Research Seminar in Mathematics
Fall 2023
All talks will take place in 509/511 Lake Hall and follow the schedule:
12:00-1:00 - First half of talk
1:00-1:30 - Break
1:30-2:30 - Second half of talk
When they're available, abstracts, slides, and notes can be found by clicking on the entries.
Braid groups, differential equations and quantum groups
Abstract: Braids on a given number of strands n can be concatenated and thereby form a group Bn. This group possesses two different incarnations: the first is algebraic, and consists in a presentation on a simple set of generators and relations due to E. Artin (1947), the second is topological, and realises Bn as the fundamental group of the space of configurations Xn of n points in the complex plane. I will explain how each of these incarnations leads to a class of representations of Bn. The topological representations arise from differential equations of Xn which are symmetric under the algebra gl_n of nxn matrices. The algebraic representations arise instead from a deformation of this algebra known as the quantum group U_q(gl_n). Finally, I will tie the knot by relating these two classes of representations.
Abstract: Algebraic geometry studies geometric shapes defined as solutions of systems of polynomial equations. The building blocks of algebraic shapes consist of three types: positively curved, flat, and negatively curved. Fano varieties are those that are positively curved. In this talk, we will first take a glimpse into the rich geometry of Fano varieties. Then, we will explore some of the main ideas of the Fano Search Program, first initiated by the Imperial School. In particular, we will see how mirror symmetry predicts that we can build ‘periodic tables’ for the shape of Fano varieties by studying Laurent polynomials and their associated Newton polytopes, using combinatorial tools like cluster algebras.
Abstract: An R-matrix is a solution to the Yang-Baxter equation. The latter was discovered in the 1970's by C.N. Yang and R.J. Baxter and is a central object in Mathematical Physics and Statistical Mechanics. R-matrices were the driving force behind the discovery of quantum groups, and have featured prominently in various areas of Mathematics and Physics, such as Integrable Systems, Knot Theory, Enumerative Geometry and Gauge Theory.
In the first part of my talk, I will explain what the Yang-Baxter equation is, and how the theory of quantum groups naturally leads to its solutions. In the second part, I will present a new method to construct meromorphic R-matrices for a family of quantum groups known as Yangians. Our method relies upon constructing an appropriate twist which abelianizes the R-matrix. Time permitting, I will present a conjectural relation between our approach and the stable bases construction of Maulik and Okounkov. The talk is based on joint works with Valerio Toledano Laredo, Curtis Wendlandt, and Andrea Appel.
Abstract: A Poisson structure on a manifold M is a Lie bracket on the algebra of smooth functions that satisfies the Leibniz identity. This bracket induces a foliation of M in which each leaf carries a symplectic form, and at each point the transverse structure of this foliation is encoded by the action of a Lie algebra. In this way, Poisson geometry is a crossroads where foliation theory, symplectic geometry, and representation theory meet.
When M has a compatible action of a Lie group G, Hamiltonian reduction gives a procedure that "simplifies" M to a smaller Poisson manifold by removing its G-symmetries. After a general introduction to Poisson spaces, we'll survey Hamiltonian reduction from several geometric perspectives. Then we'll see how this tool can be applied to construct beautiful Poisson varieties in geometric representation theory.
Abstract: The classification of algebraic varieties is one of the central goals of algebraic geometry. This often comes in the form of constructing a moduli space. Broadly speaking, the points of a moduli space represent equivalence classes of varieties of a given type, and its geometry reflects the ways these varieties deform in algebraic families. The classification question then is tantamount to understanding the geometry of the corresponding moduli space.
In this talk I will give an introduction to the moduli theory of algebraic varieties, starting with the moduli space of curves. I will then explain recent advances in the moduli theory of higher dimensional varieties, with a focus on the problem of compactifying moduli spaces.
Abstract: In the first part of the talk I will explain the "exact WKB method" for studying monodromy of ordinary differential equations on a Riemann surface. This method turns out to be connected to various pretty bits of geometry (enumerative geometry, hyperkahler geometry); I will sketch some of these applications. In the second part I will describe new work (very much in progress) which connects the exact WKB method to a particular problem in conformal field theory, namely the explicit construction of conformal blocks for the Virasoro algebra.
Introduction to the Vlasov-Maxwell equation in Plasma Kinetic Theory
Abstract: Plasma is one of the four fundamental states of matter. It consists of a gas of ions and free electrons. Most important models in the kinetic level to describe dilute plasmas include the Vlasov-Maxwell/Vlasov-Poisson systems. I will give a brief introduction of these equations, and discuss two topics of interest in the field: wellposedness, and stability of equilibrium solutions.