Posted March 5, 2024
Mathematical Physics and Representation Theory Seminar
2:30 pm – 3:20 pm 233 Lockett
Pablo Boixeda Alvarez, Yale University
Microlocal sheaves and affine Springer fibers
The resolutions of Slodowy slices $\tilde{\mathcal{S}}_e$ are symplectic varieties that contain the Springer fiber $(G/B)_e$ as a Lagrangian subvariety. In joint work with R. Bezrukavnikov, M. McBreen, and Z. Yun, we construct analogues of these spaces for homogeneous affine Springer fibers. We further understand the categories of microlocal sheaves in these symplectic spaces supported on the affine Springer fiber as some categories of coherent sheaves. In this talk, I will mostly focus on the case of the homogeneous element $ts$ for $s$, a regular semisimple element, and will discuss some relations of these categories with the small quantum group providing a categorification of joint work with R.Bezrukavnikov, P. Shan and E. Vasserot. If I have time, I will then mention some recent application of this result to the Breuil-Mezard conjecture by T. Feng and B. Le Hung.
Posted January 28, 2024
Last modified April 1, 2024
Mathematical Physics and Representation Theory Seminar
2:30 pm – 3:20 pm Lockett 233
Greg Parker, Stanford University
$\mathbb Z_2$-harmonic spinors as limiting objects in geometry and topology
$\mathbb Z_2$-harmonic spinors are singular solutions of Dirac-type equations that allow topological twisting around a submanifold of codimension 2. These objects arise as limits at the boundary of various moduli spaces in several distinct areas of low-dimensional topology, gauge/Floer theory, and enumerative geometry. The first part of this talk will introduce these objects, and discuss the various contexts in which they arise and the relationship between them. The second part of the talk will focus on the deformations of $\mathbb Z_2$-harmonic spinors when varying background parameters as a model for the novel analytic problems presented by these objects. In particular, the deformations of the singular submanifold play a role, giving the problem some characteristics similar to a free-boundary-value problem and leading to a hidden elliptic pseudo-differential operator that governs the geometry of the moduli spaces.
Posted June 4, 2024
Mathematical Physics and Representation Theory Seminar
3:30 pm – 4:30 pm Lockett 233
Mikhail Khovanov, Johns Hopkins University
Foams in algebraic K-theory and dynamics
We'll discuss a recent paper where algebraic K-theory is related to foams with a flat connection.
Posted September 27, 2024
Last modified October 1, 2024
Mathematical Physics and Representation Theory Seminar
2:30 pm – 3:20 pm Lockett 233
Tom Gannon, UCLA
Quantization of the universal centralizer and central D-modules
We will discuss joint work with Victor Ginzburg that proves a conjecture of Nadler on the existence of a quantization, or non-commutative deformation, of the Knop-Ngô morphism—a morphism of group schemes used in particular by Ngô in his proof of the fundamental lemma in the Langlands program. We will first explain the representation-theoretic background, give an extended example of this morphism for the group GL_n(C), and then present a precise statement of our theorem. Time permitting, we will also discuss how the tools used to construct this quantization can also be used to prove conjectures of Ben-Zvi and Gunningham, which predict a relationship between the quantization of the Knop-Ngô morphism and the parabolic induction functor.
Posted October 9, 2024
Mathematical Physics and Representation Theory Seminar
2:30 pm – 3:20 pm Lockett 233
Tobias Simon, University of Erlangen, Germany
Realizations of irreducible unitary representations of the Lorentz group in spaces of distributional sections over de Sitter space
In Algebraic Quantum Field Theory, one is interested in constructing nets of local von Neumann algebras satisfying the Haag Kastler axioms. Every such net defines a local net of standard subspaces in the corresponding Hilbert space by letting the selfadjoint elements in the local algebras act on a common cyclic and separating vector. In this talk, we discuss work by Frahm, Neeb and Olafsson which constructs nets standard subspaces on de Sitter space satisfying the corresponding axioms. Here the main tool is "realizing" irreducible unitary representations of the Lorentz group SO(1,d) in spaces of distributional sections over de Sitter space. These can be constructed from SO(1,d-1)-finite distribution vectors obtained as distributional boundary values of holomorphically extended orbit maps of SO(d)-finite vectors. Our main contribution is the proof of polynomial growth rates of these orbit maps, which guarantees the existence of the boundary values in the space of distribution vectors.
