavailable

Introduction to spectral methods (Fourier, Chebyshev, Fast Fourier Transform), finite element methods (Galerkin methods, energy estimates and error analysis), and mesh-free methods (Monte-Carlo, smoothed-particle hydrodynamics) for solving partial differential equations. Applications from science and engineering. Applications from science and engineering. Students are strongly encouraged to have programming skills (e.g.COMP SCI 200), undergraduate numerical analysis (e.g.MATH/COMP SCI 514orCOMP SCI 412), analysis (MATH 322and521) and linear algebra (e.g.MATH 341or equiv.)

COMPSCI/MATH/STAT 475: Introduction to Combinatorics

3 credits

Problems of enumeration, distribution, and arrangement. Inclusion-exclusion principle. Generating functions and linear recurrence relations. Combinatorial identities. Graph coloring problems. Finite designs. Systems of distinct representatives and matching problems in graphs. Potential applications in the social, biological, and physical sciences. Puzzles. Problem solving.

CURRIC/MATH 471: Mathematics for Secondary School Teachers

Credits unavailable

Capstone for future middle and high school teachers, drawing connections between higher mathematics and school mathematics.

ECE/MATH 842: Topics in Applied Algebra

Credits unavailable

Applied topics with emhasis on algebraic constructions and structures. Examples include: algebraic coding theory; codes (algebraic-geometric, convolutional, low-density-parity-check, space-time); curve and lattice based cryptography; watermarking; computer vision (face recognition, multiview geometry).

ECE/MATH/STAT 888: Topics in Mathematical Data Science

1–3 credits

Advanced topics in the mathematical foundations of data science

HISTSCI/MATH 473: History of Mathematics

3 credits

An historical survey of the main lines of mathematical development.

ISYE/MATH/OTM/STAT 632: Introduction to Stochastic Processes

3 credits

Topics include discrete-time Markov chains, Poisson point processes, continuous-time Markov chains, and renewal processes. Applications to queueing, branching, and other models in science, engineering and business.

MATH 112: College Algebra

3 credits

Properties of elementary functions, such as polynomial, absolute value, radical, rational, exponential, and logarithmic functions. Topics include equations, inequalities, functions, and their graphs. Formulating, analyzing, solving, and interpreting mathematical and real-world problems. Provides the algebra skills required for calculus.

MATH 113: Trigonometry

3 credits

Covers the graphs, properties and geometric significance of trigonometric functions of a real variable. Other topics include trigonometric equations and identities, trigonometric form of complex numbers, DeMoivre's theorem, and polar and parametric equations. Formulating, analyzing, solving, and interpreting mathematical and real-world problems using trigonometry.

MATH 114: Precalculus

5 credits

Properties of elementary functions, such as polynomial, absolute value, radical, rational, exponential, logarithmic, and trigonometric functions. Topics include equations, inequalities, functions, and their graphs. Other topics include trigonometric equations and identities and the geometric significance of trigonometry. Formulating, analyzing, solving, and interpreting mathematical and real-world problems. Provides the algebra and trigonometry skills required for calculus.

MATH 141: Quantitative Reasoning and Problem Solving

3 credits

Develops a habit of mind, competency, and comfort in working with numerical data. Learn to reason and solve quantitative problems from a wide array of authentic contexts and everyday life situations, develop the ability to reason mathematically, and make and evaluate logical arguments supported by quantitative evidence. This course is for students who need to satisfy part A of the Quantitative Reasoning requirement and prepare for QR-B courses, but do not want to continue in the calculus sequence.

MATH 198: Directed Study

Credits unavailable

Directed study projects as arranged with a faculty member.

MATH 207: Study Abroad: Intermediate Mathematics

Credits unavailable

A study abroad equivalency for intermediate-level mathematics that does not equate to a UW-Madison course. Requires current enrollment in a UW-Madison sponsored international program.

MATH 211: Survey of Calculus 1

4 credits

Essential concepts of differential and integral calculus; exponential and logarithmic functions; functions of several variables. Primarily for students in prebusiness and some social sciences. Students preparing for advanced study in mathematics, physics, engineering and other sciences (including most biological sciences) should takeMATH 221.

MATH 213: Survey of Calculus 2

3 credits

First order differential equations, introduction to multivariable calculus and constrained optimization, infinite sequences and series, methods of approximation, and a brief introduction to probability. Models and applications from business and the social sciences.

MATH 221: Calculus and Analytic Geometry 1

5 credits

Introduction to differential and integral calculus and plane analytic geometry; applications; transcendental functions.

MATH 222: Calculus and Analytic Geometry 2

4 credits

Techniques of integration, improper integrals, first order ordinary differential equations, sequences and series, Taylor series, vector geometry in two and three dimensions.

