2024
University of North Carolina at Chapel Hill
Mondays/Fridays 4pm, PH-367 or PH-385
The aim of this seminar is to bring speakers from this area and outside to speak on topics related to Representation Theory (specially geometric and topological methods employed in Representation Theory). The speakers are expected to give their talks at a level suitable for graduate students. The seminar is organized by Prakash Belkale, Jiuzu Hong, Shrawan Kumar and Richárd Rimányi.
2024 Fall
| Date | Speaker | Affiliation | Title |
|---|---|---|---|
| Nov 15 | Giacomo Nanni | Universita di Bologna | Lagrangian Planes on Kummer-Type Manifolds |
| Nov 1 | Syu Kato | Kyoto | Geometry of Dyck paths |
| Oct 25 | Mikhail Kapranov | Kavli IPMU | Operadic analysis of parabolic induction |
| Oct 4 | Phil Tosteson | UNC-CH | Homology of spaces of curves on blowups |
| Sept 27 | Nicola Tarasca | Virginia Commonwealth | Higher Rank Series and Root Puzzles for Plumbed 3-Manifolds |
| Sept 13 | Jakub Koncki | IMPAN Warsaw & UNC-CH | Multiplicative structure of the K-theoretic McKay correspondence for Hilbert schemes of points |
| Aug 30 | Tom Gannon | UCLA | Quantization of the Ngô morphism |
| Giacomo Nanni: Lagrangian Planes on Kummer-Type Manifolds Abstract: The birational geometry of K3 surfaces can be understood through their intersection theory, and Hassett and Tschinkel initiated a program to achieve a similar understanding for their higher-dimensional analogs: irreducible holomorphic symplectic (IHS) manifolds. Recent developments in Bridgeland stability theory have brought new insights to this problem. In this talk, I will provide a brief introduction to IHS manifolds, followed by an overview of the theory of moduli spaces of Bridgeland-stable objects on K3 and abelian surfaces. I will then present a characterization of the homology class of lines contained in Lagrangian planes within IHS manifolds of K3^[n] or generalized Kummer type. |
| Syu Kato: Geometry of Dyck paths Abstract: There are two major research trends in the theory of symmetric functions arising from Dyck paths. One is the theory of Catalan symmetric functions and its geometric realization conceived by Chen-Haiman, following the works of Broer and Shimozono-Weyman. The symmetric function part of this story was established by a series of works of Blasiak-Morse-Pun-Summers. Another is the study of chromatic symmetric functions of graphs, that utilize the fact that Dyck paths roughly correspond to unit interval graphs.We first exhibit a family of smooth algebraic varieties that realize Catalan symmetric functions, and explain how it imply the geometric predictions of Broer, Shimozono-Weyman, Chen-Haiman, and Blasiak-Morse-Pun. We then exhibit that our variety also naturally realize the chromatic symmetric functions of unit interval graphs. Thus, in a sense, our varieties might be understood as a geometric realization of the world of Dyck paths (which are related but not the same as the Hessenberg varieties).This talk is based on arXiv:2301.00862 and arXiv:2410.12231. |
| Mikhail Kapranov: Operadic analysis of parabolic induction Abstract: The construction of various types of Hall algebas can be seen as an instance of the parabolic induction formalism. More precisely, this formalism is used to arrange various linear data associated to groups GL_n for all n (e.g., cohomology of the moduli spaces of vector bundles) into an associative algebra. But parabolic induction makes sense for arbitrary reductive groups. The talk, based on joint work in progress with V. Schechtman, O. Schiffman and J. Yuan, proposes an operad-style approach to formalize the algebra-like structures provided by all reductive groups taken together. The corresponding “double” structures (reducing to graded bi/Hopf algebras for the GL_n series) are important for the problem of classifying perverse sheaves on the adjoint quotients h/W. |
| Phil Tosteson: Homology of spaces of curves on blowups Abstract: Let C be a smooth projective curve and X be a smooth projective variety. We will consider the space degree d algebraic (or holomorphic) maps from C to X.When X is a projective space, Segal discovered an interesting phenomenon: as the degree increases, the homology of the space of algebraic maps approximates that of the space of continuous maps. Recently, Ellenberg-Venkatesh observed that this phenomenon is related to Manin’s conjectures about rational points on Fano varieties, suggesting it holds more generally.I will talk about joint work with Ronno Das considering the case where X is a blowup of a projective space at finitely many points (in particular the case of del Pezzo surfaces). |
| Nicola Tarasca: Higher Rank Series and Root Puzzles for Plumbed 3-Manifolds Abstract: The Witten-Reshetikhin-Turaev (WRT) invariants provide a powerful framework for constructing a family of invariants for framed links and 3-manifolds. An ongoing pursuit in quantum topology revolves around the categorification of these invariants. Recent progress has been made in this direction, particularly through a physical definition of a new series invariant for negative definite plumbed 3-manifolds. These invariants exhibit a convergence towards the WRT invariants in their limits. In this talk, I will present a refinement of such series invariants and show how one can obtain infinitely many new series invariants starting from the data of a root lattice of rank at least 2 and a solution to a combinatorial puzzle defined on that lattice. This is joint work with Allison Moore. |
