Posted October 4, 2021

Last modified November 5, 2021

Mathematical Physics and Representation Theory Seminar

3:30 pm - 4:20 pm Zoom: https://lsu.zoom.us/j/7376728101
Colleen Delaney, Indiana University Bloomington

Zesting and Witten-Reshetikhin-Turaev invariants

I’ll discuss the ribbon zesting construction on pre-modular categories from a diagrammatic point of view and show that Witten-Reshetikhin-Turaev invariants of framed knots and links decouple under zesting. As an application, I will explain how the Mignard-Schauenburg ``modular isotopes” can be understood through zesting. This talk is based on joint work with Cesar Galindo, Julia Plavnik, Eric Rowell, and Qing Zhang as well as Sung Kim.

Posted October 3, 2021

Last modified November 5, 2021

Mathematical Physics and Representation Theory Seminar

2:30 pm - 3:20 pm Lockett 233
Gurbir Dhillon, Yale University

Kazhdan--Lusztig theory for affine Lie algebras at critical level

Formulas for simple characters have a long and rich history in representation theory, and our main result is one more such formula, originally conjectured by Feigin--Frenkel. However, in the majority of the talk, we will provide a general survey of such results for non-specialists. After recalling Weyl's character formula for highest weight modules for simple algebraic groups, we will discuss the Kazhdan--Lusztig character formula for highest weight modules for simple Lie algebras. In particular, we will convey some of the striking ideas involved in its proof via localization, due to Beilinson--Bernstein and Brylinski-Kashiwara, which birthed the subject of geometric representation theory. Moving beyond simple Lie algebras and groups, we will recall that associated to each simple Lie algebra is a one parameter family of infinite dimensional Lie algebras, the affine Lie algebras, which appear repeatedly in algebraic geometry and mathematical physics. By work of Kashiwara--Tanisaki, the highest weight characters at all points in the family save one were understood by the mid 1990s. At this remaining point, the critical level, the representation theory of affine Lie algebras undergoes a phase transition, and the remarkable phenomena present at this point have deep connections to the geometric Langlands program. An analog of the Kazhdan--Lusztig conjecture for affine Lie algebras at critical level was proposed by Feigin--Frenkel in the early 1990s. We have proven this conjecture in forthcoming work joint with David Yang, using localization theory at critical level as developed by Beilinson--Drinfeld and Frenkel--Gaitsgory. The main emphasis throughout will be on basic ideas and simple examples, and we will not presume familiarity with any of these subjects beyond the finite dimensional representations of SL2.

Posted September 28, 2021

Last modified November 14, 2021

Mathematical Physics and Representation Theory Seminar

3:30 pm - 4:20 pm Lockett 233
Mee Seong Im, United States Naval Academy

Iterated wreath products and foams, with applications

I will explain a new perspective of foams with connections to the representation theory of iterated wreath products. If I have time, I will discuss the connections of foams to field extensions, Sylvester sums, and matrix factorizations. This is joint work with Mikhail Khovanov, with Appendix joint with Lev Rozansky.

Posted September 29, 2021

Last modified January 30, 2022

Mathematical Physics and Representation Theory Seminar

3:30 pm - 4:20 pm Zoom: https://lsu.zoom.us/j/98489192227, Lockett 233
Iva Halacheva, Northeastern University

Welded tangles and the Kashiwara-Vergne group

Welded or w-tangles are a higher dimensional analogue of classical tangles, which admit a yet further generalization to welded foams, or w-trivalent graphs, a class of knotted tubes in 4-dimensional space. Welded foams can be presented algebraically as a circuit algebra. Together with Dancso and Robertson we show that their automorphisms can be realized in Lie theory as the Kashiwara-Vergne group, which plays a key role in the setting of the Baker-Campbell-Hausdorff series. In the process, we use a result of Bar-Natan and Dancso which identifies homomorphic expansions for welded foams, a class of powerful knot invariants, with solutions to the Kashiwara-Vergne equations.

Posted January 31, 2022

Last modified March 6, 2022

Mathematical Physics and Representation Theory Seminar

2:30 pm - 3:20 pm Lockett 233, https://lsu.zoom.us/j/91730960984
Peter Koroteev, UC Berkeley

DAHA Representations and Branes

I will describe our recent geometric representation theory construction for the double affine Hecke algebra (DAHA) of rank one. The spherical subalgebra of DAHA can be understood as flat one-parameter deformation (geometric quantization) of the SL(2, C) character variety X of a one-punctured torus. This variety for rank one DAHA is described by an affine cubic surface which is an elliptic fibration of Kodaira type I_0^*. Our main result provides an equivalence between the Fukaya category of X and the category of finite-dimensional modules of DAHA. Upon this correspondence, compact Lagrangian submanifolds of X are related to finite-dimensional representations of DAHA. This is a work in progress with S. Gukov, S. Nawata, D. Pei, and I. Saberi.