MATH 228: Wes Calculus Supplement

Credits unavailable

Topics in differential, integral and multi-variable calculus and analytic geometry.

MATH 234: Calculus--functions of Several Variables

4 credits

Introduction to calculus of functions of several variables; calculus on parameterized curves, derivatives of functions of several variables, multiple integrals, vector calculus.

MATH 298: Directed Study in Mathematics

Credits unavailable

Directed study projects as arranged with a faculty member.

MATH 319: Techniques in Ordinary Differential Equations

3 credits

Review of linear differential equations; series solution of linear differential equations; boundary value problems; Laplace transforms; possibly numerical methods and two dimensional autonomous systems.

MATH 320: Linear Algebra and Differential Equations

3 credits

An introduction to linear algebra and differential equations with emphasis on the relationship between the theory of linear algebra and analytical and numerical techniques for solving differential equations. Linear algebra topics include linear systems, matrices and their algebra, vector spaces and linear transformations, eigenvalues and eigenvectors. Topics from differential equations include first order ODE, homogeneous and nonhomogeneous linear systems, and numerical methods.

MATH 321: Applied Mathematical Analysis 1: Vector and Complex Calculus

3 credits

Vector analysis: algebra and geometry of vectors, index notation, 3D rotations, Euler angles; vector differential and integral calculus, theorems of Green, Gauss, and Stokes; grad, div, curl, and Laplacian in cylindrical and spherical coordinates; Complex analysis: analytic functions, Cauchy-Riemann equations, conformal mapping, complex integrals, residue theorem.

MATH 322: Applied Mathematical Analysis 2: Partial Differential Equations

Credits unavailable

Sturm-Liouville theory; Fourier series, including mean convergence; initial and boundary value problems for linear second order partial differential equations, including separation of variables and eigenfunction expansions; fundamental solutions and Green's functions in multiple dimensions.

MATH 331: Introductory Probability

3 credits

Topics covered include axioms of probability, random variables, the most important discrete and continuous probability distributions, expectation and variance, conditional probability and conditional expectations, Markov's and Chebyshev's inequalities, laws of large numbers, and the central limit theorem. Includes a brief introduction to techniques of multivariate integration.

MATH 340: Elementary Matrix and Linear Algebra

3 credits

An introduction to linear algebra. Topics include matrix algebra, linear systems of equations, vector spaces, sub-spaces, linear dependence, span, basis, rank of matrices, determinants, linear transformations, coordinate representations, kernel, range, eigenvalues and eigenvectors, diagonalization, inner products and orthogonal vectors, symmetric matrices. Covers linear algebra topics in greater depth and detail thanMATH 320. Formal techniques in mathematical argument [MATH 341] not covered.

MATH 341: Linear Algebra

3 credits

The theory of linear algebra with an introduction to proofs and proof writing. Topics include vector spaces, linear dependence, span, basis, linear transformations, kernel, image, inner products and inner product spaces, geometry, eigenvalues, eigenvectors, standard matrix factorizations. Other content includes basic set theory, logical operations, quantifiers, direct and indirect arguments, and induction. Differential equations [MATH 320] not covered.

MATH 345: Linear Algebra and Optimization

4 credits

Introduction to linear algebra, differential calculus in several variables, and basic optimization theory with applications to data science and related topics. Vectors, analytic geometry, matrices, linear functions, linear independence, orthogonality, inverses, partial derivatives and gradients, Taylor approximation, gradient descent, Lagrange multipliers, clustering, regression, classification. Implementation in Python.

MATH 375: Topics in Multi-variable Calculus and Linear Algebra

5 credits

Vector spaces and linear transformations, differential calculus of scalar and vector fields, determinants, eigenvalues and eigenvectors, multiple integrals, line integrals, and surface integrals. Freshmen students are invited to enroll by the Department of Mathematics.

MATH 376: Topics in Multi-variable Calculus and Differential Equations

Credits unavailable

Topics in multi-variable calculus and introduction to differential equations.

MATH 390: Undergraduate Research with Madison Experimental Mathematics Lab

3 credits

An introduction to mathematical research. Instruction in ancillary skills such as literature review, mathematical software use, technical writing and communication, etc. Requires acceptance to the Madison Experimental Mathematics undergraduate research lab.

MATH 407: Study Abroad: Advanced Mathematics

Credits unavailable

A study abroad equivalency for advanced-level mathematics that does not equate to a UW-Madison course. Requires current enrollment in a UW-Madison sponsored international program.

MATH 415: Applied Dynamical Systems, Chaos and Modeling

3 credits

An introduction to nonlinear dynamical systems including stability, bifurcations and chaos. Provides underlying mathematical ideas, but emphasizes applications from many scientific fields.

