Past Seminar Talks
Fall 2025
September 9 (in-person)
Elijah Bodish (IU Bloomington)
Title: Categorified quantum symmetric pairs
Abstract: Quantum symmetric pairs are the quantum group analog of symmetric spaces. It was discovered by Khovanov-Lauda and Rouquier that (Drinfeld-Jimbo) quantum groups can be categorified. These categorifications have had applications to representation theory, topology, and algebraic geometry. The purpose of the talk will be to introduce quantum symmetric pairs (i-quantum groups) via examples and concrete realizations, and then to discuss some instances in which we are beginning to understand how these algebras can be categorified as well.
September 16 (online)
Shon Ngo (ISTA Austria)
Title: Hausel's big algebras, commuting differential operators and Bethe subalgebras of the Yangian
Abstract: Recently, Hausel inroduced the notion of big algebra associated to a representation of a reductive Lie algebra. These algebras are commutative and capture a lot of representation-theoretic information, and also related to the equivariant cohomology of certain singular varieties. In our work, we consider a specific GL_n-module, namely the coordinate ring of the affine space Mat(n,r) and calculate its big algebra generators in terms of differential operators with polynomial coefficients. This construction allows to prove the commutativity of type A big algebras and relate them to the Bethe subalgebra in the Yangian Y(gl_n). Time permitting, we might discuss an analogue of this construction for the ring of odd differential operators.
October 21 (in-person)
Keshav Dahiya (IU Indianapolis)
Title: R-matrices for vector representations of quantum affine algebras
Abstract: In this talk, I will discuss a method for finding explicit expressions of R-matrices acting in representations of quantum affine algebras. The method relies on q-characters and some useful properties of the R-matrices. I will illustrate the method for vector representations of some quantum affine algebras, in particular, for G_2^(1) and D_4^(3). The answer is written in terms of projectors related to the decomposition of the tensor squares with respect to non-affine quantum algebras.
October 28 (in-person)
Yu Li (University of Notre Dame)
Title: Cluster structures, integrable systems and symplectic groupoids
Abstract: We introduce two operations that can be applied to a compatible cluster structure \Phi on a Poisson variety (Y, \pi). (1) If there exists a point y \in Y where the Poisson bivector \pi vanishes, then taking the lowest degree terms of \Phi gives rise to a set \Phi^{\rm low} of pairwise Poisson commutative functions on the tangent space T_y(Y) equipped with the Poisson bivector \pi_0 which is the linearization of \pi. We present a sufficient condition, in terms of the degree of the log-volume form of \Phi, under which \Phi^{\rm low} is an integrable system on (T_yY, \pi_0). When (Y, \pi) is the Poisson dual group of the standard Poisson Lie group GL_n and \Phi is the generalized cluster structure of Gekhtman-Shapiro-Vainshtein, our operation produces an integrable system on gl_n^* which is different than the celebrated Gelfand-Zeitlin integrable system. (2) If the Poisson variety (Y, \pi) integrates into a symplectic groupoid s,t:(G, \Omega) \implies (Y, \pi), then s^* \Phi \cup t^* \Phi is a set of logcanonical functions on (G, \Omega), which, in many examples of representation theoretical interest, can be completed to a compatible cluster structure \Psi. We explain how the mutable variables, frozen variables and cluster mutations of \Psi are related to those of \Phi. When (Y, \pi) is the standard Poisson Lie group GL_n, our operation produces a compatible cluster structure on the Heisenberg double of GL_n. Time permitting, we explain how to relate these two operations by taking the lowest degree terms of \Psi along the identity section of G.
November 4 (in-person)
Siwei Xu (Ohio State University)
Title: Affine Yangians and quantum toroidal algebras in type A
Abstract: The main objects of my talk are the affine Yangian and the quantum toroidal algebra of type A with two deformation parameters, as introduced in the work of Bershtein and Tsymbaliuk (2019). These algebras first appeared in Maulik and Okounkov’s 2012 work, where they were realized in a purely geometric form through the equivariant cohomology of quiver varieties. In this talk, I construct an explicit functor from an appropriate category of representations of the affine Yangian associated with sl(mn) to that of the quantum toroidal algebra associated with sl(m) for all positive integers m and n, which extends the construction of Gautam and Toledano-Laredo (2016).
November 11 (online)
Deniz Kus (TU Munich)
Title: Representations of parabolic quantum affine algebras
Abstract: Quantized enveloping algebras of Kac-Moody algebras and their representation theory have played a significant role in mathematics and physics over the past decades. In this talk, I will discuss the first attempt to quantize a class of equivariant map algebras that realize parabolic subalgebras of affine Kac-Moody algebras. After presenting some structural results, I will introduce the classification of finite-dimensional irreducible representations over fields of characteristic zero, assuming the deformation parameter is not a root of unity. The classification is formulated in terms of Drinfeld polynomials, revealing new phenomena - for instance, for certain maximal parabolic subalgebras, certain divisibility conditions will appear.
