Graduate Courses

Below are extended descriptions of certain courses to help you plan your academic pathway in our graduate program.

6000

MATH-6100: Real Analysis I

A central part of this course is the study of measure theory, which generalizes the concept of length, area, and volume to more abstract spaces. This leads to the development of integral, in particular, the Lebesgue integral, a more powerful and versatile tool than the Riemann integral from undergraduate calculus. Measure theory is essential for many advanced fields like functional analysis, harmonic analysis and ergodic theory and the rigorous foundation of modern probability theory. The Lebesgue measure theory can be used to characterize when a bounded function on a closed interval [a,b] is Riemann integrable and to remedy several deficiencies of Riemann integration in Calculus such as that the Riemann integration does not handle functions with many discontinuities; the Riemann integration does not handle unbounded functions; and the Riemann integration does not work well with limits.

Topics include σ-algebras, measurable spaces, measure spaces, outer measures, Borel sets on R¹, Lebesgue sets, Lebesgue measure on R1, the Littlewood three principle: 1. Every measurable set is nearly a finite union of intervals; 2. Every measurable function is nearly continuous (Luzin’s theorem); 3. Every pointwise convergent sequence of measurable functions
is nearly uniformly convergent (Egorov’s theorem), approximation by simple functions, integration, three powerful convergence theorems: monotone convergence theorem, Fatou’s Lemma, and the dominated convergence theorem; Vitali covering lemma, Hardy-Littlewood maximal function, Lebesgue differentiation theorem, Lebesgue Density theorem, product measure, Monotone class theorem, Tonelli’s theorem, Fubini’s theorem, Borel sets in Rⁿ, Lebesgue measure and Lebesgue integral on Rⁿ.
MATH-6101: Real Analysis II

This course is introduction to functional analysis and Fourier analysis. Functional analysis is built on the foundation of measure theory to explore the study of infinte-dimensional vector spaces and the linear operators acting on them. The core idea is to generalize our intuition from finite-dimensional linear algebra to infinite-dimensional spaces of functions, equipping them with norms that allow us to talk about limits. Functional analysis takes messy, infinite-dimensional problems from analysis and provides a clean, geometric, and algebraic framework to solve them. It turns analysis (calculus, limits) into algebra (linear operators, spectra) and provides the rigorous foundation for modern physics and engineering.

Fourier analysis is a field that builds directly on the concepts of functional analysis and measure theory. It provides a language to decompose complex signals, functions, or data into fundamental, oscillatory components. It is used in modern science and engineering such as signal processing, differential equations, probability theory, crystallography and MRI.

Topics include metric spaces, Banach spaces, Hahn-Banach theorem, Baire’s theorem, uniform boundedness principle, open mapping theorem, closed graph theorem, L^p spaces, Hölder’s inequality, Minkowski’s inequality, Hilbert spaces, Pythagorean theorem, Cauchy-Schwartz inequality, Riesz representation theorem, Bessel’s inequality, Gram-Schmidt process, Real and Complex Measures (Hahn Decomposition Theorem, Jordan Decomposition Theorem, Lebesgue Decomposition Theorem, Radon Nikodym Theorem), bounded linear operators on a Hilbert space, spectrum of an operator, self-adjoint operator, normal operator, compact operator, Fredholm operator, spectral theorem for self-adjoint compact operators, spectral theorem for normal compact operators, singular values, singular value decomposition of a compact operator, Fourier Series, Poisson integral, Fourier coefficients, Riemann-Lebesgue lemma, harmonic function, convergence of Fourier series in L², Fourier transform, convolution, Poisson kernel, Fourier inversion formula, Plancherel’s Theorem.
MATH-6200: Topology I

This is the first semester of a year-long sequence covering the
foundations of topology. The first semester focuses on developing the
theory of topological and smooth manifolds. The course starts with a
review of basic point-set topology, such as continuity, convergent
sequences, the Hausdorff property, countability axioms, compactness, and
connectedness. We also review basic constructions like the subspace,
product and quotient topologies, with emphasis on the universal
properties defining these constructions. The course proceeds to more
advanced topics including topological groups, function spaces, the
Tychonoff theorem, paracompactness, and partitions of unity. Some
attention is paid to technical considerations that are essential to the
theory of manifolds, including second countability, local
path-connectedness, and local compactness. After discussing topological
manifold and manifolds with boundary and analyzing many examples, the
course transitions to the smooth setting and the theory of smooth
manifolds. Preliminary topics here include coordinate charts and smooth
atlases, smooth structures, smooth manifolds, and smooth partitions of
unity. The course concludes by studying the information encoded in
tangent spaces of smooth manifolds and the local behavior of maps; this
includes the definition of tangent vectors in terms of derivations,
vector fields, the tangent bundle, the differential of a smooth map,
submersions, immersions, the inverse function theorem, and the regular
value theorem.
MATH-6201: Topology II