MATH 421: The Theory of Single Variable Calculus

3 credits

Covers material in first and second semester calculus but it is intended to teach math majors to write and understand proofs in mathematics in general and in calculus in particular.

MATH 443: Applied Linear Algebra

Credits unavailable

Review of matrix algebra. Simultaneous linear equations, linear dependence and rank, vector space, eigenvalues and eigenvectors, diagonalization, quadratic forms, inner product spaces, norms, canonical forms. Discussion of numerical aspects and applications in the sciences.

MATH 444: Graphs and Networks in Data Science

Credits unavailable

Mathematical foundations of networks with an emphasis on their applications in modern data science, using tools from algorithmic graph theory and linear algebra. Topics include: basics of graph theory, network statistics, graph traversal algorithms and implementation, matrix methods, community detection, PageRank, simulation of random graph models.

MATH 461: College Geometry I

Credits unavailable

An introduction to Euclidean or non-Euclidean geometry.

MATH 467: Introduction to Number Theory

3 credits

An introduction to proof writing techniques through a study of classical topics in elementary number theory. Topics include the divisibility, basic properties of primes, congruences, Fermat's theorem.

MATH 490: Undergraduate Seminar

Credits unavailable

Intermediate or upper level topics course in mathematics. Topics vary.

MATH 491: Topics in Undergraduate Mathematics

Credits unavailable

Intermediate or upper level topics course in mathematics. Topics vary.

MATH 519: Ordinary Differential Equations

Credits unavailable

Provides a rigorous, proof-based introduction to ordinary differential equations and dynamical systems.

MATH 521: Analysis I

3 credits

The real numbers, elements of set theory, metric spaces and basic topology, sequences and series, limits, continuity, differentiation, integration, sequences and series of functions, uniform convergence.

MATH 522: Analysis II

3 credits

Special functions, power series, Fourier series, approximation, contraction principle, characterizations of compactness in metric spaces, applications to differential equations. Differential calculus in normed spaces, including implicit and inverse function theorems.

MATH 531: Probability Theory

Credits unavailable

A rigorous introduction to probability theory at an advanced undergraduate level. Only a minimal amount of measure theory is used, in particular, the theory of Lebesgue integrals is not needed. It is aimed at math majors and Master's degree students, or students in other fields who will need probability in their future careers. Gives an introduction to the basics (Kolmogorov axioms, conditional probability and independence, random variables, expectation) and discusses some classical results with proofs (DeMoivre-Laplace limit theorems, the study of simple random walk on the one dimensional lattice, applications of generating functions).

MATH 535: Mathematical Methods in Data Science

3 credits

A rigorous introduction to mathematical concepts important for modern data science. Topics include: matrix factorizations, optimization theory and algorithms, probabilistic models, finite Markov chains. Mathematical techniques are motivated by and illustrated on a range of applied problems from machine learning and statistics.

MATH 540: Linear Algebra II

3 credits

Review of linear transformations, duality. Diagonalization of linear transformations. The Cayley Hamilton Theorem. Minimal polynomials. The Jordan canonical form. Exponential function. Inner product spaces, orthonormal bases and Gram-Schmidt orthogonalization. Operators on inner product spaces. Self-adjoint, unitary and positive operators. Spectral theorem. Singular value decomposition. Bilinear and quadratic forms. Norms, bounded linear operators, matrix norms. Basic multilinear algebra.

MATH 541: Modern Algebra 1

3 credits

Groups, normal subgroups, Cayley's theorem, rings, ideals, homomorphisms, polynomial rings, abstract vector spaces.

MATH 542: Modern Algebra 2

3 credits

Field extensions, roots of polynomials, splitting fields, simple extensions, linear transformations, matrices, characteristic roots, canonical forms, determinants.

MATH 551: Point-set Topology

3 credits

Topological spaces, connectedness, compactness, separation axioms, metric spaces.

MATH 552: Introduction to Algebraic Topology

Credits unavailable

Introduction to algebraic topology. Emphasis on geometric aspects, including two-dimensional manifolds, the fundamental group, covering spaces, basic simplicial homology theory, the Euler-Poincare formula, and homotopy classes of mappings.

MATH 561: Differential Geometry

Credits unavailable

Theory of curves and surfaces by differential methods.

MATH 567: Modern Number Theory

Credits unavailable

A course in number theory covering fundamentals and modern applications in topics of recent interest: Modular arithmetic, quadratic reciprocity, arithmetic functions, zeta function, Diophantine equations, transcendental numbers, Roth's theorem, continued fractions, and the circle method. Optional material from probability including random matrix theory.

MATH 570: Fundamentals of Set Theory

Credits unavailable

Introduces the basic concepts of Set Theory including: Set-theoretical paradoxes and means of avoiding them, sets, relations, functions, orders and well-orders, proof by transfinite induction and definitions by transfinite recursion, cardinal and ordinal numbers and their arithmetic, construction of the real numbers, the axiom of choice and its consequences.

