Physically inspired mathematics seminar at the Department of Mathematics at the University of North Carolina at Chapel Hill

The mathematics that appears in physics oftentimes has a characteristic internal structure, often captured by the word integrable. In many examples, this at first mysterious integrability structure was later connected with the representation theory of the corresponding algebraic objects: quantum groups, Virasoro algebras, or other infinite dimensiona Lie algebas. The study of these mathematical objects brings insights back into modern mathematical physics, as demonstrated in examples of conformal field theory and string theory. The goal of the seminar is to study this physically inspired mathematics.

Organizers: Lev Rozansky, Alexander Varchenko, Andrey Smirnov, and Slava Naprienko.

Guidance for Speakers: We strongly urge speakers to ensure that their presentations are lucid and accompanied by simple, illustrative examples that would be easy accessible by graduate students. We would appreciate if speakers dedicated at least half of their talk to the motivation behind their research, previous findings, and fundamental ideas.

The seminar is organized by the Department of Mathematics at the University of North Carolina at Chapel Hill.

The seminar meets at Phillips Hall, Room 385.

Date Speaker Title Abstract and Materials
3:00 pm, September 15 Slava Naprienko Integrable lattice models and symmetric functions, part 1 I will talk about how integrable lattice models from statistical mechanics unexpectedly became about the most powerful tool to study symmetric functions from representation theory and combinatorics of affine flag varieties. The talk will be accessible for graduate students and will feature multiple examples.
3:00 pm, September 22 Philip Tosteson Stability in the homology of the discriminant 1 hypersurface I will talk about the hypersurface in C^n defined by the equation $\prod_{i < j} (x_i - x_j) = 1$. This hypersurface has interesting symmetries, from the action of the alternating group permuting the variables and from the action of roots of unity by rescaling variables. I will describe how, as $n$ varies, these actions can be promoted to the action of a category that is closely related to the category of finite sets and injections. This category governs the behavior of the homology groups of these hypersurfaces for large values of $n$.
3:00 pm, September 29 Slava Naprienko Integrable lattice models and symmetric functions, part 2 The talk was canceled.
3:00 pm, October 6 (Zoom) Masha Vlasenko Calabi-Yau differential operators

These are ordinary linear differential operators with strong arithmetic properties, whose notion goes back to the discovery of mirror symmetry in the 1990s. I will speak about proving the integrality of instanton numbers as well as some recent conjectures in this field. The talk is based on our joint work with Frits Beukers.

Write to slava@unc.edu to get the Zoom link for the talk.