November 18 (online)
Clément Delcamp (IHES)
Title: Non-invertible symmetries in one-dimensional quantum lattice models
Abstract: I will present a framework for studying non-invertible symmetries in one-dimensional quantum lattice models. Within this framework, we can systematically construct representatives of symmetric gapped phases, define corresponding gauging operations, and identify the resulting duality relations. I will illustrate these constructions with several concrete examples.
Spring 2025
January 29 (online)
Alexander Voronov (University of Minnesota)
Title: Schwarz’s extended super Mumford form on the super Sato Grassmannian
Abstract: In 1987, Albert Schwarz suggested a formula which extends the super Mumford form from the supermoduli space of super Riemann surfaces into the super Sato Grassmannian. His formula is a remarkably simple combination of super tau functions. The super Mumford form on the supermoduli space is believed to be the main ingredient in computing the superstring partition function and scattering amplitudes. The supermoduli space is known to sit in the super Sato Grassmannian as an orbit of the Neveu-Schwarz (NS) algebra. I will introduce these notions and present a formula for the action of the NS-algebra on super tau functions and show that Schwarz's extended Mumford form is invariant under the NS action, which strengthens Schwarz's proposal that a locus within the Grassmannian can serve as a universal moduli space with applications to superstring theory. This is a joint work with Katherine A. Maxwell, https://arxiv.org/abs/2412.18585.
February 12 (in-person)
Kyungtak Hong (Purdue University)
Title: Orthosymplectic R-matrices
Abstract: In this talk, we present formulas for the finite and affine orthosymplectic R-matrices associated with any parity sequence, evaluated on the first fundamental representation. The finite R-matrices factorize as ordered products of q-exponents, parametrized by positive roots in the reduced root system. Using combinatorial methods in shuffle superalgebras, we compute the evaluation of each q-exponent, thereby deriving the formulas for the finite R-matrices. We then guess the formulas for the affine R-matrices by applying the Yang-Baxterization process to the finite R-matrices, which produces solutions to the Yang-Baxter equation. Finally, we verify their intertwining properties, confirming that the obtained solutions are indeed affine R-matrices. This is joint work with Sasha Tsymbaliuk.
February 19 (online)
Khrystyna Serhiyenko (University of Kentucky)
Title: Consistent dimer models on surfaces with boundary
Abstract: A dimer model is a quiver with faces embedded in a surface, which gives rise to certain Jacobian algebras called dimer algebras. Consistent dimer models on tori have been studied extensively in the physics literature, in relation to phase transitions in solid state physics, while those on the disk are related to Grassmannian cluster algebras. We define and investigate various notions of consistency for dimer models on surfaces with boundary, generalizing various results from the torus and the disk cases. This is joint work with Jonah Berggren.
February 26 (online)
Ryan Kinser (University of Iowa)
Title: Hopf actions and algebras in tensor categories
Abstract: The classical notion of group actions (symmetries) has a natural generalization as Hopf algebra actions (quantum symmetries). At the same time, the classical study of finite-dimensional associative algebras has a natural generalization to algebras in finite tensor categories. This talk will be an introductory overview of these concepts, along with some proposed ideas for extending the fundamentals of the theory of finite dimensional algebras to the setting of finite tensor categories. I'll cover in a little more detail recent joint work with Elise Askelsen about Hopf actions and Ore extensions.
March 5 (online)
Xiaowen Zhu (University of Washington)
Title: Bulk edge correspondence of topological insulators with curved interfaces
Abstract: Topological insulators are one of the central objects in condensed matter physics. They refer to insulators (i.e. Hamiltonians with a spectral gap) to which one can associate a non-trivial topological invariant. When two insulators with distinct topological invariants are glued together, the material becomes a conductor and currents flow only along the edge. Furthermore, the edge conductance is also quantized, and equals the difference of the bulk topological invariants when the edge is straight. This is the well-known bulk-edge correspondence (BEC). In this talk, we discuss BEC for topological insulators with curved interfaces and discuss several related results, including absolutely continuous edge spectrum, Z2 topological insulator cases, etc. The talk is based on a series of joint work with Alexis Drouot and Jacob Shapiro.