This is the second part of a year-long sequence on the foundations of
topology. This semester focuses on algebraic topology, that is, on
developing algebraic invariants for topological spaces and understanding
the information encoded in them. The course begins with the concept of
homotopy, including contractibility, homotopy equivalence, deformation
retractions and the homotopy extension property, as well several basic
topological constructions such as the cone, suspension, join, wedge
sums, CW complexes, mapping cones, and mapping cylinders. With this
foundation in place, the course next introduces the fundamental group of
a space and its close relationship to covering spaces and the universal
cover. Topics here include the homotopy lifting property and the lifting
criterion, van Kampen’s theorem, deck transformations, covering group
actions, and the Galois correspondence between covers and subgroups of
the fundamental group. The next main goal is to develop the machinery of
homology. After introducing the intuitive idea and the notion of chain
complexes, the course dives into the details of simplicial and singular
homology; emphasis is placed on exact sequences and the long exact
sequence associated to a short exact sequence of chain complexes and
implications for relative homology and Mayer-Vietoris sequences. Other
topics include homotopy invariance, excision and barycentric
subdivision, local homology, degree theory, and cellular homology. The
course concludes with the dual theory of cohomology, which parallels
that of homology in many ways but has important differences. Key topics
here are the universal coefficient theorem, the cup product and
cohomology ring, and Poincaré duality.
MATH-6210: Differential Topology

The course begins with a brief review of smooth manifolds, cotangent spaces, and tangent spaces. The main topics discussed include vector bundles, with emphasis on tensor bundles and exterior algebra, differential forms, Lie derivatives, integration of forms, and de Rham cohomology. The course concludes with an introduction to characteristic classes.

Students are expected to have a background in smooth manifolds, as well as homology and cohomology, such as that provided by recent versions of Math 6200 and Math 6201. A solid background in multilinear algebra is also required.
MATH-6300: Algebra I

This is the first semester of a year-long course covering the foundations of abstract algebra. The first semester focuses on group theory, together with the fundamentals of ring and field theory needed for further study. After a brief review of prerequisite material from elementary group theory, including basic definitions and examples, Lagrange’s theorem, and the isomorphism theorems, the course treats group actions, Sylow’s theorems, and the structure theorem for finitely generated abelian groups. Further group-theoretic topics include nilpotent and solvable groups, free groups, and group presentations. The course concludes with a review of the basic theory of rings, integral domains, and fields needed for the second semester, including isomorphism theorems, prime and maximal ideals, polynomial rings, divisibility, irreducibility, and the Euclidean algorithm.
MATH-6301: Algebra II

This is the second part of a year-long course covering the foundations of abstract algebra. The course begins with field extensions and Galois theory, including algebraic extensions and algebraic closures, splitting fields, finite field theory, separability, normal extensions, the fundamental theorem of Galois theory, and applications to solvability of equations in radicals. It also treats more advanced topics in ring theory, including Euclidean rings, unique factorization domains, Noetherian rings, and the Hilbert basis theorem. The course then studies basic module theory, with emphasis on the structure theorem for finitely generated modules over principal ideal domains and its applications to rational and Jordan normal forms. The final part of the course introduces the language of associative algebras and establishes the foundations needed for representation theory; covered results include Frobenius’s classification of finite-dimensional real division algebras and the Artin-Wedderburn theorem.
MATH-6620: Linear Optimization

Linear optimization (also known as linear programming) involves minimization and maximization problems with a very simple structure, but it is of great practical and theoretical importance. In practice, linear optimization can be used to solve problems in operations research as well as discrete optimization problems that arise in computer science. From a theoretical perspective, linear optimization is important because it has a very strong duality theory with many nice consequences. Math 6620 covers the fundamental theory of linear optimization in conjunction with the development of a classic algorithm known as the Simplex Method. Applications to integer linear programming and discrete optimization are discussed, as well as newer algorithms such as the Ellipsoid Method and interior point methods.
MATH-6630: Nonlinear Optimization

Nonlinear optimization deals with general maximization or minimization problems, using ideas from both multivariable calculus and linear algebra. Math 6630 covers theory and techniques for both unconstrained and constrained optimization. The emphasis is on methods that are useful for solving real problems, although important theoretical ideas such as convexity and the Karush-Kuhn-Tucker conditions are also covered. The material includes methods for both one- and many-dimensional problems. Methods involving no derivatives, first derivatives (gradient-based methods, which are the foundation for many machine-learning algorithms) and higher derivatives are discussed.
MATH-6700: Combinatorics