MATH 607: Study Abroad: Proof-based Advanced Mathematics

Credits unavailable

A study abroad equivalency for proof-based advanced-level mathematics that does not equate to a UW-Madison course. Requires current enrollment in a UW-Madison sponsored international program.

MATH 616: Data-driven Dynamical Systems, Stochastic Modeling and Prediction

Credits unavailable

An introduction to data-driven dynamical systems, including mathematical theory, methodology, numerical algorithms, applications and the use of a programming language to solve related coding problems. Topics include stochastic toolkits for dynamical systems and data science, linear Gaussian processes, nonlinear stochastic systems, elementary stochastic differential equations, data assimilation, parameter estimation, forecasting and prediction.

MATH 619: Analysis of Partial Differential Equations

3 credits

A rigorous introduction to the theoretical underpinnings of the basic methods and techniques in the modern theory of PDEs. It is aimed at math majors, but will also be useful to some students in the sciences, engineering and economics who feel the need for a deeper understanding of the theory of PDEs. The emphasis is on the exposure to a number of different methods of solution of PDEs and their connection to physical phenomena modeled by the equations. The goals include both learning to solve some basic types of PDEs as well as to understand the motivation behind and inner workings of the techniques involved.

MATH 621: Introduction to Manifolds

Credits unavailable

Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.

MATH 623: Complex Analysis

Credits unavailable

Elementary functions of a complex variable; conformal mapping; complex integrals; the calculus of residues.

MATH 627: Introduction to Fourier Analysis

Credits unavailable

Fourier series and integrals, and their applications.

MATH 629: Introduction to Measure and Integration

Credits unavailable

Lebesgue integral and measure, abstract measure and integration, differentiation, spaces of integrable functions.

MATH 635: An Introduction to Brownian Motion and Stochastic Calculus

Credits unavailable

Presents an introduction to Brownian motion and its application to stochastic calculus. Sample path properties of Brownian motion, Ito stochastic integrals, Ito's formula, stochastic differential equations and properties of their solutions, and various applications will be included.

MATH 681: Senior Honors Thesis

3 credits

Individual study for honors math majors writing a thesis in mathematics.

MATH 682: Senior Honors Thesis

3 credits

Individual study for honors math majors writing a thesis in mathematics.

MATH 691: Undergraduate Thesis

2–4 credits

Individual study for students writing a thesis in mathematics.

MATH 692: Undergraduate Thesis

Credits unavailable

Individual study for students writing a thesis in mathematics.

MATH 698: Directed Study

1–3 credits

Directed study projects as arranged with a faculty member.

MATH 699: Directed Study

1–6 credits

Directed study projects as arranged with a faculty member.

MATH 703: Methods of Applied Mathematics 1

3 credits

Study of the linear algebraic structure underlying discrete equilibrium problems. Boundary value problems for continous equilibria: Sturm-Liouville equations, Laplace's equation, Poisson's equation, and the equations for Stokes flow. Contour integration and conformal mapping. Applications of dynamics leading to initial value problems for ODEs and PDEs. Green's functions for ODEs and introduction to asymptotic methods for ODEs, e.g. WKB analysis. Separation of variables and eigenfunction expansions for linear PDEs. Examples from physics and engineering throughout. Knowledge of undergraduate linear algebra, analysis and complex analysis is strongly recommended.

MATH 704: Methods of Applied Mathematics-2

Credits unavailable

Derivation, nature and solution of canonical partial differential equations of applied mathematics. Conservation laws, advection, diffusion. First order PDEs, characteristics, shocks. Traffic flow, eikonal and Hamilton-Jacobi equations. Higher order PDEs: classification, Fourier analysis, well-posedness. Series solutions and integral transforms. Green's functions and distributions. Similarity solutions. Asymptotics of Fourier integrals. Laplace's method, stationary phase. Ship waves. Perturbation methods. Knowledge of undergraduate linear algebra, analysis and complex analysis is strongly recommended.

MATH 705: Mathematical Fluid Dynamics

Credits unavailable

Advanced introduction to fluid dynamics. Basic concepts; elementary viscous flow; Navier-Stokes equations. Elementary airfoil theory; boundary layers. Vortex motion. Waves. Very viscous flow. Compressible flows. Instabilities, bifurcations, turbulence. Requires working knowledge of multivariate calculus, differential equations and mechanics.

MATH 716: Ordinary Differential Equations

Credits unavailable

Existence, uniqueness, and continuous dependence theorems, linear systems, stability, singular points, and boundary value problems. Qualitative behavior of nonlinear equations, stability, Lyapunov functions, invariant manifolds, bifurcation theory, periodic orbits, and connecting orbits.