| Jakub Koncki: Multiplicative structure of the K-theoretic McKay correspondence for Hilbert schemes of points Abstract: The Hilbert scheme of points in the complex plane is a classical object of study in algebraic geometry. McKay correspondence provides an isomorphism between its K-theory (or cohomology) and the space of symmetric functions, creating a bridge between geometry and combinatorics. Multiplication by a class in the K-theory induces an endomorphism of the space of symmetric functions. In the cohomological case, compact formulas for such maps were found by Lehn and Sorger. The K-theoretical case was studied by Boissiere using torus equivariant techniques. He proved a formula for multiplication by the class of the tautological bundle and stated a conjecture for the remaining generators of K-theory. In the talk, I will show how torus action simplifies this problem and prove the conjectured formula using restriction to a one-dimensional subtorus. This is a joint work with M. Zielenkiewicz. |
| Tom Gannon: Quantization of the Ngô morphism We will discuss work, joint with Victor Ginzburg, which proves a conjecture of Nadler on the existence of a quantization (non-commutative deformation) of the Ngô morphism, a morphism of group schemes constructed by Ngô in his proof of the fundamental lemma in the Langlands program. We will first explain the construction of the Ngô morphism and discuss an extended example of this map for the group of invertible n x n complex matrices. Then, we will give a precise statement of our main theorem and discuss some of the tools used in proving this theorem, including a quantization of Moore-Tachikawa varieties. Time permitting, we will also discuss how the tools used to construct this morphism can be used to prove conjectures of Ben-Zvi—Gunningham, which predict a “spectral decomposition” of DG categories with an action of a reductive group. |
2024 Spring
| Date | Speaker | Affiliation | Title |
|---|---|---|---|
| April 12 | Syu Kato | Kyoto | Semi-infinite flag manifolds, space of maps, and quantum K-groups of flag manifolds | April 5 | Syu Kato | Kyoto | Algebraic models of semi-infinite flag manifolds |
| Mar 22 | Thomas Haines | Maryland | Cellular pavings of convolution fibers and applications |
| Feb 23 | Jayce Getz | Duke | On the Poisson summation conjecture |
| Feb 9 | Luke Conners | UNC | Row-Column Mirror Symmetry for Colored Torus Knot Homology |
| Syu Kato Talk 1: Algebraic models of semi-infinite flag manifolds Abstract: We first recall the classical Borel-Weil-Bott theorem and examine its representation-theoretic meaning. Then, we exhibit affine analogue of their representation-theoretic meaning and form several schemes. Finally, we identify the resulting schemes with so-called semi-infinite flag manifolds set-theoretically known from 1980s and its Schubert and Richardson varieties. This talk is mainly based on arXiv:1810.07106. |
| Syu Kato Talk 2: Semi-infinite flag manifolds, space of maps, and quantum K-groups of flag manifolds Abstract: We first recall the Plucker relations, that yield a description of a(n open) space of maps from a projective line to a (partial) flag manifolds. Then, we observe a close relationship between Richardson varieties of semi-infinite flag manifolds and the space of stable maps from a projective line to a flag manifold. This makes us possible to describe quantum K-groups of flag manifolds with the K-group of semi-infinite flag manifolds. This talk is mainly based on arXiv:1805.01718. If time permits, we also explain a natural isomorphism between the K-groups of semi-infinite flag manifolds and that of affine Grassmannians conjectured by Lam-Li-Shimozono-Mihalcea in order to complete the picture. |
| Tom Haines: Cellular pavings of convolution fibers and applications Abstract: A convolution morphism is the geometric analogue of a convolution of functions in a Hecke algebra. The properties of fibers of convolution morphisms are used in a variety of ways in the geometric Langlands program and in the study of Schubert varieties. I will explain a very general result about cellular pavings of fibers of convolution morphisms in the setting of partial affine flag varieties, as well as applications related to the very purity and parity vanishing of cohomology of Schubert varieties over finite fields, rationality of the BBD Decomposition Theorem over finite fields, structure constants for parahoric Hecke algebras, and the (motivic) geometric Satake equivalence. If time permits, I will describe a new combinatorial model for generalized Mirkovic-Vilonen intersections and the branching to Levi subgroups. |
| Jayce Getz: On the Poisson summation conjecture Abstract: Braverman, Kazhdan, Lafforgue, Ngo, and Sakellaridis have conjectured that Fourier analysis on a vector space is but the first example of a larger phenomenon. More generally, one should have Schwartz spaces, Fourier transforms, and, crucially, Poisson summation formulae for affine spherical varieties. I will give an account of the few cases in which the conjecture is known, and describe some techniques for proving new cases from old cases. |
| Luke Conners: Row-Column Mirror Symmetry for Colored Torus Knot Homology Abstract: The HOMFLYPT polynomial is a 2-variable link invariant generalizing the celebrated Jones polynomial and other Type A quantum link polynomials. Its construction passes through a Hecke algebra representation of the braid group, and by making use of certain idempotent elements in the Hecke algebra, one can extend the invariant to links with components labeled by arbitrary Young diagrams. The resulting invariant, called the colored HOMFLYPT polynomial, has a well-known “mirror symmetry” property describing its behavior under exchanging each such Young diagram with its transpose. One categorical level up, Khovanov and Rozansky constructed a triply-graded homological link invariant that recovers the HOMFLYPT polynomial upon taking Euler characteristic. Various authors have constructed colored versions of triply-graded Khovanov-Rozansky homology, and these invariants are conjectured to satisfy a categorical lift of the polynomial mirror symmetry described above. In this talk, we will formulate this conjecture precisely and outline a recent proof in the special case of a positive torus knot colored by a single row or column of arbitrary length. |