Posted September 27, 2024
Last modified October 16, 2024
Mathematical Physics and Representation Theory Seminar
2:30 pm – 3:20 pm Lockett 233
Xinchun Ma, University of Chicago
Cherednik algebras, Torus knots and flag commuting varieties
In this talk, we will explore how the Khovanov-Rozansky homology of the (m,n)-torus knot can be extracted from the finite-dimensional representation of the rational Cherednik algebra at slope m/n, equipped with the Hodge filtration. Our approach involves constructing a family of coherent sheaves on the Hilbert scheme of points on the plane, arising from cuspidal character D-modules. In describing this family of coherent sheaves, the geometry of nilpotent flag commuting varieties naturally emerges, closely related to the compactified regular centralizer in type A.
Posted September 27, 2024
Last modified October 24, 2024
Mathematical Physics and Representation Theory Seminar
2:30 pm – 3:20 pm Lockett 233
Nikolay Grantcharov, University of Chicago
Infinitesimal structure of BunG
Given a semisimple group G and a smooth projective curve X over an algebraically closed field of arbitrary characteristic, let Bun_G(X) denote the moduli space of principal G-bundles over X. For a bundle P without infinitesimal symmetries, we describe the n^th order divided-power infinitesimal jet spaces of Bun_G(X) at P for each n. The description is in terms of differential forms on the Fulton-Macpherson compactification of the configuration space, with logarithmic singularities along the diagonal divisor. We also briefly discuss applications into constructing Hitchin's flat connection on the vector bundle of conformal blocks.
Posted March 9, 2025
Mathematical Physics and Representation Theory Seminar
12:30 pm – 1:20 pm 233 Lockett Hall
David Boozer, Indiana University
Student Seminar on Instanton Homology and Foam Evaluations
This is to help prepare graduate students for David Boozer's talk at 2:30pm on the same day. He will discuss some of the basic definitions behind his 2:30pm talk and take questions from graduate students on the objects of study in his talk.
Posted February 10, 2025
Last modified February 24, 2025
Mathematical Physics and Representation Theory Seminar
2:30 pm – 3:20 pm 233 Lockett Hall
David Boozer, Indiana University
The combinatorial and gauge-theoretic foam evaluation functors are not the same
Kronheimer and Mrowka have outlined a new approach that could potentially lead to the first non-computer based proof of the four-color theorem. Their approach relies on a functor J-sharp, which they define using gauge theory, from a category of webs in R^3 to the category of finite-dimensional vector spaces over the field of two elements. They have also suggested a possible combinatorial replacement J-flat for J-sharp, which Khovanov and Robert proved is well-defined on a subcategory of planar webs. We exhibit a counterexample that shows the restriction of the functor J-sharp to the subcategory of planar webs is not the same as J-flat.
Posted February 10, 2025
Last modified March 9, 2025
Mathematical Physics and Representation Theory Seminar
2:30 pm – 3:20 pm Lockett 233
Maarten Mol, University of Toronto
Constructibility of momentum maps and variation of singular symplectic reduced spaces
Proper maps in various categories studied in singularity theory (for example, the real analytic category) are known to be constructible, in the sense that the image of the map can be stratified in such a way that the map is a topological fiber bundle over each stratum. Such stratifications provide insight into how the fibers of the map vary. In this talk we will discuss the existence of such a stratification for momentum maps of Hamiltonian Lie group actions (a natural class of maps studied in symplectic/Poisson geometry), which provides insight into how the so-called symplectic reduced spaces of the Hamiltonian action vary. Along the way we will also try to give an overview of some more classical results on the geometry of such maps.
Posted February 10, 2025
Last modified March 13, 2025
Mathematical Physics and Representation Theory Seminar
2:30 pm – 3:20 pm Monday, March 17, 0025 Lockett 233
Sam Gunningham, Montana State University
Geometric Satake Revisited
The geometric Satake equivalence is a fundamental result in the geometric Langlands program. It can be understood as a kind of Fourier transform, relating different flavors of sheaves on a dual pair of spaces. Just like the usual Fourier transform, the equivalence exchanges the structures of convolution and pointwise product on each side. In this talk, I will discuss a circle of ideas relating pointwise tensor product of sheaves on the affine Grassmannian, the Knop-Ngo action for the group scheme of regular centralizers, and Moore-Tachikawa varieties. This builds on past joint work with D. Ben-Zvi and some current work in progress with D. Ben-Zvi and S. Devalapurkar.
Posted February 10, 2025
Mathematical Physics and Representation Theory Seminar
2:30 pm – 3:20 pm Lockett 233
Joshua Mundinger, University of Wisconsin
TBA