MATH 717: Stochastic Computational Methods

3 credits

Introduction to computational methods that use stochastic algorithms and/or methods that are applied to random or stochastic mathematical problems. The main emphasis will be placed on learning practical tools, while some aspects of theoretical foundations will also be covered (e.g., basic error analysis for numerical solution of stochastic differential equations (SDEs), and basic convergence of Monte Carlo methods). Topics include Monte Carlo methods, Bayesian inference and Bayesian sampling, simulation of Markov chains, numerical analysis for SDEs, data assimilation / state estimation, stochastic optimization methods and random sketching. Applications to science, engineering, finance, data science, and other practical problems also included.

MATH 718: Randomized Linear Algebra and Applications

Credits unavailable

Random solvers have been playing increasingly crucial roles in the modern computational tasks. The recent breakthroughs in applied and computational linear algebra that incorporate techniques of randomization have proven to be of great importance in modern applied math, computational sciences and data science, such as inverse problems, machine learning and scientific computing. The guiding principle is that one may greatly reduce computational and storage expenses at the cost of a small probability of failure. Systematic study of these modern methods of randomized linear algebra solvers will be provided, presenting mathematical backgrounds, algorithms, and concrete applications. Core theoretical topics include randomized Kaczmarz and its generalization to stochastic gradient descent, randomized singular value decomposition, random sketching, matrix completion, and compressive sensing, and corresponding applications.

MATH 719: Partial Differential Equations

3 credits

Classical theory of partial differential equations, together with an introduction to the modern theory based on functional analysis. Familiarity with basic measure theory (e.g.MATH 629or721) or concurrent registration inMATH 721is strongly recommended.

MATH 720: Partial Differential Equations

Credits unavailable

Linear elliptic, parabolic and hyperbolic equations, continuing with calculus of variations and then nonlinear initial value problems.

MATH 721: A First Course in Real Analysis

3 credits

Real analysis concentrating on measures, integration, and differentiation and including an introduction to Hilbert spaces. Knowledge of undergraduate analysis (e.g. the sequenceMATH 521and522) is strongly recommended.

MATH 722: Complex Analysis

Credits unavailable

The basic theory of functions of one complex variable including Cauchy formula, singularities and residues, meromorphic functions, conformal mappings, harmonic functions, approximation and the nonhomogeneous d-bar equation. Requires knowledge of undergraduate analysis (e.g. the sequenceMATH 521and522).

MATH 725: A Second Course in Real Analysis

Credits unavailable

Continuation ofMATH 721. An introduction to further topics in real analysis: Banach spaces, Fourier transforms, elements of distribution theory, and applications.

MATH 735: Stochastic Analysis

Credits unavailable

Foundations of continuous time stochastic processes, semimartingales and the semimartingale integral, Ito's formula, stochastic differential equations, stochastic equations for Markov processes, application in finance, filtering, and control. The course relies on measure theoretic probability theory that can be reviewed at the beginning of the semester.

MATH 740: Enumerative Combinatorics/symmetric Functions

Credits unavailable

Inclusion-exclusion principle, permutation statistics, sieve methods, unimodal sequences, posets, lattice theory, Mobius functions, generating functions, bases and transition matrices for symmetric functions, Young tableaux, plane partitions, polytopes, poset homology, Stanley-Reisner rings.

MATH 741: Abstract Algebra

3 credits

Usually a study of finite groups and noncommutative rings. Group theoretic topics may include: permutation groups, Lagrange's theorem, Cauchy's theorem and the Sylow theorems, solvable and nilpotent groups. Ring theoretic topics may include: Artinian rings and modules, the Wedderburn theorems, the Hopkins-Levitzki theorem, the Jacobson radical and density theorem. The basic prerequisite for all advanced graduate courses in algebra. Familiarity with topics in undergraduate algebra (e.g.MATH 541and542) is strongly recommended.

MATH 742: Abstract Algebra

Credits unavailable

Continuation ofMATH 741. Usually the study of commutative rings and fields. Ring theoretic topics may include: modules over PIDs, Noetherian rings and the Hilbert basis theorem, the Lasker-Noether theorem, the Krull intersection theorem, integrality and the Hilbert Nullstellensatz. Field theoretic topics may include: algebraic extensions, Galois theory, solvability of polynomials and classical constructability problems.

MATH 746: Topics in Ring Theory

Credits unavailable

Will alternate between commutative and noncommutative ring theory. Commutative topics include localization; local rings; dimension theory; Cohen-Macaulay rings. Noncommutative topics include projective modules; injective modules; flat modules; homological and global dimension; Wedderburn and Goldie rings. Basic graduate algebra courses (e.g.MATH 741and742) are strongly recommended.