March 12 (in-person)
Trung Vu (Yale University)
Title: De Concini-Kac quantum groups at roots of unity revisited
Abstract: Quantum groups were introduced by Drinfeld and Jimbo in mid 1980's. Soon after, quantum groups became a subject of great interest and revealed deep connections with many areas of mathematics and physics, in particular, representation theory. In this talk, we will discuss some aspects of representation theory of quantum groups at roots of unity, in particular, the De Concini-Kac form of quantum groups. This form has been studied in the series of papers by De Concini-Kac-Procesi around 90's in the case when the order of the roots of unity is odd. We will discuss our modification to the De Concini-Kac form to deal with the case when the order of roots of unity is even. This is a joint work with Ivan Losev and Alexander Tsymbaliuk.
March 26 (in-person)
Benjamin Brubaker (University of Minnesota)
Title: A universal lattice model for Hecke-Grothendieck polynomials
Abstract: Many of the special functions we study in geometry, algebra, and the representation theory have recursive definitions via divided difference operators. These include Schubert and Grothendieck polynomials (representing cohomological and K-theoretic classes in the flag variety) and various generalizations and specializations of Macdonald polynomials (including Iwahori Whittaker functions). Kirillov unified all such examples with a five-parameter family known as Hecke-Grothendieck polynomials. We present a new (and strange) family of solvable lattice models whose partition functions realize every such family of Hecke-Grothendieck polynomials, leading to new positivity results resolving conjectures of Kirillov from 2015. What's missing is a complete (multi-parameter) quantum group module interpretation for the lattice model, though we know some special cases, and we invite audience participation to resolve this! Along the way, we'll review how quantum groups give rise to special functions via solvable lattice models, so no prior familiarity is necessary. This is joint work with my PhD student Suki Dasher and prior work of ours with the 2023 Polymath Jr. Program.
April 2 (in-person)
Giovanny Mora (UCLA)
Title: Braided Zestings of Verlinde Modular Categories and Their Modular Data
Abstract: In this talk, we will discuss the procedure of “zesting” in braided fusion categories, a technique that enables the construction of new modular categories from an existing modular category with non-trivial invertible objects. We will present a classification and construction of all possible braided zesting data for modular categories associated with quantum groups at roots of unity. Additionally, we will present the formulas that we have found, based on the root system associated with the quantum group, for the modular data of these new modular categories. This talk is a joint work with Eric Rowell and Cesar Galindo and is based on our paper “Braided Zestings of Verlinde Modular Categories and Their Modular Data".
April 9 (online)
Charles Wang (University of Michigan)
Title: Explicit Landau-Ginzburg models for cominuscule homogeneous spaces
Abstract: While Rietsch has constructed Landau-Ginzburg (LG)models for all partial flag varieties G/P in terms of Lie-theoretic data, it is often desirable to have a further description of these LG models in terms of natural coordinates on these spaces as well. There have been several works which make use of Plucker coordinate descriptions of Rietsch's LG models obtained using heavily type-dependent methods for particular cases such as the Grassmannians Gr(k,n). In this talk, we will present a uniform, type-independent construction of Plucker coordinate LG models for all cominuscule homogeneous spaces. One of our main tools for this is an order-theoretic description of Plucker coordinates which enables us to avoid type-specific arguments.
April 16 (online)
Gabrielle Rembado (Universities of Montpellier)
Title: Deformations and quantizations of moduli spaces of meromorphic connections on curves
Abstract: The de Rham/Betti spaces of wild nonabelian Hodge theory parameterize isomorphism classes of meromorphic connections on principal bundles (or their monodromy/Stokes data). When the base of the bundle is a Riemann surface, they assemble into local systems of (complex) Poisson/symplectic algebraic varieties over any `admissible' deformation of the base: this goes beyond the deformations of pointed Riemann surfaces, due to the additional local moduli of irregular-singular connections at each pole, which lead in particular to cabled braid groups. Moreover, it possible to quantize the moduli spaces of meromorphic connections to construct (projectively) flat vector bundles over the same base, which instead lead to conformal blocks in the WZW model. In this talk we will aim at a review of (a small) part of this story, and then describe extensions in the direction of: (i) deformation quantization (based on work with D. Calaque, G. Felder, R. Wentworth); and (ii) the twisted/ramified case (based on work with P. Boalch, J. Douçot, M. Tamiozzo, D. Yamakawa).
April 30 (in-person)
Shamgar Gurevich (University of Wisconsin-Madison)
Title: How you think on a function defined on 0,1,…,N-1?
Abstract: Between thousand to million times per day, your cellphone calculates the Fourier Transform (FT) of certain functions defined on 0,1,…,N-1, with N large (order of magnitude of thousands and more). The calculation is done using the Fast Fourier Transform (FFT) - discovered by Cooley-Tukey in 1965 and by Gauss in 1805. In the lecture I want to advertise a beautiful way - due to Auslander-Tolimieri—to obtain the FFT as a natural consequence of an answer to the following:
Question. How to think on the space of functions on the set 0,1,…,N-1?