Posted February 8, 2022

Last modified March 16, 2022

Mathematical Physics and Representation Theory Seminar

2:30 pm - 3:20 pm Lockett 233
Akos Nagy, UC Santa Barbara

BPS monopoles with arbitrary symmetry breaking

Magnetic monopole solutions of Maxwell's equations have been known since Dirac's famous paper in 1931. In the mid-1970s, Bogomolny and, independently, Prasad and Sommerfeld, gave nonabelian generalizations to these solutions in Yang-Mills theory, which are now called Bogomolny-Prasad-Sommerfeld (BPS) monopoles. These are gauge theoretic field configurations over 3-dimensional backgrounds. In this talk I will introduce BPS monopoles both from the mathematical and physical points of view and recall some of the most important results about them, with an emphasis on BPS monopoles over the euclidean 3-space. In particular, I will introduce the concept of symmetry breaking for these fields. While a lot is known about monopoles with maximal symmetry breaking, the general case has been much less understood. After the general introduction, I will present my recent results on the construction of monopoles with arbitrary, nonmaximal symmetry breaking. This is achieved by understanding the analytic behavior of harmonic spinors associated to Dirac operators twisted by monopoles. This is a joint work with Benoit Charbonneau.

Posted January 30, 2022

Last modified April 3, 2022

Mathematical Physics and Representation Theory Seminar

2:30 pm - 3:20 pm Lockett 233
Tudor Padurariu, Columbia University

Categorical Hall algebras in Donaldson-Thomas theory

Kontsevich-Soibelman defined the cohomological Hall algebra (CoHA) of a quiver with potential. By a result of Davison-Meinhardt, CoHAs are deformations of the universal enveloping algebra of the BPS Lie algebra of the quiver with potential. One can also define categorical and K-theoretic Hall algebras of a quiver with potential. Examples of such Hall algebras are (positive parts of) quantum affine algebras. I will introduce the categorical and K-theoretic replacements of the BPS spaces and explain how to prove analogues of the Davison-Meinhardt theorem in these contexts. These results have applications in Donaldson-Thomas theory and in the study of Hall algebras of surfaces.

Posted February 8, 2022

Last modified April 16, 2022

Mathematical Physics and Representation Theory Seminar

2:30 pm - 3:20 pm Lockett 233
Vasily Krylov, MIT

Symplectic duality and equivariant Hikita-Nakajima conjecture for ADHM spaces

We will discuss the general notion of symplectic duality between symplectic resolutions of singularities and give examples. Equivariant Hikita-Nakajima conjecture is a general conjecture about the relation between the geometry of symplectically dual varieties. We will consider the example of the Hilbert scheme of points on the affine plane and briefly discuss the proof of the equivariant Hikita-Nakajima conjecture in this particular case. We will also briefly discuss the generalization of this proof to the case of ADHM spaces (moduli spaces of instantons on R^4). Time permitting we will say about the possible approach towards the proof of Hikita-Nakajima conjecture for other symplectically dual pairs (such as Higgs and Coulomb branches of quiver gauge theories). The talk is based on the joint work with Pavel Shlykov arXiv:2202.09934.

Posted February 6, 2022

Last modified April 25, 2022

Mathematical Physics and Representation Theory Seminar

2:30 pm - 3:20 pm Lockett 233
Eugene Rabinovich, University of Notre Dame

Classical Bulk-Boundary Correspondences via Factorization Algebras

A factorization algebra is a cosheaf-like local-to-global object which is meant to model the structure present in the observables of classical and quantum field theories. In the Batalin-Vilkovisky (BV) formalism, one finds that a factorization algebra of classical observables possesses, in addition to its factorization-algebraic structure, a compatible Poisson bracket of cohomological degree +1. Given a ``sufficiently nice'' such factorization algebra on a manifold $N$, one may associate to it a factorization algebra on $N\times \mathbb{R}_{\geq 0}$. The aim of the talk is to explain the sense in which the latter factorization algebra ``knows all the classical data'' of the former. This is the bulk-boundary correspondence of the title. Time permitting, we will describe how such a correspondence appears in the deformation quantization of Poisson manifolds.