slides

4:00 pm, October 27 Mykola Dedushenko Physics and geometry of Bethe/Gauge correspondence I will give a (biased) overview of the program, known as the Bethe/Gauge correspondence, revealing integrability structures (e.g., representations of Yangians) inside supersymmetric gauge theories. Geometric approach to the subject was initiated by Maulik and Okounkov and further developed in many subsequent works. Reinterpreting their ideas in terms of quantum field theory will play a major role in this talk. Based on joint works with N.Nekrasov.
4:00 pm, October 30 Mikhail Kapranov Perverse sheaves and resurgence Perverse sheaves provide a topological counterpart of regular holonomic D-modules, whose solutions are multivalued functions of certain restricted type. Now, much more general multivalued functions (on the complex plane C) have been studied in J. Ecalle's theory of resurgence using, as one of the main tools, additive convolution of analytic functions. The talk, based on joint work in progress with Y. Soibelman, will propose a topological counterpart to the theory of resurgence based on perverse sheaves on C which are algebras with respect to (middle) additive convolutions. Such sheaves typically have infinitely many singular points. In particular, we will argue that the cohomological Hall algebra in a 3-Calabi-Yau situation localizes to such a "resurgent perverse sheaf".
4:00 pm, November 6 Hunter Dinkins q-Hypergeometric Functions and the Geometry of Quiver Varieties Vertex functions are special functions associated to Nakajima quiver varieties. They generalize basic hypergeometric functions, which are q-deformations of classical hypergeometric functions, whose study began with Gauss. In recent years, conjectures originating in physics known as "3d mirror symmetry" have uncovered new properties of these functions. Our main result relates the vertex function of a type A quiver variety with that of the cotangent bundle of a complete flag variety. As a consequence, we are able to prove 3d mirror symmetry of vertex functions for a certain class of type A quiver varieties. Time permitting, we will explain ongoing work to extend these results to bow varieties. No prior knowledge of these topics will be assumed.
Unusual location: Phillips Hall, Room 383
4:00 pm, November 10 Leonid Petrov Stationarity for Colored Interacting Particle Systems on the Ring Recently, there has been much progress in understanding stationary measures for colored (also called multi-species or multi-type) interacting particle systems motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials). I will describe a unified approach to constructing stationary measures for colored ASEP, q-Boson, and q-PushTASEP systems based on integrable stochastic vertex models and the Yang-Baxter equation. Stationary measures become partition functions of new "queue vertex models" on the cylinder, and stationarity is a direct consequence of the Yang-Baxter equation. Our construction recovers and generalizes known stationary measures constructed using multiline queues and the Matrix Product Ansatz. In the quadrant, Yang-Baxter implies a colored version of Burke's theorem, which produces stationary measures for particle systems on the line. We also compute the colored particle currents in stationarity. Joint work with Amol Aggarwal and Matthew Nicoletti.
November 17 No meeting No meeting -- conflict with Workshop on Geometric Representation Theory and Moduli spaces
3:00 pm, February 9, 2024 Jae Hee Lee Quantum Steenrod operations of symplectic resolutions Quantum connections of symplectic manifolds X are flat connections defined from Gromov—Witten theory of X, acting on singular cohomology of X. When the target manifold X is a symplectic resolution, the quantum connection of X is known to recover and generalize well-known flat connections from representation theory. On cohomology of X with coefficients in positive characteristic, Fukaya and Wilkins have defined new operations that deform the action of the classical Steenrod operations using symplectic Gromov—Witten theory. I will discuss results from the study of these quantum Steenrod operations for (conical) symplectic resolutions, in particular their relationship with the mod p quantum connection and its p-curvature. If time permits, I will discuss potential generalizations to quantum K-theory.
3:00 pm, February 16, 2024 Marco Castronovo Curved Fukaya algebras and the Dubrovin spectrum The cohomology ring of a compact symplectic manifold can be deformed by counting pseudo-holomorphic spheres. Fukaya-Oh-Ohta-Ono developed an obstruction theory that explains when Lagrangian submanifolds yield modules over such deformation, based on counting pseudo-holomorphic disks. I will focus on a piece of numerical data coming out of their theory and called curvature, explaining why it must be an eigenvalue of an operator studied by Dubrovin much earlier. I will also outline how this could simplify the calculation of the Fukaya category of G/P.
4:00 pm, March 1, 2024 Prakash Belkale Motivic factorization of KZ local systems and deformations of representation and fusion rings. Let g be a simple Lie algebra over C. The KZ connection is a connection on the constant bundle associated to a set of n finite dimensional irreducible representations of g and a nonzero k, over the configuration space  of n-distinct points on the affine line. Via the work of Schechtman–Varchenko and Looijenga, when k is rational the associated local systems can be seen to be realizations of naturally defined motivic local systems. We prove a basic factorization for the nearby cycles of these motivic local systems as some of the n points coalesce.   This leads to the construction of a family of deformations  of the representation ring of g—we call these enriched representation rings—which allows one to compute the ranks of the Hodge filtration of the associated variations of mixed Hodge structure; in turn, this has applications to both the local and global monodromy of the KZ connection, and of the associated quotient of conformal blocks. In the case of special linear groups we give an explicit algorithm for computing all products in the enriched representation rings.   This is joint work with N. Fakhruddin and S. Mukhopadhyay.
TBD, April 19, 2024 Kiran Kedlaya TBA TBA