Engineers tell us that there are two answers for this question:
(A) as functions on that set, where 0,1,…,N-1 regarded as times; and,
(B) as functions on that set, where 0,1,…,N-1 regarded frequencies;
and then the FT is an operator translating between the two spaces.
In the lecture, I will explain that there is another answer, i.e., a not so well-known third space (C), of arithmetic nature - discovered by the quantum physicists Joshua Zak, that also gives an answer to the above question, and then the FFT appears simply as the composition of two operators: the one translating between spaces(A) and (C), and the one that translates (C) to (B).
Fall 2024
September 24 (online)
Linhui Shen (Michigan State University)
Title: Cluster Nature of Quantum Groups
Abstract: We present a rigid cluster model to realize the quantum group U_q(g) for g of type ADE. We prove that there is a natural Hopf algebra isomorphism from the quantum group to a quotient algebra of the Weyl group invariants of a Fock-Goncharov quantum cluster algebra. Applying the quantum duality of cluster algebras, we show that the quantum group admits a cluster canonical basis \Theta whose structural coefficients are in N[q^{\pm 1/2}}]. The basis \Theta satisfies an invariance property under the braid group action, the Dynkin automorphisms, and the star anti-involution.
October 1 (in-person)
Vu Trung (University of Illinois Urbana-Champaign)
Title: Arctic Curves of T-system with Slanted Initial Data
Abstract: We study the 𝑇𝑇 -system of type 𝐴∞, also known as the octahedron recurrence/equation, viewed as a 2 + 1-dimensional discrete evolution equation. Generalizing earlier work on arctic curves for the Aztec Diamond obtained from solutions of the octahedron recurrence with “flat" initial data, we consider initial data along parallel “slanted" planes perpendicular to an arbitrary admissible direction (𝑟𝑟, 𝑠𝑠, 𝑡𝑡) ∈ ℤ3 + . The corresponding solutions of the 𝑇𝑇 -system are interpreted as partition functions of dimer models on some suitable “pinecone" graphs introduced by Bousquet-Melou, Propp, and West in 2009. The 𝑇𝑇 -system formulation and some exact solutions in uniform and periodic cases allow us to explore the scaling limit of the corresponding dimer models and to derive exact arctic curves separating the various phases of the system. This approach bypasses the standard general theory of dimers using the Kasteleyn matrix and uses instead the theory of Analytic Combinatorics in Several Variables.
October 15 (online)
Theo Pinet (University of Paris)
Title: Inflations for representations of shifted quantum affine algebras
Abstract: The only finite-dimensional simple Lie algebra admitting a 2-dimensional irreducible representation is sl(2). The restriction functors arising from Dynkin diagram inclusions in (classical) Lie theory are thus in general not essentially surjective on finite-dimensional simple modules. The goal of this talk is to specify whether or not this "surjectivity defect" remains in the case of Finkelberg-Tsymbaliuk's shifted quantum affine algebras (SQAAs). SQAAs are infinite-dimensional associative algebras parametrized by a simple finite-dimensional Lie algebra and a coweight in the corresponding coweight lattice. They appear naturally in the study of Coulomb branches, of quantum integrable systems and of cluster algebras. In this presentation, we will give a brief introduction to the vast representation theory of SQAAs and will state some results about the existence of remarkable modules, that we call "inflations", which are constructed as special preimages for different canonical restriction functors (associated here also to Dynkin diagram inclusions). We will finally, if time permits, discuss potential applications of our results to the study of cluster structures on Grothendieck rings.
October 22 (online)
Shivang Jindal (University of Edinburgh)
Title: Quantum groups from Cohomological Donaldson-Thomas theory
Abstract: In 2010, Kontsevich and Soibelman defined Cohomological Hall Algebras for quivers and potential as a mathematical construction of the algebra of BPS states. These algebras are modeled on the cohomology of vanishing cycles, which makes these algebras particularly hard to study but often result in interesting algebraic structures. A deformation of a particular case of them gives rise to a positive half of Maulik-Okounkov Yangians. The goal of my talk is to give an introduction to these ideas and if time permits; I will explain how for the case of tripled cyclic quiver with canonical cubic potential, this algebra turns out to be one-half of the universal enveloping algebra of the Lie algebra of matrix differential operators on the torus, while its deformation turn out be one half of an explicit integral form of the Affine Yangian of gl(n).
