Non-Qual Graduate Course Descriptions for Fall 2020 Math 8000 - Algebra I (Rider) This is the standard Qualifying Exam Course in Algebra, see department course descriptions. Math 8100 - Real Analysis I (Magyar) This is the standard Qualifying Exam Course in Real Analysis, see department course descriptions. Math 8190 - Algebraic Groups (Nakano) A major problem in modular representation theory has been the determination of a character formula and the dimensions of finite dimensional simple modules over a connected reductive algebraic group. Lusztig conjectured that the character could be computed recursively in terms of ``Kazhdan-Lusztig'' polynomials. These polynomials arise from the Hecke algebra associated with the Weyl group of the given algebraic group. One of the goals of this course is to intoduce the terminology and ideas used in formulating the Lusztig conjecture (which has been proved for p very large). For this purpose most of the material will be taken from Jantzen's ``Representations of Algebraic Groups''. The possible topics include: Kempf's vanishing theorem, Bott-Borel-Weil theorem, Weyl's Character Theorem, representations of Frobenius kernels $G_{r}$ and related module categories $G_{r}T$ and $G_{r}B$, Strong Linkage Principle, injective $G_{r}$ modules, Steinberg's twisted tensor product formula, and the Translation Principle. Other related topics might also include cohomology for infinitesimal Frobenius kernels and Friedlander and Parshall's support varieties for restricted Lie algebras. In this course we will use an interesting mix of tools from both algebra and algebraic geometry. Math 8210 - Topology of Manifolds (Gay) This course serves jointly as an introduction to smooth manifolds and as a second course in algebraic topology (covering various versions of cohomology and Poincare duality, among other fundamental topics). It covers material important for early-to-middle-stage students in both geometry and topology, and the topology group accordingly tries to run it every year. Math 8230 - Four-manifolds and Kirby calculus (K. Larson) Smooth 4-manifolds exhibit properties reminiscent of both high- and low-dimensional manifolds (and indeed there are certain features unique to dimension 4). Like manifolds in higher dimensions, there exist manifolds with exotic smooth structures in dimension 4 (this follows from work of Donaldson and Freedman, each separately winning a Fields medal for their contributions). On the otherhand, as with lower dimensions we are able to concretely visualize handle decompositions for 4-manifolds and diffeomorphisms between 4-manifolds. The techniques used to do this are often referred to collectively as Kirby calculus, and they provide a powerful tool for studying 4-manifolds as well as the 3-manifolds arising as their boundaries. This course will give a general introduction to 4-manifold theory, provide a detailed description of Kirby calculus, and then use it to discuss a variety of applications in 3-manifold and 4-manifold topology. Math 8300 - Introduction to Algebraic Geometry (Engel) An invitation to algebraic geometry through a study of examples. Affine and projective varieties, regular and rational maps, Nullstellensatz. Veronese and Segre varieties, Grassmannians, algebraic groups, quadrics. Smoothness and tangent spaces, singularities and tangent cones. Math 8320 - Algebraic Curves (Clark) This course will concern the geometry of algebraic curves over an arbitrary ground field k. The foundational perspective will be that of valuations on an algebraic function field in one variable, as is done for instance in Stichtenoth's GTM "Algebraic Function Fields and Codes." In particular we will prove the Riemann-Roch theorem using Weil differentials. We should be able to treat some non-foundational aspects of algebraic curves, e.g.: hyperelliptic and superelliptic curves, automorphism groups, automorphism groups, Weierstrass points, Artin-Schreier coverings, and/or Hasse-Weil zeta functions. This course should be useful and interesting for students of algebraic geometry, arithmetic geometry and algebraic number theory (for the latter group, it is a nice chance to see algebraic methods applied in a geometric context). Math 8430 - Etale Fundamental Groups and Etale Cohomology (Litt) The proof of the Weil conjectures by Dwork, Grothendieck, and Deligne was one of the greatest achievements of 20th century mathematics, and heralded the import of tools from algebraic topology into arithmetic geometry. The goal of this course will be to develop those tools (in particular, the etale fundamental group and etale cohomology) and work towards their applications, ultimately proving (part of) the Weil conjectures. We will also discuss other arithmetic and algebro-geometric applications of these tools. Math 8450 - Cryptography (Lorenzini) We will discuss the main public key protocols and signature schemes currently in use, such as RSA, Diffie-Hellman, and ElGamal, as well as known attacks on them. We will also look at quantum resistant protocols, including NTRU and recent schemes based on lattice techniques. Math 8510 - Numerical Analysis II (M-J Lai) We will discuss triangulation of points in 2D setting as well as triangulation of a polygonal domain. Then we explain how to define bivariate splines over triangulation, in particular, how to evaluate, how to take derivative, how to integrate a bivariate spline function. Interpolation and approximation will be explained. MATLAB programing will be shown. Next we will explain tetrahedral partition of points in 3D and tetrahedral partition of polyhedral domain in 3D. Then we explain trivariate splines. Approximation properties will be discussed. This class does not need any prerequisite. Math 8850 - Harmonic Analysis (Lyall) This will be a seminar course serving as an introduction to the basic tools, results and conjectures in Harmonic Analysis. Non-Qual Graduate Course Descriptions for Spring 2021 Math 8030 - Homological Algebra (Boe or Rider) This is a standard course, see department course descriptions. Math 8030 - Quiver Representations (Wilbert) This course provides an introduction to the