MATH 747: Lie Algebras

Credits unavailable

Lie algebras and matrix groups. Topics: tangent spaces; exponentials; Baker-Campbell-Hausdorff formula; (nilpotent, solvable, semisimple) Lie algebras; Engel's and Lie's theorems; Levi decomposition; Killing form; sl(2)-representations; root systems; Dynkin diagrams; Weyl groups; Cartan and Borel subalgebras; Serre's theorem. Basic graduate algebra courses (e.g.MATH 541and542orMATH 741and742) are strongly recommended.

MATH 748: Algebraic Number Theory

Credits unavailable

A rigorous introduction to the arithmetic of number fields; algebraic integers, geometry of numbers, Dirichlet's Unit Theorem, ideal class groups, first case of Fermat's Last Theorem; prime decompositions, Galois automorphisms. Basic graduate algebra courses (e.g.MATH 741and742) are strongly recommended.

MATH 749: Analytic Number Theory

3 credits

An introduction to the Riemann zeta function and Dirichlet L-functions, and their application to topics such as the distribution of prime numbers and the arithmetic of quadratic fields. Familiarity with elementary number theory [such asMATH 567] and complex analysis [such asMATH 623] is strongly recommended.

MATH 750: Homological Algebra

Credits unavailable

Topics include: complexes, cohomology, double complexes, spectral sequences; abelian categories, derived categories, derived functors; Tor and Ext, Koszul complexes; group cohomology; sheaf cohomology, hypercohomology. Basic graduate algebra courses (e.g.MATH 741and742) are strongly recommended

MATH 751: Introductory Topology I

3 credits

An introduction to algebraic and differential topology. Elements of homotopy theory, fundamental group, covering spaces. Differentiable manifolds, tangent vectors, regular values, transversality, examples of compact Lie groups. Homological algebra, chain complexes, cell complexes, singular and cellular homology, calculations for surfaces, spheres, projective spaces, etc. Familiarity with undergraduate algebra and topologyMATH 541or551) is strongly recommended.

MATH 752: Introductory Topology II

Credits unavailable

Continuation ofMATH 751. Cohomology, Universal Coefficient Theorem, Kunneth Formula, cup and cap products, applications to manifolds, orientability, Poincare Duality. Differential forms, integration and Stokes Theorem, De Rham Theorem. Calculations, further duality theorems, Euler class, Lefschetz Fixed-Point Theorem.

MATH 753: Algebraic Topology I

Credits unavailable

Higher homotopy groups, elements of obstruction theory, fibrations, bundle theory, classifying spaces, applications to smooth mainfolds, differential forms, vector bundles, characteristic classes, cobordism, applications and calculations. Basic graduate topology courses (e.g.MATH 751and752) are strongly recommended.

MATH 758: Introduction to Ergodic Theory and Dynamics

Credits unavailable

An introduction to ergodic theory and dynamics covering fundamental theorems of ergodic theory, classical examples of one and two dimensional dynamics as well as applications to study of group actions.

MATH 761: Differentiable Manifolds

3 credits

Differentiable manifolds, vector bundles, implicit function theorem, submersions and immersions, vector fields and flows, foliations and Frobenius theorem, differential forms and exterior calculus, integration and Stokes' theorem, De Rham theory, Riemannian metrics. Familiarity with basic undergraduate analysis coursesMATH 521and522orMATH 621) is strongly recommended.

MATH 763: Introduction to Algebraic Geometry

3 credits

Algebraic preliminaries, including local rings; valuation theory, and power series rings; geometry of algebraic varieties with emphasis on curves and surfaces.

MATH 764: Introduction to Algebraic Geometry

Credits unavailable

Continuation ofMATH 763.

MATH 765: Differential Geometry

Credits unavailable

Covers the metric properties of Riemannian manifolds. The following topics will be covered: Vector bundles and connections, Riemannian metrics, submanifolds and second fundamental form, first variation of arc length, geodesics, Hopf-Rinow theorem, second variation of arc length, Jacobi fields and index lemmas, Bonnet-Meyer theorem, Rauch comparison theorem, spaces of constant curvature, Hodge-de Rham theory. Familiarity with the topics in a differential manifolds course (e.g.MATH 761) is strongly recommended.

MATH 770: Foundations of Mathematics

3 credits

First-order logic syntax and semantics, Completeness and Compactness Theorems, Lowenheim-Skolem Theorem, computable and computably enumerable sets, Incompleteness Theorem, axioms of Zermelo-Fraenkel set theory with choice, ordinal and cardinal arithmetic.

MATH 771: Set Theory

Credits unavailable

Martin's Axiom, Suslin and Aronszajn trees, diamond principle, absoluteness and reflection, constructible universe, and one-step forcing constructions. Familiarity with the topics in a basic Foundations course such asMATH 770is strongly recommended.