October 29 (online)
Tudor Dimofte (University of Edinburgh)
Title: Tannakian QFT
Abstract: In a topological quantum field theory (TQFT), extended operators are expected to organize themselves into categories, with various additional algebraic structures. However, from a physical perspective, these categories are often difficult to compute or identify directly. I'd like to present one systematic approach to analyzing line-like extended operators in 3d TQFT, as representations of Hopf algebras and generalized quantum groups - by exploiting boundary conditions in the TQFT's to explicitly compute/identify the quantum groups themselves in physics. I'll discuss some applications, from rederiving old results in Chern-Simons theory to new results in supersymmetric gauge theories. Mathematically, our methods are based on an implementation of Tannaka duality in TQFT. Based on joint work with Wenjun Niu.
November 5 (in-person)
Iordanis Romaidis (University of Edinburgh)
Title: On the holonomicity of skein modules
Abstract: Skein theory forms a once-categorified 3d TQFT and assigns skein algebras to surfaces and skein modules to 3-manifolds. Motivated by physics, these modules are expected to satisfy a certain holonomicity property, generalizing Witten's finiteness conjecture of skein modules. In this talk, we will recall the basic notions of skein theory as a deformation quantization theory, and then state and discuss the generalized Witten's finiteness conjecture.
November 14 (in-person)
Sachin Gautam (Ohio State University)
Title: Lattice operators of quantum affine algebra
Abstract: Let g be a finite-dimensional, simple Lie algebra over the field of complex numbers, and U be the quantum, untwisted affine algebra, associated to g. Via Lusztig's q-exponential formulae, it is well known that the affine braid group of g acts on any integrable representation of U. In particular, one obtains an action of the coroot lattice of g on such a representation. In this talk, I will present an explicit formula for these lattice operators on finite-dimensional representations of U, in terms of the generators of its maximal commutative subalgebra in Drinfeld's loop presentation. This formula was obtained in a recent joint work with Valerio Toledano Laredo.
November 19 (in-person)
Joshua Mundinger (University of Wisconsin–Madison)
Title: Hochschild homology of algebraic varieties in characteristic p
Abstract: Hochschild homology is an invariant of noncommutative rings. When applied to a commutative ring, the Hochschild-Kostant-Rosenberg theorem gives a formula for Hochschild homology in terms of differential forms. This formula extends to the Hochschild-Kostant-Rosenberg decomposition for complex algebraic varieties. In this talk, we quantitatively explain the failure of this decomposition in positive characteristic.
December 3 (in-person)
Chris Bairnsfather (Purdue University)
Title: Band Unfolding via the Quadratic Pseudospectrum
Abstract: We use pseudospectral methods to find approximate joint eigenvectors for collections of non-commuting quantum operators. This leads to an efficient technique for unfolding the dispersion curves of a crystalline material to the primitive Brillouin zone when the Hamiltonian might only be invariant under the generators of a sublattice of the crystal lattice. This is illustrated with the 1d and 2d SSH model and graphene.
Spring 2024
January 23 (in-person)
Eric Samperton (Purdue University)
Title: Computing the TVBW 3-manifold invariants from Tambara-Yamagami categories
Abstract: I'll give a quick intro to spherical fusion categories and the Turaev-Viro-Barrett-Westbury construction, which associates an invariant of oriented 3-dimensional manifolds to each such category (and more generally give rise to 3-dimensional topological quantum field theories). Some of the simplest spherical fusion categories are the so-called Tambara-Yamagami categories, which depend on the data of a finite abelian group A, a choice of isomorphism between A and its dual, and a sign +1 or -1. Despite their fairly simple definition, these categories are known to give rise to TVBW invariants that are NP-hard to compute. I'll explain what this means, and then describe my recent work with Colleen Delaney and Clement Maria that establishes an efficient algorithm for computing these invariants on 3-manifolds with bounded first Betti number. As motivation, I will also say a few things about why such complexity/algorithm results are interesting in the context of 3-manifold topology and quantum computation.
January 30 (in-person)
Michael Gekhtman (University of Notre Dame)
Title: Poisson maps as a toll for constructing cluster structures
Abstract: We present a construction for cluster charts in simple Lie groups compatible with Poisson structures in the Belavin-Drinfeld classification. The latter are described by combinatorial data associated with the corresponding root system. We use this data to construct a birational Poisson map from the group to itself that transform a Poisson bracket associated with a nontrivial Belavin-Drinfeld data into the standard one. It allows us to obtain a cluster chart as a pull-back of the Berenstein-Fomin-Zelevinsky cluster coordinates on the open double Bruhat cell. This is a joint work with M. Shapiro and A. Vainshtein.