representation theory of finite-dimensional, associative algebras over algebraically closed fields. The general goal is to classify the finitely-generated modules over a given such algebra and understand the homomorphisms between them. It turns out that this classification problem can be translated into a problem about representations of quivers. A quiver is a finite, oriented graph. A representation of a quiver consists of a collection of finite-dimensional vector spaces (one for each vertex) together with a bunch of linear maps (one for each arrow). By using quivers, one can thus turn the problem of classifying modules over an associative algebra into a hands-on linear algebra problem amenable to explicit computations. The first goal of the course will be to give a detailed proof of Gabriel’s theorem, which classifies all quivers with only finitely many indecomposable representations (up to isomorphism). Indecomposable representations can be thought of as the basic building blocks of all representations in the sense that any given representation decomposes as a direct sum of indecomposable representations. We will then cover the basics of Auslander-Reiten theory and discuss the so-called “knitting algorithm” which can be used to iteratively construct the representation category of a given quiver in many interesting cases. If time permits, we will discuss Kac’s theorem (a generalization of Gabriel’s theorem) and establish connections to Ringel-Hall algebras and (quantum groups of) Kac-Moody algebras. Math 8150 - Complex Analysis (S. Wang) This is the standard Qualifying Exam Course in Complex Analysis, see department course descriptions. Math 8200 - Algebraic Topology (Lambert-Cole) This is the standard Qualifying Exam Course in Topology, see department course descriptions. Math 8230 - Heegaard Floer homology (Alishahi) This course is an introduction to Heegaard Floer homology, as defined by Peter Ozsváth and Zoltan Szabó around 2000. Heegaard Floer homology is a package of powerful invariants of smooth 3- and 4-manifolds, knots/links and contact structures. Over the last 19 years, it has become a central tool in low-dimensional topology. It has been used extensively to study and resolve important questions concerning unknotting number, slice genus, knot concordance, Dehn surgery. It has been employed in critical ways to study taut foliations, contact structures, smooth 4-manifolds. There are also many rich connections between Heegaard Floer homology and other manifold and knot invariants coming from gauge theory as well as representation theory. We will learn the basic construction of Heegaard Floer homology, starting with the definition of the 3-manifold invariant. In the second half of this course, we will turn to applications of the theory to low-dimensional topology. Math 8330 - Birational geometry, Minimal model program, and Moduli (Alexeev) Birational geometry and Minimal model program are some of the most important fields of algebraic geometry, noted by two Fields Medals (last in 2018) and a Breakthrough prize. Note also the 2019 semester on Birational geometry and Moduli spaces at the MSRI. I will give a general introduction to this field using Kollár-Mori's book and include some of the latest applications to the KSBA moduli spaces generalizing Deligne-Mumford's moduli of stable curves. Math 8330 - Algebraic surfaces and 4-manifolds (Engel) This course will overview the theory of algebraic surfaces (two complex-dimensional manifolds of solutions to polynomial equations), developed in the early 20th century by the Italian school of geometry. We will also explore connections to the theory of 4-manifolds. Possible topics include the classification of algebraic surfaces following Beauville's book, and the intricate relationships between homeomorphism, diffeomorphism, and algebraic deformation equivalence. Math 8400 - Algebraic Number Theory I (Pollack) MATH 8400 is a first course in algebraic number theory. Our focus will be on establishing the three "fundamental theorems" for rings of integers of number fields: unique factorization into ideals, finiteness of the class group, and Dirichlet's description of the unit group. The tools you will be introduced to in this course are part of the standard tool chest of every working number theorist. As such, this course is strongly recommended for anyone pursuing further studies in number theory. The primary course text will be the book "A conversational introduction to algebraic number theory", by the instructor. This will be supplemented by material from Pierre Samuel's book "Algebraic theory of numbers". Math 8430 - Class Field Theory and Galois Cohomology (Clark) This course will concern local and global class field theory from the cohomological perspective, and it will also contain some development of Galois cohomology for its own sake. A reasonable reference for the class field theory portion is James Milne's lecture notes (available online). This course should be accessible to those who have taken Number Theory I and II. Math 8550 - Numerical PDE (Mu) Numerical Solution to partial differential equations. Several numerical methods including finite elements method, discontinuous Galerkin method, weak Galerkin method and spline method will be explained. Poisson equations and other linear and nonlinear PDE will be solved. Math 8710 - Calculus of Variations (Hu) Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals. Many important problems arise in this way across pure and applied mathematics and physics. They range from the problem in geometry of finding the minimal surfaces of revolution to finding the configuration of a piece of elastic that minimises its energy. As time allows, topics will include Euler-Lagrange conditions, Hamilton-Jacobi equations, Pontryagin maximum principle, and an introduction to control theory in the context of calculus of variations. Students are assumed to have some background in elementary differential equations and analysis. Math 8850 - Harmonic Analysis (Lyall) This will be a seminar course serving as an introductions to the basic tool, results and conjectures in Harmonic Analysis.