Combinatorics deals with counting finite structures. A simple example is the theory of combinations and permutations. Math 6700 discusses several general techniques that can handle a wider variety of structures. Recurrence relations provide counting information for structures that can be described in a recursive way, and can be solved using techniques similar to those used for differential equations. Inclusion-exclusion techniques can count structures (such as objects with particular properties) where there is a partial order. Generating functions represent counting information as formal power series, which allows the use of techniques from algebra, calculus and complex analysis. Redfield-Polya theory applies the theory of group actions to count objects with symmetry.
MATH-6710: Graph Theory

Graphs are mathematical models of networks, which show up in many applications and also in many other areas of mathematics. Graphs consist of vertices (points or nodes) joined by edges (links). Math 6710 studies fundamental concepts for graphs such as traversability, flows, connectivity, colorings, and embeddability in surfaces (particularly the plane or sphere). Traversability problems involve issues such as efficiency (shortest paths) or coverage (hamilton cycles). Flow problems involve moving something through a network, such as fluids, vehicles, or information, and are related to connectivity: a highly connected graph is one in which something can be moved from one vertex to another along many disjoint paths. Coloring problems can be used to model real-world scheduling problems. Embeddability in the plane provides many nice properties that are stronger than those of general graphs.

7000

MATH-7100: Theory of Functions of a Complex Variable
MATH-7120: Functional Analysis

This is a standard second year graduate course that will start off with a quick
review of basic principles of functional analysis such as the Hahn-Banach
theorem, the open mapping theorem, the closed graph theorem and the uniform
boundedness principle.

The course will include a chapter on (locally convex) topological vector
spaces (i.e. infinite dimensional vector spaces provided with a topology
that makes the vector space operations continuous), including
separation versions of the Hahn-Banach theorem, weak topologies and Alaoglu’s
theorem. Weak topologies play an important role in PDE’s (weak solutions). If
time permits, there will be a section on fixed point theorems and the
Krein-Milman theorem.

In the next part of the course basic Hilbert space techniques will be
discussed. These techniques are important in many areas of mathematics,
physics and computer science such as quantum physics, quantum information
theory, operator algebras, (group) representation theory, harmonic analysis,
game theory, Fourier analysis, signal processing and wavelets. Topics include
bounded linear operators on Hilbert space, such as projections, (partial)
isometries and unitaries, the polar decomposition of operators, and basic
spectral theory. There will be a chapter on compact and Fredholm operators. If
there is enough time, the spectral theorem for normal operators on Hilbert
space will be presented. This leads to a powerful functional calculus for
normal operators, a very important tool in applications.

Depending on student interest, the course will take a few detours into
special topics such as quantum physics, operator algebras, group
approximation properties (e.g., amenability, property (T)), distribution
theory and weak solutions of PDEs, noncommutative measure theory &
noncommutative topology, harmonic analysis, or other topics.

Prerequisites: The equivalent of a first year graduate course in real
analysis (includes measure theory and integration, L^p spaces, Riesz
representation theorem etc.).
MATH-7130: Harmonic Analysis

The course begins with the development of the Fourier transform on the space of absolutely integrable functions on the real line, including the key operators of translation, modulation, dilation, and involution, and the Riemann–Lebesgue lemma. A substantial portion of the early part of the course is devoted to convolution and its many facets: Young’s inequality, convolution as filtering and averaging, smoothing and differentiation, and the Banach algebra structure of the absolutely integrable functions. The fundamental duality between smoothness and decay is developed carefully. Approximate identities are treated in detail, including their role in Lebesgue space approximation, pointwise and uniform convergence, and Gibbs’s phenomenon. The inversion formula is established via the Fejér kernel, and the section concludes with the Schwartz space and its topology.

The course then extends the Fourier transform to the broader setting of locally compact abelian (LCA) groups, including the notions of Haar measure, characters, and dual groups, illustrated through the real line, the torus, the integers, and the finite cyclic groups. Fourier series on the torus are developed in depth: partial sums and the Dirichlet and Fejér kernels, Cesàro summability, completeness and the square-integrable theory, the Weyl equidistribution theorem, basis properties of the trigonometric system, norm and pointwise convergence of Fourier series, the Poisson summation formula, and Wiener’s lemma.