February 13 (online)
Ryo Fujita (Kyoto University)
Title: Isomorphisms among quantum Grothendieck rings and their cluster theoretical interpretation
Abstract: Quantum Grothendieck ring in this talk is a deformation of the Grothendieck ring of the monoidal category of finite-dimensional modules over the quantum loop algebra, endowed with a canonical basis consisting of the so-called simple (q,t)-characters. We discuss a collection of isomorphisms among the quantum Grothendieck rings of different Dynkin types respecting the canonical bases, via which the (q,t)-characters of non-simply-laced type inherit several good properties from those of the unfolded simply-laced type. We also discuss their cluster theoretical interpretation, which particularly yields non-trivial birational relations among the (q,t)-characters of
different Dynkin types. This is a joint work with David Hernandez, Se-jin Oh, and Hironori Oya.
February 20 (online)
Ellizabeth Kelley (University of Illinois Urbana-Champaign)
Title: Snake Graphs for Graph LP Algebras
Abstract: Graph LP algebras are a generalization of cluster algebras that were introduced by Lam and Pylyavskyy. In joint work with Esther Banaian, Sunita Chepuri, and Sylvester. With Zhang, we provide a combinatorial proof of positivity for certain cluster variables in these algebras. Our proof uses a hypergraph generalization of snake graphs, which were introduced by Musiker, Schiffler, and Williams to prove positivity for cluster algebras from surfaces. In this talk, I will explain our construction without assuming prior knowledge about cluster algebras or snake graphs.
February 28 (in-person)
Taylor Hughes (University of Illinois Urbana-Champaign)
Title: Defeating the Chiral Doubling Theorem: Three paths to new phenomena
Abstract: Over forty years ago, Nielsen and Ninomiya proved that chiral modes cannot exist in discrete systems without a matching anti-chiral partner. While this theorem was proved in the context of simulating the weak interactions of the standard model on a lattice, it is far more general and has implications across a broad spectrum of physics sub-fields. In this talk I will discuss three ways to violate the theorem by avoiding the assumptions. I will illustrate that each type of violation is responsible for a new class of topological phenomena. If time permits I will also discuss a new type of chirality that can appear in only non-relativistic systems in two spatial dimensions and its realization in topo-electric circuits.
March 5 (in-person)
Corey Lunsford (Northwestern University)
Title: Lax Matrices & Clusters in Type A & C q-Deformed Open Toda Chain
Abstract: At the turn of the century, Etingof and Sevostyanov independently constructed a family of quantum integrable systems, quantizing the open Toda chain associated to a simple Lie group. The elements of this family are parameterized by Coxeter words of the corresponding Weyl group. Twenty years later, in the works of Finkelberg, Gonin, and Tsymbaliuk, this was generalized to a family of quantum Toda chains parameterized by pairs of Coxeter words. We show that this family is actually a single cluster integrable system written in different clusters associated to cyclic double Coxeter words. Furthermore, if we restrict the action of Hamiltonians to its positive representation, these systems become unitary equivalent.
March 19 (in-person)
Amanda Young (University of Illinois Urbana-Champaign)
Title: A bulk gap for a truncated Haldane pseudopotential
Abstract: In this talk, we discuss recent work which proved a bulk gap for a truncated version of the 1/3-filled pseudopotential in the infinite cylinder geometry that is expected to well-approximate the original model for small radii. One of the main obstacles for proving this result was the presence of edge states for finite cylinders, which
produce spectral gap estimates that do not accurately reflect the behavior of the bulk gap. To overcome this challenge, a new scheme based off separating edge states and ground states into distinct invariant subspaces was developed, which allowed for a more refined application of gap estimating techniques. This is based on joint work with S. Warzel.
March 26 (in-person)
Katrina Barron (University of Notre Dame)
Title: Irrational vertex operator algebras and graded pseudo-traces for indecomposable non simple modules
Abstract: Irrational vertex operator algebras and graded pseudo-traces for indecomposable non simple modules
Abstract: To construct a Conformal Field Theory (CFT) one needs a category of modules for a vertex operator algebra (VOA) with certain ``nice” properties. If the VOA satisfies a certain finiteness condition, called C_2-cofinite, and has semi-simple representation theory (is rational), then its category of N-gradable modules forms a
Modular Tensor Category (MTC) in which the graded traces are q-series that form an invariant space under the action of the modular group SL2(C). In 2004, Miyamoto proved that much of this structure carries through for C_2-cofinite irrational VOAs if one includes graded ``pseudo-traces” in addition to the graded traces. However very few
such C_2-cofinite irrational VOAs have been constructed, and the machinery necessary to carry out Miyamoto’s construction is very involved. We will present a setting in which the C_2-cofiniteness of V is not necessary, graded pseudo-traces are easily defined, and they satisfy linearity, symmetry, and the logarithmic derivative property.
We show that this setting has interesting applications to, for instance, the two most common VOAs, namely the Heisenberg and Virasoro VOAs.
