Graduate Course Descriptions for Fall 2021 Math 8000 - Algebra I (Rider) This is the standard Qualifying Exam Course in Algebra, see department course descriptions. Math 8030 - Flag varieties, equivariant cohomology, and K-theory (S. Larson) This course will cover flag varieties and representation theory, focusing on applications of equivariant cohomology and K-theory. Topics include Demazure and Weyl character formulas, the Borel-Weil-Bott theorem, T-equivariant cohomology of flag varieties, and geometry of Schubert varieties. Basic understanding of finite dimensional Lie algebras and algebraic groups will be helpful. Math 8100 - Real Analysis I (Lyall) This is the standard Qualifying Exam Course in Real Analysis, see department course descriptions. Math 8230 - Characteristic Classes (Alishahi) This course covers the Stiefel-Whitney, Chern, and Pontrjagin classes, following the classic book of Milnor and Stasheff on Characteristic Classes and chapters 1 and 3 of Hatcher's book on Vector bundles and K-theory. Math 8310 - Schemes (Engel) Schemes were developed by Grothendieck in 1960. Roughly, a scheme is a way to "glue" rings together. As an analogy, the ring of continuous functions on a circle can be glued from the rings of continuous functions on two intervals covering the circle. The introduction of schemes revolutionized mathematics, providing a uniform language to approach problems in both number theory and algebraic geometry. For instance, the theory of schemes was the necessary foundation for Deligne to prove the Weil conjectures. This famously led to a proof of the Ramanujan conjecture, whose statement is largely unrelated to schemes! Topics include the spectrum of a ring, gluing spectra to form schemes, products, quasi-coherent sheaves of ideals, and the functor of points. Math 8430 - Rational points on algebraic varieties (Litt) This course will function as an introduction to the modern theory of diophantine equations, loosely following Bjorn Poonen's book "Rational points on varieties." We will try to understand local-global principles and explanations for their failure (e.g. the Brauer-Manin obstruction and its generalizations), as well as methods for finding rational points or proving their non-existence. Ideally the content of the course will be dictated by the interests of enrolled students. Math 8440 – Analytic number theory (Magyar) The aim of this course is to introduce analytic methods to study integer solutions of Diophantine equations, that is of systems of integral polynomial equations. We start by developing the classical circle method of Hardy-Littlewood and Hua for the Waring problem, on the number of representations of a large number as a sum of k-th powers. We also plan to discuss modern developments such as Wooley’s efficient congruencing method related to Vinogradov’s mean value theorem. We will also study solutions to systems of non-diagonal Diaphantine equations, in particular we develop the Birch-Davenport version of the circle method, as well as- time permitting- discuss some recent improvements. Math 8500 - Numerical Analysis I (Mu) This is the standard Qualifying Exam Course in Numerical Analysis, see department course descriptions. Math 8630 - Stochastic Analysis (Q. Zhang) A probabilistic approach to data analysis. Math 8770 - Partial Differential Equations (Tie) The theory of partial differential equations (PDE) forms today a vast branch of mathematics and mathematical physics that uses methods of all the remaining parts of mathematics (from which the PDE theory is inseparable). In turn, PDE influence numerous parts of mathematics, physics, chemistry, biology and other sciences. Here are the topics that I will cover: 1. Linear differential operators 2. Wave Equations 3. Heat Equations 4. Laplace Equations 5. Distributions, convolution and Fourier transform 6. Sobolev Spaces 7. Heisenberg Groups and sub elliptic equations Graduate Course Descriptions for Spring 2022 Math 8020 - Commutative Algebra (Litt) This course is taught every other year, see department course descriptions. Math 8130 - Cohomology and Representation Theory (Nakano) The course will be aimed at introducing students to homological methods in the representation theory of groups and algebras. Information about the extensions between representations (modules) enables one to understand how representations are constructed. Calculations involving cohomology groups are essential for determining how representations are “glued” together. Over the past 40 years, it has been useful to introduce geometric methods into the subject. The cohomology rings involved are often finitely generated algebras, and this allows one to define an affine algebraic variety associated to each representation. These varieties are called support varieties. Computation of support varieties has also been an important ingredient in our understanding of the representation theory. The ultimate goal of the course is to reach the modern interfaces between representation theory, geometry and homological algebra. Collecting all the representations for a certain algebraic object forms a tensor triangulated category. Ideas from homological algebra via the use of support theories can be used to build a bridge between tensor triangulated categories and geometric objects. Uncovering this “hidden geometry” often leads to new and beautiful insights about the algebraic object and its representations. The basic material of the course will be from D.J. Benson, Representations and Cohomology, Volumes I and II, Cambridge University Press, 1991. The prerequisite is Math 8000 (graduate algebra). Math 8150 - Complex Analysis (S. Wang) This is the standard Qualifying Exam Course in Complex Analysis, see department course descriptions. Math 8170 - Functional Analysis (Hu) Introduction to Hilbert spaces and Banach spaces, topological vector spaces, duality theory, spectral theory, bounded and bounded operators, and related applications in optimization and PDEs. For example, Variational Formulation of Boundary Value Problems, etc. I may also introduce Lp and Sobolev Spaces as needed. Text: Reed and Simon's book on Functional Analysis (Vol.1) together with Brezis's book on Functional Analysis, Sobolev Spaces and PDEs. Math 8200 - Algebraic Topology (Gay) This is the standard Qualifying Exam Course in Topology, see department course descriptions. Math 8230 - Symplectic Geometry (Wu) We will introduce basic concepts in symplectic geometry, including the concept of symplectic manifolds, Lagrangian submanifolds, the Moser method, neighborhood theorems, Hamiltonian actions and reductions. Time permitted, we will discuss holomorphic curves in symplectic geometry and some of their applications. Math 8230 - Contact Topology (Lambert-Cole) Contact geometry first appeared several centuries ago in the work of Huygens and others on geometric optics. Today, it is an active subject of research with profound implications for low-dimensional topology, symplectic topology, singularity theory and dynamics. In Arnold's words: "contact geometry is all geometry". This course will be an introduction to contact structures and related geometric objects, such as transverse and Legendrian knots, open book decompositions and symplectic fillings. We will also discuss applications to important problems in low-dimensional topology, such as Cerf's Theorem; Property P; and the Thom conjecture. Math 8315 - Sheaves and cohomology / Derived categories (Alexeev) Standard course, see department course descriptions. Math 8440 - Additive Combinatorics (Petridis) The course will develop the combinatorial basics in additive combinatorics and will also introduce basic tools from combinatorial geometry. Topics will include: sumset inequalities (Ruzsa calculus), the Balog-Szemeredi-Gowers theorem, Freiman's theorem, the Szemeredi-Trotter theorem, the sum-product phenomenon, Szemeredi's regularity lemma.
