Graduate Course Descriptions for Academic Year 2024-2025 Open configuration options Graduate Course Descriptions for Fall 2024 8000 Algebra- Introductory graduate level algebra (preparation for qualifying exams) 8100-Real Analysis I Introductory graduate level real analysis (preparation for qualifying exams) 8080-Lie Algebras Nilpotent and solvable Lie algebras, structure and classification of semisimple Lie algebras, roots, weights, finite-dimensional representations 8300-Introduction to Algebraic Geometry Algebraic geometry concerns the study the geometry of spaces given as zero sets of polynomials. It is one of the main branches of mathematics with strong connections to topology, algebra and number theory. This course will be an elementary introduction to the topic, following the book of Shafarevich Basic Algebraic Geometry 1. 8330-Topics in Algebraic Geometry (Algebraic Surfaces) Theory of algebraic surfaces is one of the most beautiful and useful topics in algebraic geometry, with applications to many (most?) fields of mathematics. A crowning achievement of the Italian school of the beginning of 20th century, it was critically reworked in the 1960s using modern cohomological methods. The theory includes many special surfaces i.e. rational, elliptic, abelian, K3, Enriques, Godeaux, Campedelli; Minimal Model Theory (whose higher-dimensional generalization has been the central topic in algebraic geometry for the last 30 years); singularities, and other topics. As the main text, I am going to use Beauville's book. 8130-Topics in Analysis (Continuous vs Discrete Analysis) Continuous vs discrete analysis • Introducing Christ’s dyadic cubes in a setup of general spaces of homogeneous type. Showing L ́epingle’s type inequalities for martingales, i.e. variational, jump and oscillation estimates for the dyadic martingale in the context of Christ’s dyadic cubes. • Establishing Calder ́on–Zygmund theory using the martingale approach. • Lp boundedness of the maximal operator based on continuous Radon average operators. • Almost orthogonality method(s) and its applications to harmonic analysis problems. Ap- plications: 1. Showing appropriate decay of the Fourier transform of the spherical surface measure. This allows to obtain sharp range of p’s for which the spherical maximal oper- ator is bounded on Lp(Rd). 2. Direct proof of L2 boundedness of the Hardy–Littlewood maximal operator. 3. Connection with the differencing method in Number Theory and proving Weyl’s type inequality, which is an important tool when proving asymptote formula in the Waring problem, see below. • Introducing the concept of major/minor arcs in the case of Waring problem. Proving Bourgain’s theorem, i.e. L2 boundedness of the maximal oparator based on discrete Radon average operators. If time permits we want to cover also the case of general Lp spaces, the latter requires introducing theory of Ionescu–Wainger projections. • Showing differences between analysis in the continuous and discrete worlds. In particular, showing sharp Lp bounds for the discrete spherical maximal operator and compare this range with the one in a continuous setup. Dimension-free estimates in analysis • Introducing a problem of dimension-free estimates for a general Hardy–Littlewood max- imal averaging operator related to a convex symmetric body in Rd. Connecting this problem with the isotropic constant conjecture. • Showing an appropriate bound for the Fourier transform of a general symmetric convex body. Proving Bourgain and Carbery result saying that for a general convex symmet- ric body the associated Hardy–Littlewood maximal averaging operator is bounded on Lp(Rd) for p > 3/2 with a constant independent of d and the related convex symmetric body. Discussing sharpness of their method. • Showing dimension-free estimates for all p > 1 in case of particular symmetric convex bodies. Here we are planning to look at lq balls, Bq = {(x1,...,xd) : |x1|q +...+|xd|q ≤ 1},1≤q≤∞. Thecaseofq<∞wasprovedbyMu ̈llerwhereasq=∞wasprovedby Bourgain. • Following Aldaz article we want to show that the constant in the corresponding weak type (1, 1) estimate is not dimension-free if we consider Hardy–Littlewood maximal operator based on cubes in Rd. We plan to give some positive results for the classical Hardy– Littlewood maximal operator restricted to radial functions. • Introducing analogous problem of dimension-free estimates for a general Hardy–Littlewood maximal operator in a discrete setup and discussing why this problem cannot be true in such generality. Proving some positive results in a discrete setup. 8770-Partial Differential Equations Classification of second order linear partial differential equations, modern treatment of characteristics, function spaces, Sobolev spaces, Fourier transform of generalized functions, generalized and classical solutions, initial and boundary value problems, eigenvalue problems. 8250-Differential Geometry I This is a first course on differential geometry and covers differentiable manifolds, vector bundles, tensors, flows, and Frobenius' theorem along with an introduction to Riemannian geometry at the end of the semester 8400-Number Theory I This is a first graduate course in algebraic number theory that uses graduate level algebra. The prerequisite is Math 8000; prior experience with or concurrent enrollment in Math 8020 (Commutative Algebra) is not required but will be helpful. We will develop the theory in the level of generality of extensions of Dedekind domains, drawing on the structure theory of finitely generated modules over a Dedekind domain. However a significant percentage of the course will be devoted to the "classical case" of rings of integers of number fields, including some discussion of Geometry of Numbers. We will also mention -- how much depends upon the interest of the participants -- the "geometric case" of affine coordinate rings of curves. The course will be taught from the following set of notes (possibly with some additions or modifications): http://alpha.math.uga.edu/~pete/ANT22.pdf 8230-Topics in Topology (Vector Bundles and Characteristic Classes) Vector bundles over topological spaces are an essential structure across many fields of geometry, algebraic geometry and topology, and characteristic classes are an essential tool for classifying and distinguishing vector bundles. This introduction to the subject will follow Milnor and Stasheff's book "Characteristic classes" and should be of interest to students across a range of fields. 