MATH 773: Computability Theory

Credits unavailable

Turing degree and jump, strong reducibilities, arithmetic hierarchy, index sets, simple and (hyper)hypersimple sets, easy forcing arguments in computability theory, finite and infinite injury, Friedberg-Muchnik and Sacks Splitting Theorem, Sacks Jump and Sacks Density Theorems, computable ordinals. Familiarity with the topics in a basic Foundations course such asMATH 770is strongly recommended.

MATH 776: Model Theory

Credits unavailable

Review of compactness and some consequences. Quantifier elimination with examples. The omitting types theorem. Categoricity. Baldwin-Lachlan theory. Strongly minimal and o-minimal theories. Saturated models. Morley's theorem. Familiarity with the topics in a basic Foundations course such asMATH 770is strongly recommended.

MATH 790: Masters Thesis

1–3 credits

Work on a Master's thesis under the supervision of a faculty member.

MATH 801: Topics in Applied Mathematics

Credits unavailable

Selected topics in applied mathematics, applied analysis or numerical analysis and scientific computing.

MATH 807: Dynamical Systems

Credits unavailable

Treats the qualitative behavior of continuous and discrete dynamical systems, including Hamiltonian systems of differential equations. Typical topics include periodic and almost periodic solutions, the fixed point theorem of Poincare and Birkhoff, invariant curves and KAM theory, celestial mechanics, and chaotic behavior.

MATH 821: Advanced Topics in Real Analysis

Credits unavailable

Topics in partial differential equations and real analysis.

MATH 823: Advanced Topics in Complex Analysis

Credits unavailable

Several complex variables. Basic several complex variables or more special topics.

MATH 825: Selected Topics in Functional Analysis

Credits unavailable

Topics will vary and may include spectral theory, nonlinear functional analysis or abstract harmonic analysis.

MATH 826: Advanced Topics in Functional Analysis and Differential Equations

Credits unavailable

Topics in functional analysis and differential equations.

MATH 827: Fourier Analysis

Credits unavailable

Introduction to Fourier analysis in Euclidean spaces and related topics that may include singular and oscillatory integrals and trigonometric series.

MATH 828: Advanced Topics in Harmonic Analysis

Credits unavailable

Continuation ofMATH 827. Advanced topics in harmonic analysis.

MATH 843: Representation Theory

Credits unavailable

Introduction to the representation theory of Lie groups and their combinatorics. Universal enveloping algebras, highest weight modules, induction, restriction, weights, characters, multiplicity formulas, tensor products, Shapovalov forms, filtrations, Kazhdan-Zetlin patterns, Littelmann paths.

MATH 844: Arithmetic Geometry

Credits unavailable

An introduction to arithmetic geometry with emphasis on arithmetic of elliptic curves.

MATH 845: Class Field Theory

Credits unavailable

Introduction to local and global class field theory. Theory of local fields; local and global class field theory; complex multiplication, adeles, ideles, idele class characters, Tchebotarev's Density Theorem, CM elliptic curves, construction of class fields of imaginary quadratic fields.

MATH 846: Topics in Combinatorics

Credits unavailable

Topics in algebraic combinatorics such as (but not limited to) association schemes, hypergeometric series, classical orthogonal polynomials, codes, lattices, invariant theory, alternating sign matrices and domino tilings, statistical mechanical models, 6j-symbols, buildings and diagram geometries, matroids.

MATH 847: Topics in Algebra

Credits unavailable

Topics may include: Lie groups, algebraic groups, Chevalley groups, simple groups and associated geometries, group cohomology, group rings, Hopf algebras, enveloping algebras, quantum groups, infinite-dimensional Lie algebras, Hecke algebras, automorphic forms, Galois representations, zeta and L-functions, abelian varieties.

MATH 848: Advanced Topics in Number Theory

Credits unavailable

This is an advanced graduate topic course in number theory. Topics will vary. Target audience: Advanced graduate students in number theory, representation theory, and algebraic geometry.

MATH 849: Automorphic Forms

3 credits

Classical and/or modern theory of automorphic forms. Representation theory of GL(2).

MATH 851: Topics in Geometric Topology

Credits unavailable

Advanced Topics in Geometric Topology.

MATH 853: Topics in Algebraic Topology

Credits unavailable

Topics in Algebraic Topology.

MATH 856: Topics in Differential Topology

Credits unavailable

The theory of differential manifolds such as differential forms and de Rham theorem, cobordism groups, Lie groups, homogeneous spaces, fiber bundles, characteristic classes.