April 2 (online)
Alexander Garballi (University of Melbourne)
Title: Shuffle algebras and lattice paths
Abstract: In recent years many interesting results in mathematical physics have been obtained with the help of shuffle algebras. A notable example is the trigonometric Feigin—Odesskii (FO) shuffle algebra. In this talk I will discuss a connection between Yang—Baxter integrable vertex models and the commutative subalgebra of the FO shuffle algebra. This connection allows us to compute a family of partition functions of vertex models in terms of elements of the commutative FO shuffle algebra. Our results imply similar relations between other shuffle algebras and integrable vertex models.
April 16 (in-person)
Andrew Hardt (University of Illinois Urbana-Champaign)
Title: Solvable lattice models and the boson-fermion correspondence
Abstract: Consider the (infinite-dimensional) vector space whose basis vectors are indexed by integer partitions. We will discuss two sources of linear operators on this vector space. The first source, solvable lattice models, are ice-like rectangular grids that originated in statistical physics, and which satisfy the famous Yang-Baxter equation. The second source, the boson-fermion correspondence, has connections to soliton solutions of the KP hierarchy. We will discuss the intersection of these two approaches, along with applications to symmetric functions.
April 23 (in-person)
Anthony Giaquinto (Loyola University Chicago)
Title: Some Diagram Algebras and their Schur-Weyl Dualities
Abstract: Many diagram algebras have been well-studied in the context of Schur-Weyl Duality. These include the symmetric group algebras, Brauer algebras, partition algebras and Temperley-Lieb algebras. We will review these contexts with an emphasis on examples and origins of these algebras. Recent new instances of Schur-Weyl duality associated to braid and twin groups will be presented in the context of their Burau representations. Application to the representation theories of these algebras will be given. This is joint work with Stephen Doty.
Fall 2023
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Spring 2023
February 16 (in-person)
Ivan Karpov (MIT)
Title: Nonlinear SRA
Abstract: Let G be a finite subgroup of Sp(V) for V being a symplectic vector space. A simplectic reflection algebra H(V, G, c) is, by definition, an unobstructed deformation of the smash product of G with the Weyl algebra W(V). Etingof and Dolgushev (2005) have conjectured an existence of the similar deformation in the case of V being a general smooth affine symplectic variety. We will prove their conjecture using the results of Losev, Bezrukavnikov-Kaledin, Namikawa, et al.; and discuss the similar statement for V/G replaced by an arbitrary symplectic singularity. This is a joint work with P. Etingof and A. Vitanov.
March 8 (in-person)
Joshua Wen (Northeastern University)
Title: Wreath Macdonald operators
Abstract: Defined by Haiman, wreath Macdonald polynomials are a certain generalization of Macdonald polynomials to wreath products of the symmetric groups with a fixed cyclic group. Through an analogue of the Frobenius characteristic, they can be viewed as partially-symmetric functions. Most of the standard aspects of Macdonald theory have yet to be developed: Pieri rules, norm formulas, etc. In this talk, I will introduce wreath analogues of Macdonald operators. They were discovered by computing the action of certain elements of a quantum toroidal algebra via integral formulas which may be of independent interest. The wreath Macdonald operators seem much more complicated than the original Macdonald operators, and I will give a sense in which this is reflective of the subtler combinatorics present in the wreath setting. This is joint work with Daniel Orr and Mark Shimozono.
March 22 (online)
Ilya Dumanski (MIT)
Title: Geometric approach to Feigin-Loktev fusion product
Abstract: Feigin-Loktev's fusion product of cyclic graded modules over the current algebra was introduced in 1998. Although its definition is elementary, very little is known about its properties in general. We introduce a way to study it geometrically. This approach also gives Borel-Weil-type theorems for the Beilinson-Drinfeld Grassmannian and the global convolution diagram. We will also discuss a relation to the quantum loop group representations.
March 29 (in-person)
Philippe Di Francesco (University of Illinois Urbana-Champaign)
Title: Cluster Algebras, Networks and Integrability
Abstract: Cluster algebras are combinatorial models of discrete time evolution of abstract variables, attached to the nodes of an evolving quiver, and play crucial roles in the geometry of Teichmuller spaces, dimer statistical models, representation theory, and more. In this talk we concentrate on the so-called T-system cluster algebra and the related octahedron recurrence. We show how to explicitly solve such recurrences by means of flat connections thus displaying the hidden integrable nature of the models. This gives an alternative viewpoint on the Lax matrices of Coxeter-Toda quantum systems and their integrable structure (based on joint works with Rinat Kedem).