8440-Topics in Number Theory (Arithmetic Statistics) In this class, we are going to cover a couple of main topics in arithmetic statistics. We will discuss conjectures/results/tools towards distribution of class groups and number fields. A tentative list of topics is as following:-Cohen Lenstra heuristics-Random Matrix-Malle's Conjecture-Counting abelian extensions-Geometry of Numbers-Genus theory-Nilpotent extensions-negative Pell equation?-Function Field analogue-homological stability Graduate Course Descriptions for Spring 2025 8150-Complex Analysis Introductory graduate level complex analysis (preparation for qualifying exams) 8200-Algebraic Topology Introductory graduate level algebraic topology (preparation for qualifying exams) 8030-Topics in Algebra (Category Theory for Representation Theorists) Category theory can be a useful framework all sorts of mathematics, but in modern representation theory (and related fields) it is an essential part of the language of the field. This course will be a tour of a number of categorical concepts coming from, or useful in, representation theory. Possible topics include: Morita equivalence, highest weight categories, tensor categories, FI-modules and representation stability, quantum knot invariants, categorification, triangulated categories, derived equivalences, Tannakian formalism. 8330-Topics in Algebraic Geometry (Hodge Theory) Hodge theory is a central area of algebraic geometry which is focused on the study of cohomology groups of complex algebraic varieties. Hodge theory plays an important role in the study of moduli problems and algebraic cycles in algebraic geometry, but also in representation theory, combinatorics, and number theory. In the first part of this course, we will cover the basics of Hodge theory, including the Hodge decomposition and the hard Lefschetz theorem. In the second part, we will cover more advanced topics, such as variations of Hodge structures, moduli spaces of abelian varieties, or applications to algebraic cycles, depending on the interest of the students. The main reference for the course will be the books “Hodge theory and complex algebraic geometry I-II” by Claire Voisin. 8170-Functional Analysis I Hilbert spaces and Banach spaces, spectral theory, topological vector spaces, convexity and its consequences, including the Krein-Milman theorem. 8630-Stochastic Analysis Conditional expectation, Brownian motion, semimartingales, stochastic calculus, stochastic differential equations, stochastic control, stochastic filtering. 8440-Combinatorial Number Theory The course will be an introduction to Ramsey Theory, with an aim to reach recent spectacular advances. 8230-Topics in Topology (Applied Topology, Topological Data Analysis) This course introduces topological techniques that offer new insights into data analysis, which has traditionally been grounded in statistical approaches. By leveraging these methods, we can analyze complex data structures in ways that conventional machine learning methods cannot. Starting with fundamental topological concepts, the course will focus on developing Persistent Homology (PH) - a key tool in topological data analysis (TDA). Then we will examine real-world applications of PH, such as in image recognition and cancer detection/genotyping etc. Topics covered include clustering, a visual introduction to topology, data modeling and visualization, as well as selected subjects such as nonlinear dimensionality reduction, graph-based data models, Reeb graphs, multi-scale data approaches, and persistent homology. This course is a great way to see how advanced mathematics is applied in solving real life problems 8030-Homological Algebra This is a course taught at UGA every other year, and intended to serve the needs of graduate students interested in Algebraic Topology, Algebraic Geometry, Algebra, and Number Theory. I plan to teach a standard course on the topic, and one of the recommended textbooks for it will be J. Rotman, An Introduction to Homological Algebra, Springer, 2008. The prerequisites for the course includes a graduate course in Abstract Algebra. Beyond that, I will try to tailor the course to the participants. 8230-Topics in Geometry (groups acting on hyperbolic spaces) First part of the course: PSL(2,R) and its Discrete groups: classification of transformations, the hyperbolic metric, Fuchsian groups and their fundamental domains, closed Riemann surfaces and their automorphism groups, automorphic functions and uniformization. Second part of the course: This pretty much depends on the audience background and interest. My plan is to either continue with: i- The modular group: Lattices, tori and moduli, the discriminant of a cubic polynomial, the modular group, the modular function, the Riemann surface of $\sqrt{p(z)}$, p(z) a cubic polynomial, quotient surfaces for the modular group. or ii-groups acting on Trees, where my main goal is to explain the structure of group acting on a tree (being a discrete counterpart of hyperbolic space) as a powerful interplay between geometry and geometric group theory.