MATH 858: Introduction to Dynamics on Moduli Spaces

Credits unavailable

An introduction to ergodic theory and dynamics on the moduli space of translation surfaces with applications to counting problems in rational billiards and on flat surfaces. Covers dynamical techniques inspired by homogeneous dynamics including the use of Margulis functions to establish quantitative recurrence results; the use of random walks, inspired by work of Benoist and Quint, to prove measure rigidity statements; and the use of Zimmer's theory of the algebraic hull to prove finiteness results.

MATH 863: Advanced Topics in Algebraic Geometry

Credits unavailable

Geometry of several complex variables; algebraic groups, abelian varieties; topological aspects of algebraic geometry, including sheaf theory and homology theory; advanced theory of local rings; intersection theory of algebraic varieties.

MATH 865: Advanced Topics in Geometry

Credits unavailable

Selected from advanced projective geometry, non-Euclidean geometry, Riemannian geometry, distance geometry and the geometry of convex surfaces, geometry of numbers.

MATH 873: Advanced Topics in Foundations

Credits unavailable

Advanced topics from all areas of mathematical logic.

MATH 890: Pre-dissertator Research

1–8 credits

Reading and research in all areas of Mathematics.

MATH 900: Graduate Teaching Seminar

1 credits

Focuses on theory and practical skills relevant to teaching mathematics at the graduate or post-secondary level.

MATH 921: Seminar in Analysis

1–3 credits

Selected topics in Analysis.

MATH 941: Seminar-algebra

1–3 credits

Selected topics in Algebra.

MATH 951: Seminar in Topology

1–3 credits

Selected topics in Topology.

MATH 96: Preparatory Algebra

3 credits

Covers the necessary mathematical tools needed to succeed in our algebra course and provides fundamental mathematical skills. Topics include real numbers, linear equations and inequalities, integral and fractional exponents, polynomials and their arithmetic, polynomial equations and equations with fractional exponents, the quadratic formula and completing the square, systems of two linear equations, graphing, and problem solving using algebra and graphs. All students must pass an assessment on basic mathematical skills to complete the course. The course does not count for degree credit.

MATH 967: Seminar in Number Theory

1–3 credits

Selected topics in Number Theory.

MATH 975: Seminar-the Foundations of Mathematics

1–3 credits

Selected topics in Mathematical Logic.

MATH 990: Reading and Research

1–3 credits

Reading and research in all areas of Mathematics.

MATH/PHILOS 571: Mathematical Logic

3 credits

Basics of logic and mathematical proofs; propositional logic; first order logic; undecidability.

MATH/STAT 309: Introduction to Probability and Mathematical Statistics I

3 credits

Probability and combinatorial methods, discrete and continuous, univariate and multivariate distributions, expected values, moments, normal distribution and derived distributions, estimation.

MATH/STAT 310: Introduction to Probability and Mathematical Statistics II

3 credits

Mathematical statistical inference aims at providing an understanding of likelihood's central role to statistical inference, using the language of mathematical statistics to analyze statistical procedures, and using the computer as a tool for understanding statistics. Specific topics include: samples and populations, estimation, hypothesis testing, and theoretical properties of statistical inference.

MATH/STAT 409: Probability and Statistical Theory I

Credits unavailable

Provide a grounding in the theory underlying Statistics geared towards understanding why statistical methods work and to critically evaluate their performance. Probability theory highlighting the aspects of the field most relevant to statistical theory. Topics include: probability models, combinatorial methods, discrete and continuous random variables, univariate and multivariate distributions, expected values, moments, normal distribution and derived distributions, large sample theory and probability simulation.

MATH/STAT 431: Introduction to the Theory of Probability

3 credits

Topics covered include axioms of probability, random variables, the most important discrete and continuous probability distributions, expectation and variance, moment generating functions, conditional probability and conditional expectations, multivariate distributions, Markov's and Chebyshev's inequalities, laws of large numbers, and the central limit theorem.

MATH/STAT 709: Mathematical Statistics I

4 credits

An introduction to measure theoretic probability, random variables, and high-dimensional statistics; conditional expectation, sufficiency, and unbiased estimation; methods of large sample theory including laws of large numbers and central limit theorems.

MATH/STAT 710: Mathematical Statistics II

Credits unavailable

Estimation theory, efficiency, Neyman-Pearson theory of hypothesis testing, confidence regions, decision theory, likelihood ratio theory, current research topics in mathematical statistics.

MATH/STAT 733: Theory of Probability I

3 credits

An introduction to measure theoretic probability and stochastic processes.Topics include foundations, independence, zero-one laws, laws of large numbers, convergence in distribution, characteristic functions, central limit theorems, random walks, conditional expectations. Familiarity with basic measure theory (e.g.MATH 629or721) or concurrent registration inMATH 721is strongly recommended.

Mathematics course catalog