Mathematical Physics

Unimodularity is a classical notion shows up in various fields like linear algebra, lattices, Poisson algebras, etc. In this talk, we focus on unimodular Hopf algebras and unimodular tensor categories. We will introduce unimodular module categories and use them to construct Frobenius algebras and unimodular tensor categories. These ideas will be illustrated with examples drawn from Hopf algebras.

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Zoom link https://pitp.zoom.us/j/98477599322?pwd=UDJmTklMTGxGODJZTm1Xc1VhL2tDdz09

A tower is an infinite sequence of deloopings of symmetric monoidal ever-higher categories. Towers are places where extended functorial field theories take values. Towers are a "deeper" version of commutative rings (as opposed to "higher rings" aka E∞-spectra). Notably, towers have their own opinions about Galois theory, and think that usual Galois groups are merely shallow approximations of deeper homotopical objects. In this talk, I will describe some steps in the construction and calculation of the deeper Galois group of a characteristic-zero field. In particular, I'll explain a homotopical version of the Kummer description of abelian extensions. This is joint work in progress with David Reutter.

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Zoom link: https://pitp.zoom.us/j/97950701035?pwd=Wk9FRSt2MkN3eWptTVltRVJncnFHdz09

In this talk, I will consider the centralizer of the quantum group U_q(sl_2) in the tensor product of three identical spin representations. The case of spin 1/2 (fundamental representation) is understood within the framework of the Schur-Weyl duality for U_q(sl_N), and the centralizer is known to be isomorphic to a Temperley-Lieb algebra. The case of spin 1 has also been studied and corresponds to the Birman-Murakami-Wenzl algebra. For a general spin, I will explain how to describe explicitly the centralizer (by generators and relations) using a combination of the braid group algebra and the Askey-Wilson algebra, which has been introduced in the context of orthogonal polynomials.

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Zoom link: https://pitp.zoom.us/j/98471794356?pwd=NjZFdjRFaDFON05HNkdTZS9hZTUvQT09

I will describe a correspondence between hyperkähler/quaternion-Kähler geometry and two dimensional chiral algebras that arises from twistor theory. As an application, I will explain how to use correlators of these chiral algebras to compute gravitational scattering amplitudes in four dimensional general relativity.

Zoom Link: https://pitp.zoom.us/j/98706048161?pwd=a3k1U3l2UXJlUjlqUElkOEZhZGwwUT09

In recent work, Aganagic proposed a categorification of quantum link invariants based on a category of A-branes, which is solvable explicitly. The theory is a generalization of Heegaard-Floer theory from gl(1|1) to arbitrary Lie algebras. I will describe in the detail the two simplest cases: the su(2) theory, categorifying the Jones polynomial, and the gl(1|1) theory, categorifying the Alexander polynomial. I will give an explicit algorithm for computing link homologies in these cases. I will also briefly describe the generalization to other simple Lie algebras and to Lie superalgebras of type gl(m|n). This talk is based on work to appear with Mina Aganagic and Miroslav Rapcak.

Zoom Link: TBD

This talk is based on joint work with Owen Gwilliam and Brian Williams. We define factorization homology of factorization envelopes valued in a collection of generalized Weyl modules supported on a cycle in a smooth complex projective variety X. When X is a smooth projective curve, and the cycle a collection of points, we recover the space of coinvariants studied in CFT. 

Zoom link:  https://pitp.zoom.us/j/97473973419?pwd=QjRQUklac2lTRlduSVc3S0FKQW9IQT09

The advent of topological materials has brought with it unexpected new connections between physics and pure mathematics. In particular, algebraic topology has played a significant role in the classification of topological materials. In this talk, I will offer a brief look at an emerging chapter in this story in which algebraic geometry — in particular the algebraic geometry of moduli spaces associated with complex curves — is used to anticipate new forms of quantum matter arising from 2-dimensional hyperbolic lattices.  In the process, I will explain my recent joint works with each of J. Maciejko, E. Kienzle, and A. Nagy that establishes an electronic band theory for 2-dimensional hyperbolic matter.

Zoom link:  https://pitp.zoom.us/j/98074477672?pwd=bmVScWx1M09EaGx2ZXZrRit6NXF5dz09

I will first review the construction of quiver Yangians for the quiver and superpotential from string theory on general toric Calabi-Yau threefolds; they serve as BPS algebras of these systems and their characters reproduce the unrefined BPS indices. I will then explain how to generalize the construction to arbitrary quivers and how to incorporate refined BPS indices. The entire construction allows for straightforward generalizations to trigonometric, elliptic, and generalized cohomologies. Time permitting, I will discuss some applications, such as on BPS counting and on Gauge/Bethe correspondence. 

Zoom link:  https://pitp.zoom.us/j/96812116687?pwd=RjlsSVhDaUVBREtUUnd5S04rQjhpZz09

Cluster structure on a quantum group allows one to work with its positive representations, a special class of modules similar in spirit to principal series representations but closed under tensor multiplication. On the other hand, cluster techniques proved inadequate for the study of finite-dimensional representation theory. I will discuss how one can reconcile positive and finite-dimensional representations into one theory by studying moduli spaces of local systems with non-generic monodromies.

Zoom link:  https://pitp.zoom.us/j/97677905324?pwd=Tkxlei9wN1llYVpncHA2cnpYKzlhUT09

I will describe a general categorical approach to constructing Hamiltonian actions on moduli spaces. In particular cases, this specializes to give a ``universal" Hitchin integrable system as well as the Calogero-Moser system.  Moreover, I will describe a generalization to higher dimensions of a classical result of Goldman which says that the Goldman Lie algebra of free loops on a surface acts by Hamiltonian vector fields on the character variety of the surface.  A key input is a description of deformations of Calabi-Yau structures, which is of independent interest. This is joint work with Chris Brav.

Zoom link:  https://pitp.zoom.us/j/92929253744?pwd=WGFNQmRJck5NdzFFdU8xcXRlN3RRQT09

There is a by now standard procedure for calculating twisted spin, spin^c, and string bordism groups for applications in physics: realize the twist as arising from a vector bundle, which allows one to split the corresponding Thom spectrum and greatly simplify the Adams spectral sequence computation. Not all twists arise from vector bundles, but Matthew Yu and I noticed that if you ignore this fact and pretend that everything is OK, you still get the right answer! In this talk, I'll discuss a theorem Matthew and I proved explaining this, by calculating the input to Baker-Lazarev's version of the Adams spectral sequence. Then I will discuss applications to anomalies of some quantum field theories and supergravity theories.

Zoom link:  https://pitp.zoom.us/j/93508575689?pwd=YVV6VlRwL1RGSG55V0cwTzdpUWROUT09