Spring 2018 Graduate Courses

The following is a tentative schedule of graduate courses for spring 2018 with course descriptions included below the table. Minor scheduling changes may occur before the start of the term. If you have questions or concerns, please send them to Michelle Klinger.

Course Descriptions

MATH 6120 - Complex Analysis

Prerequisite: Strong performance in an undergraduate analysis course at the level of MATH 4140, or permission of instructor.

MATH 6110-6120 are the core analysis courses in the mathematics graduate program. MATH 6120 covers complex analysis, Fourier analysis, and distribution theory.

MATH 6160 - Partial Differential Equations

Prerequisite: MATH 4130, MATH 4140, or the equivalent, or permission of instructor. Offered alternate years.

This course highlights applications of functional analysis to the theory of partial differential equations (PDEs). It covers parts of the basic theory of linear (elliptic and evolutionary) PDEs, including Sobolev spaces, existence and uniqueness of solutions, interior and boundary regularity, maximum principles, and eigenvalue problems. Additional topics may include: an introduction to variational problems, Hamilton-Jacobi equations, and other modern techniques for non-linear PDEs.

MATH 6270 - Applied Dynamical Systems

(also MAE 7760)

Prerequisite: MAE 6750, MATH 6260 (formerly MATH 6170), or equivalent.

Topics include review of planar (single-degree-of-freedom) systems; local and global analysis; structural stability and bifurcations in planar systems; center manifolds and normal forms; the averaging theorem and perturbation methods; Melnikov’s method; discrete dynamical systems, maps and difference equations, homoclinic and heteroclinic motions, the Smale Horseshoe and other complex invariant sets; global bifurcations, strange attractors, and chaos in free and forced oscillator equations; and applications to problems in solid and fluid mechanics.

MATH 6320 - Algebra

Prerequisite: MATH 6310, or permission of instructor.

MATH 6310-6320 are the core algebra courses in the mathematics graduate program. MATH 6320 covers Galois theory, representation theory of finite groups, introduction to homological algebra.

MATH 6350 - Homological Algebra

Prerequisite: MATH 6310

A first course on homological algebra. Topics will include a brief introduction to categories and functors, chain and cochain complexes, operations on complexes, (co)homology, standard resolutions (injective, projective, flat), classical derived functors, Tor and Ext, Yoneda’s interpretation of Ext, homological dimension, rings of small dimensions, introduction to group cohomology.

MATH 6370 - Algebraic Number Theory

Prerequisite: an advanced course in abstract algebra at the level of MATH 4340

An introduction to number theory focusing on the algebraic theory. Topics include, but are not limited to, number fields, Dedekind domains, class groups, Dirichlet's unit theorem, local fields, ramification, decomposition and inertia groups, and the distribution of primes.

MATH 6410 - Enumerative Combinatorics

An introduction to enumerative combinatorics from an algebraic, geometric and topological point of view. Topics include, but are not limited to, permutation statistics, partitions, generating functions, various types of posets and lattices (distributive, geometric, and Eulerian), Möbius inversion, face numbers, shellability, and relations to the Stanley-Reisner ring.

MATH 6510 - Algebraic Topology

Prerequisite: strong performance in an undergraduate abstract algebra course at the level of MATH 4340 and point-set topology at the level of MATH 4530, or permission of instructor.

MATH 6510–MATH 6520 are the core topology courses in the mathematics graduate program. MATH 6510 is an introductory study of certain geometric processes for associating algebraic objects such as groups to topological spaces. The most important of these are homology groups and homotopy groups, especially the first homotopy group or fundamental group, with the related notions of covering spaces and group actions. The development of homology theory focuses on verification of the Eilenberg-Steenrod axioms and on effective methods of calculation such as simplicial and cellular homology and Mayer-Vietoris sequences. If time permits, the cohomology ring of a space may be introduced.

MATH 6630 - Symplectic Geometry

Prerequisite: MATH 6510 and MATH 6520, or permission of instructor.