Posted January 9, 2023

Last modified February 28, 2023

Mathematical Physics and Representation Theory Seminar

2:30 pm - 3:20 pm 233 Lockett
Nadia Ott, University of Pennsylvania

Period maps for super Riemann surfaces with Ramond punctures

Super Riemann surfaces and their moduli spaces $\mathfrak{M}_g$ are supersymmetric generalizations of the classical notions of Riemann surfaces. In physics, super Riemann surfaces are the worldsheets of propagating superstrings and once can define important quantities in superstring theory, e.g., vacuum and scattering amplitudes, as integrals over $\mathfrak{M}_g$. However, computing these integrals has proved to be extremely difficult and answers are known only up to genus 2. D’Hoker and Phong’s computation in genus 2 relied on the super period map for genus 2 super Riemann surfaces. Naturally, we look to the period map for answers in the higher genus. However, the period map in $g > 2$ behaves in ways quite distinct from its ordinary counterpart. Most dramatically, it blows up along a certain divisor in supermoduli space. It is also not fully defined for either the Ramond punctured variants of super Riemann surfaces, or for the compactification. In both cases, one expects the map to blow up along a divisor, called the bad locus. In my talk, I will discuss some of the open problems as well as recent progress concerning the super period map and its relation to the supermeasure. In addition, I will talk about my joint work with Ron Donagi in which we describe the “bad locus” on the supermoduli space with Ramond punctures.

Posted March 23, 2023

Mathematical Physics and Representation Theory Seminar

2:30 pm - 3:20 pm Lockett 233
Simon Riche, Université Clermont Auvergne

Characters of modular representations of reductive algebraic groups

One of the main questions in the representation theory of reductive algebraic groups is the computation of characters of simple modules. A conjectural solution to this problem was proposed by G. Lusztig in 1980, and later shown to be correct assuming the base field has large characteristic. However in 2013 G. Williamson found (counter)examples showing that this answer is not correct without this assumption. In this talk I will explain a new solution to this problem, obtained in a combination of works involving (among others) P. Achar and G. Williamson, which is less explicit but has the advantage of being valid in all characteristics.

Posted January 11, 2023

Last modified April 9, 2023

Mathematical Physics and Representation Theory Seminar

2:30 pm - 3:20 pm Lockett 233
Melissa Sherman-Bennett, MIT

The m=2 amplituhedron and the hypersimplex

I'll discuss a curious correspondence between the m=2 amplituhedron, a 2k-dimensional subset of Gr(k, k+2), and the hypersimplex, an (n-1)-dimensional polytope in R^n. The amplituhedron and hypersimplex are both images of the totally nonnegative Grassmannian under some map (the amplituhedron map and the moment map, respectively), but are different dimensions and live in very different ambient spaces. I'll talk about joint work with Matteo Parisi and Lauren Williams in which we give a bijection between decompositions of the amplituhedron and certain matroidal decompositions of the hypersimplex (originally conjectured by Lukowski—Parisi—Williams). We also give a new decomposition of the m=2 amplituhedron into Eulerian-number-many chambers, inspired by an analogous hypersimplex decomposition. No background knowledge on the amplituhedron, the totally nonnegative Grassmannian, or the hypersimplex will be assumed.

Posted January 21, 2023

Last modified March 8, 2023

Mathematical Physics and Representation Theory Seminar

2:30 pm - 3:20 pm Lockett 233
Svetlana Makarova, University of Pennsylvania

Quiver moduli and effective global generation

Moduli problems are ubiquitous and related to all areas of mathematics in one way or another. In this talk, I will focus on the algebro-geometric picture: namely, I would like to view the set of objects of classification as a scheme, called a moduli scheme. I will provide a framework that allows to recover the algebraic structure on this set, and then I will talk about modern methods of studying moduli problems. The modern theory "Beyond GIT", introduced by Alper and being developed by Alper, Halpern-Leistner, Heinloth and others, provides a "coordinate-free" way of thinking about classification problems. Among giving a uniform philosophy, this allows to treat problems that can't necessarily be described as global quotients. Our result about moduli of quiver representations is a particularly nice example where this modern theory can be applied. After a reminder on quiver representations, I will explain how we refine a classical result of King that moduli spaces of semistable representations of acyclic quivers are projective by proving it over an arbitrary noetherian base. Our methods allow us to obtain new results about the geometry of these moduli: I will define a determinantal line bundle which descends to a semiample line bundle on the moduli space and provide effective bounds for its global generation. For an acyclic quiver, we can observe that this line bundle is ample and thus the adequate moduli space is projective over an arbitrary noetherian base. This talk is based on a preprint with Belmans, Damiolini, Franzen, Hoskins, Tajakka (https://arxiv.org/abs/2210.00033).

Posted January 9, 2023

Last modified April 22, 2023

Mathematical Physics and Representation Theory Seminar

2:30 pm - 3:20 pm Lockett 233
Anne Dranowski, University of Southern California

Canonical bases in representation theory and mathematical physics

The fusion of two Mirkovic-Vilonen cycles is a degeneration of their product, defined using the Beilinson-Drinfeld Grassmannian. We describe a conceptually friendly approach to computing this product in type A, by transferring the problem to a fusion of generalized orbital varieties using the Mirkovic-Vybornov isomorphism. We explain how using this fusion product we are able to verify a conjecture about cluster monomials and the MV basis in the coordinate ring of the upper-triangular subgroup of GL(4). Based on joint work with R. Bai and J. Kamnitzer.