Graduate Course Descriptions for Fall 2023 MATH8330 Topics in Algebraic Geometry (Arguz): Toric Geometry and Cluster Varieties Toric varieties are highly symmetric spaces which are extensively studied in algebraic geometry, symplectic geometry, and topology. The study of toric varieties exhibits a remarkable interplay between geometry and combinatorics, making the subject accessible for numerous explicit computations. The first part of this course will be an introduction to toric varieties. The second part of the course will be about generalizations of toric varieties known as cluster varieties, obtained by gluing tori together. The study of cluster varieties is a very active domain of research motivated by algebra and representation theory. In this part, we will particularly cover the recent work of Gross-Hacking-Keel-Kontsevich motivated by mirror symmetry. This work, roughly put, generalizes the combinatorial objects appearing in the study of toric varieties, to the set-up of cluster varieties. MATH8230 Topics in Topology and Geometry (Gay): Morse Theory This course will be an introduction to finite dimensional Morse theory, as a way to understand the topology of smooth manifolds. We will particularly focus on using Morse theory to get handle decompositions and surgery descriptions of manifolds and learn about how to manipulate such handle decompositions. Most of our examples will come from low dimensions, and along the way we will learn the basics of Kirby calculus. MATH8110 Real Analysis II (Lyall): A second course in analysis and beyond (ultimately covering very different topics to those taught in Fall 2023 some more basic and some more specialized): Representation theorems and basic functional analysis, Basic theory of Fourier series, the Fourier transform and Oscillatory integrals, basic Geometric Measure Theory, and an introduction to Geometric Ramsey theory. MATH8190 Lie Groups (Nakano): (The on-the-books course title is "Lie Groups" but the course proposal I have calls it "Algebraic Groups".) The course will cover the basics of algebraic groups including the structure and representation theory for reductive groups. Depending on the clientele, the course will cover topics that relate algebraic group representations to other areas of algebra, algebraic geometry and perhaps number theory. MATH8440 Comb/anly Number Th (Pollack): Actual title: Topics in Analytic Number Theory This is intended as a seminar course, with lectures to be given both by the instructor and the student participants. The goal is to collectively work through important papers that develop and/or utilize techniques of analytic number theory (including sieve methods, anatomy of integers, probabilistic number theory, and the theory of multiplicative functions). MATH8430 Arithmetic Geometry (Lorenzini) Actual Title: Elliptic Curves Elliptic curves (entry point tailored to suit the audience) Graduate Courses for Spring 2024 8030 Topics in Algebra (Nilpotent orbits and geometric representation theory), Bill Graham This course will discuss how geometric methods play a role in understanding questions in representation theory related to reductive algebraic groups (such as the general linear group). The nilpotent cone of this group plays a key role in this connection, and also provides a concrete and interesting setting to study varieties with group actions and the role played by resolutions of singularities. Sources will include the book by Collingwood and McGovern on nilpotent orbits, as well as Jantzen's paper on the topic. 8230 Topics in Topology and Geometry (h-principles), Eduardo Fernandez Fuertes In 1958 Stephen Smale discovered that it is possible to turn a sphere inside out in the 3 dimensional euclidean space, creating self-intersections but without creating corners. This is quite a remarkable and non intuitive fact. The idea of Smale was to reduce the problem of understanding spheres without corners, a condition defined by a differential inequality and, therefore geometric a priori; into a purely algebro-topological problem. Around ten years later Mikhael Gromov realized that the construction of Smale fits in a more general framework, leading him to the introduction of his famous theory of h-principles. Here the h stands for homotopy. Shortly, for an a priori geometric problem, i.e. defined by some partial differential relation, it is said that the h-principle holds if it can be understood in purely topological terms. In this case it is said that the problem is flexible, and in the opposite case it is said to be rigid. It turned out that there are many a priori geometric problems that are governed by an h-principle and therefore are flexible, making Gromov's h-principle a groundbreaking event in the mathematics of the 20th Century. In this course we will give an introduction to the h-principle and explain general methods to prove the existence of h-principles. We will illustrate the ideas through several examples, many of them from Symplectic and Contact Topology in which the interaction between flexible and rigid problems is prominent and an object of research nowadays. Anyone with a background in the theory of Differentiable Manifolds should be able to follow this course; a basic knowledge of Symplectic and Contact Topology would be useful but not strictly necessary. 8230 Topics in Topology and Geometry (Symplectic Geometry), Mike Usher This will be a general introduction to symplectic geometry, aimed at students with some familiarity with smooth manifolds. (MATH 8210 or 8250 would be more than enough background.) Topics will include symplectic linear algebra, various kinds of submanifolds of symplectic manifolds, almost complex structures, Hamiltonian flows, and some relations to Kahler and contact geometry. 8310 Geometry of Schemes, Dino Lorenzini This is a standard basic course on the language of Grothendieck's theory of schemes. Topics include the spectrum of a ring, "gluing" spectra to form schemes, quasi-coherent sheaves, properties of morphisms and base change, and the functor of points. Motivations coming from both geometry and arithmetic will be discussed. 8330 Topics in Algebraic Geometry (Enumerative Geometry), Pierrick Bousseau "Enumerative geometry is a branch of algebraic geometry, with strong connections to symplectic geometry and topology. It is mainly concerned with counting geometric objects in an algebraic variety X, that satisfy a given list of constraints. By imposing the right amount of constraints, in good situations, one obtains a number of such objects which does not change under generically varying the set of constraints. Such a number is called an enumerative invariant of X. In this course, we will give an introduction to two examples of enumerative invariants which are the subject of active current research: Gromov--Witten invariants, which are counts of curves, and Donaldson Thomas (DT) invariants, which are counts of sheaves. In the first part of this course, we will cover fundamental tools in algebraic geometry and topology which were historically motivated by enumerative geometry, such as the intersection theory of algebraic cycles and the theory of characteristic classes. In the second and more advanced part of the course, we will give an introduction to the two main modern incarnations of enumerative geometry; the theory of Gromov-Witten invariants and DT invariants. We will then explain the Gromov--Witten DT correspondence, which expresses the Gromov–Witten invariants in terms of DT invariants. We will also describe how these invariants are related to invariants appearing in symplectic geometry, and theoretical physics." 8440 Topics in Combinatorial/Analytic Number Theory (Circle Methods), Akos Magyar 8430 Topics in Arithmetic Geometry (Arithmetic Statistics), Jiuya Wang 8550 Topics in Numerical Analysis (Finite Element Methods), Lin Mu Finite element method and discontinuous Galerkin methods for numerical solution of partial differential equations 8710 Variational Methods/Perturbation Theory, Weiwei Hu Variational Methods and PDE Control