In the next part of the course, the Fourier transform is extended to the Hilbert space of square-integrable functions on the real line. Topics include the Plancherel theorem, the Hermite functions and Wiener’s construction of the square-integrable Fourier transform, interpolation and the Hausdorff–Young inequality, and the failure of the Fourier transform to extend boundedly beyond the square-integrable setting. A major component is the theory of uncertainty principles: the classical Heisenberg uncertainty principle (proved via several methods), the Paley–Wiener theorem, the Donoho–Stark uncertainty principle, and energy concentration via time-frequency limiting operators and prolate spheroidal wave functions. These topics are important in signal processing, communications engineering, and compressed sensing.

The course continues with the theory of distributions (generalized functions), for which Laurent Schwartz was awarded the Fields Medal in 1950. The three principal classes of distributions are developed as dual spaces of appropriate test-function spaces. Topics include operations on distributions defined by duality, distributional differentiation, the distributional Fourier transform on tempered distributions, the Hilbert transform, and an introduction to Sobolev spaces and the Sobolev embedding theorem. The Hilbert transform is treated in some detail as a fundamental singular integral operator, with connections to complex analysis, analytic signals in signal processing, and boundary values of analytic functions. The course concludes with the Fourier–Stieltjes transform on the space of bounded Radon measures, including the Riesz representation theorem and applications to convolution and linear time-invariant systems.

Depending on student interest, the course will include special topics such as sampling theory, the wavelet transform and multiresolution analysis as an alternative to classical Fourier methods for time-frequency localization.

Prerequisites: The equivalent of a first-year graduate course in real analysis (includes measure theory and integration, Lebesgue spaces, Riesz representation theorem, etc.).
MATH-7200: Algebraic Topology

7200 (Algebraic Topology) focuses on homotopy theory. We begin with
higher homotopy groups and then cover a selection of topics, such as
stable homotopy theory, spectral sequences, or cohomology operations.

Students are expected to have a background in the fundamental group,
homology and cohomology from recent versions of Math 6201. A solid
background in algebra is also expected.
MATH-7210: Riemannian Geometry

The course is intended for graduate students with particular interests in geometry, as well as topology and mathematical physics. It covers fundamental topics in Riemannian geometry, including vector fields and their flows, connections, Riemannian metrics, geodesics and their properties, curvature and its properties, spaces of constant sectional curvature, and Jacobi fields. Both conceptual understanding and computational techniques will be emphasized.

Students are expected to have completed a course in differential topology, such as Math 6210.
MATH-7610: Methods of Mathematical Physics
MATH-7899: Selected Advanced Topics – Computing with Splines

The course deals with a powerful and relatively new class of approximating functions called Spline Functions which have many applications in Mathematics and Science and Engineering. These include data fitting and approximation, numerical quadrature, curve and surface design and representation, and the numerical solution of both ODE’s and PDE’s, including very recent developments in IGA. The purpose of this seminar is to investigate best available algorithms for solving problems numerically based on splines. There will be no written homework problems (and no exams), and students will not be doing proofs. However, they will be asked to carry out a number of programming exercises, using Mathlab. It should be of interest to anyone interested in applications of mathematics, and should be particularly valuable to students in all branches of science and engineering.

8000

MATH-8110: Theory of Partial Differential Equations

The course covers the basic mathematical theory of partial differential equations (PDEs), with a focus on techniques needed for the study of present-day problems in nonlinear PDEs. The guiding concepts are existence, uniqueness, and properties of solutions. The necessary background in function spaces is also developed. Linear and nonlinear equations arising in physics and geometry are considered.
MATH-8300: Combinatorial and Geometric Group Theory

The course introduces students to standard combinatorial and geometric methods in group theory and prepares them for further study and research in this area. The first part of the course focuses on the combinatorial approach to understanding groups through their presentations. Topics include free groups and the Nielsen-Schreier theorem, Tietze transformations, algorithmic problems in group theory, van Kampen diagrams, small cancellation theory, amalgamated free products, HNN extensions, and applications to embedding theorems and unsolvability results such as the Adian-Rabin theorem. The course then develops the foundations of geometric group theory, including Cayley graphs, quasi-isometries, group actions on metric spaces, and the Svarc-Milnor lemma. The combinatorial and geometric approaches come together in the study of asymptotic invariants of infinite groups, such as growth functions, Dehn functions, and subgroup distortion. The final part of the course introduces hyperbolic spaces and discusses basic algebraic, geometric, and algorithmic properties of hyperbolic groups. Throughout the course, emphasis is placed on both the standard theory and the broader research landscape, including recent developments and open problems.
MATH-8997: Advanced Independent Study

9000

Selected topics courses, varying by semester