Symplectic geometry is a branch of differential geometry which studies manifolds endowed with a nondegenerate closed 2-form. The field originated as the mathematics of classical (Hamiltonian) mechanics and it has connections to (at least!) complex geometry, algebraic geometry, representation theory, and mathematical physics. In this introduction to symplectic geometry, the class will begin with linear symplectic geometry, discuss canonical local forms (Darboux-type theorems), and examine related geometric structures including almost complex structures and Kähler metrics. Further topics may include symplectic and Hamiltonian group actions, the topology and geometry of momentum maps, toric symplectic manifolds, symplectic blowups, and pseudo-holomorphic curves.

MATH 6720 - Probability Theory II

Prerequisite: MATH 6710.

Conditional expectation, martingales, Brownian motion. Other topics such as Markov chains, ergodic theory, and stochastic calculus depending on time and interests of the instructor.

MATH 6830 - Model Theory

This course will given an introduction to model theory, taught from a more algebraic perspective.  We will start by reviewing the compactness and completeness theorem.  The goal of the course is to prove Morley's Categoricity Theorem, which characterizes when the isomorphism type of an uncountable model of a given theory is determined by its cardinality.  Along the way, we will develop several concepts and tools which are important in their own right: the Lowenheim-Skolem theorem, the Vaught's test for the completeness of a first order theory, quantifier elimination, back and forth arguments, quantifier elimination, stability, and indiscernability.

Students should ideally have had some exposure to predicate logic (structures, satisfaction, proof, the completeness theorem), although a properly motivated student can pick up the relevant knowledge as it is reviewed.

MATH 7120 - Topics in Analysis: Einstein 4-Manifolds and Ricci Flow on 4-Manifolds

The first part of this course will discuss positive Einstein 4-manifolds and classifications. The second part of this course will discuss current development of the Ricci flow on 4-manifolds. Graduate student participation will be expected for the second part.

MATH 7130 - Functional Analysis

The order may change, but the list of topics planned to be covered include: The Hahn-Banach theorem and applications, topological vector spaces, Alaoglu-Bourbaki theorem, Kakutani-Markov fixed point theorem, Haar measure for compact groups, Krein-Millman theorem, compact operators and Hilbert-Schmidt operators, spectral theorem for compact normal operators, Peter-Weyl theorem for compact groups, general spectral theory, mean ergodic theorem, distributions, Hilbert transform, fundamental solutions to general PDE with constant coefficients, parametrices and regularity for elliptic equations, the Baire Category theorem and applications (uniform boundedness principle, open mapping theorem, closed graph theorem). This whole plan may already sound a bit too ambitious, but if the time will permit, we could also talk about $L^p$ spaces in harmonic analysis (interpolation theory, Hardy space, BMO), rudiments of probability theory (central limit theorem, random Fourier series, random walks), introduction to Brownian motion and applications to the Dirichlet problem.

As usual, I will do my best to keep prerequisites to a minimum. In particular, the class should be accessible to advanced undergraduate students.

MATH 7390 - Topics in Lie Groups and Lie Algebras: An Introduction to Affine Grassmannians and their Applications

Prerequisite: a one-semester course in Lie groups

Affine Grassmannians are infinite-dimensional spaces that are central objects in geometric representation theory and in the study of moduli spaces of G-bundles on algebraic curves. They have also come to play an increasingly important role in number theory. In this course, we will study their basic geometric properties and discuss some applications.

MATH 7520 - Berstein Seminar in Topology: An Introduction to Geometric Group Theory

A seminar on an advanced topic in topology or a related subject. The format is usually that the participants take turns to present.

Topic: Geometric Group Theory is the interface between algebra, geometry, and topology in the context of infinite discrete groups. This seminar will be an introduction to some of its central themes, examples, techniques, and questions.

MATH 7720 - Topics in Stochastic Processes: Random Walks on Amenable Groups

This course will be an introduction to random walks on finitely generated amenable groups.  The emphasize will be on non-abelian groups.

In the first part of the course, we will discuss how the notion of volume growth impact problems such as isoperimetry and probability of return to the starting point. We will define the notions of spectral profile and isoperimetric profile and relate these notions to certain functional inequalities.

In the second part of the course, we will explore classes of groups for which the volume growth based results are sharp and discuss specific examples of groups for which these results are not sharp.

MATH 7820 - Seminar in Logic

A twice weekly seminar in logic. Typically, a topic is selected for each semester, and at least half of the meetings of the course are devoted to this topic with presentations primarily by students. Opportunities are also provided for students and others to present their own work and other topics of interest.