April 5 (online)
Jean-Emile Bourgine (University of Melbourne)
Title: Shifted quantum groups in Algebraic Engineering
Abstract: Geometric engineering refers to the study of gauge theories as a low-energy limit of string theory in specifically constructed geometric backgrounds. Algebraic engineering combines this geometric approach with the integrable properties of some supersymmetric gauge theories to propose a construction of their BPS observables using the representation theory of quantum groups. In this talk, I will briefly review this construction. In the second part, I will recall the notion of "shifted quantum groups" which is useful to describe gauge theories with matter hypermultiplets. This subtle modification of the original definition of quantum groups brings more flexibility in the representation theory. I will present several new representations for the shifted quantum affine sl(2) and quantum toroidal gl(1) algebras, and explain how they enter in the construction of gauge theory observables.
April 12 (in-person)
Mykola Dedushenko (SCGP)
Title: Integrability structures in QFT from SUSY defects
Abstract: I will review how the algebra of supersymmetric interfaces in QFT can be used to realize interesting algebro-geometric structures, such as stable envelopes and chamber R-matrices, underpinning connections between SUSY gauge theories and integrable models, known as the Bethe/Gauge correspondence. Based on work with N. Nekrasov
Fall 2022
October 5 (online)
Huafeng Zhang (University of Lille)
Title: Associators for one dimensional representations of shifted quantum affine algebras
Abstract: To a finite dimensional complex simple Lie algebra g one attaches the quantum affine algebra in two equivalent ways, the Drinfeld-Jimbo quantum group of the affine Lie algebra of g, and the Drinfeld affinization of the quantum group of g. Modifying the Drinfeld affinization procedure Finkelberg-Tsymbaliuk defined the shifted quantum affine algebras to study K-theoretical Coulomb branches. In this talk I will explain a polynomliality property for the Drinfeld-Jimbo coproduct of the quantum affine algebra. For g of type A, this leads to nontrivial associator maps for triple tensor products of representations of shifted quantum affine algebras where the middle factor is one dimensional.
October 19 (online)
Carlo Meneghelli (University of Parma)
Title: Pre-fundamental representations for the Hubbard model and AdS/CFT
Abstract: There is a class of representations of quantum groups, referred to as prefundamental representations, that plays an important role in the solution of integrable models. The first example of such representations was given by V. Bazhanov, S. Lukyanov and A. Zamolodchikov in the context of two dimensional conformal field theory in order to construct Baxter Q-operators as transfer matrices. At the same time, there is a rather exceptional quantum group that governs the integrable structure of the one dimensional Hubbard model and plays a fundamental role in the AdS/CFT correspondence. In this talk I will introduce prefundamental representations for this quantum group, explain their basic properties and discuss some of their applications.
October 28 (in-person)
Daniele Valeri (La Sapienza University of Rome)
Title: Integrable triples in simple Lie algebras
Abstract: We define integrable triples in simple Lie algebras and classify them, up to equivalence. The classification is used to show that all (but few exceptions) classical affine W-algebras W(g,f), where g is a simple Lie algebra and f a nilpotent element, admit an integrable hierarchy of bi-Hamiltonian PDEs. This integrable hierarchy generalizes the Drinfeld-Sokolov hierarchy which is obtained when f is the sum of negative simple root vectors.
November 2 (in-person)
Noah Snyder (IU Bloomington)
Title: Towards the Quantum Exceptional Series
Abstract: Many Lie algebras fit into discrete families like GL_n, O_n, Sp_n. By work of Brauer, Deligne and others, the corresponding planar algebras fit into continuous familes GL_t and OSp_t. A similar story holds for quantum groups, so we can speak of two parameter families (GL_t)_q and (OSp_t)_q. These planar algebras are the ones attached to the HOMFLY and Kauffman polynomials. There are a few remaining Lie algebras which don’t fit into any of the classical families: G2, F4, E6, E7 and E8. By work of Deligne, Vogel, and Cvitanovic, there is a conjectural 1-parameter continuous family of planar algebras which interpolates between these exceptional Lie algebras. Similarly to the classical families, there ought to be a 2-parameter family of planar algebras which introduces a variable q, and yields a new exceptional knot polynomial. In joint work with Scott Morrison and Dylan Thurston, we give a skein theoretic description of what this knot polynomial would have to look like. In particular, we show that any braided tensor category whose box spaces have the appropriate dimension and which satisfies some mild assumptions must satisfy these exceptional skein relations.
November 30 (online)
Tomas Prochazka (Munich University)
Title: Bethe ansatz in 2d conformal field theory
Abstract: The usual approach used to study 2d CFT relies on the Virasoro algebra and its representation theory. Moving away from the criticality, this infinite dimensional symmetry is lost so it is useful to study aspects of 2d CFT which are more robust with respect to certain deformations. There is an infinite family of commuting higher spin Hamiltonians that one can construct out of Virasoro generators and perhaps surprisingly two different sets of Bethe ansatz equations that can be used to diagonalize these (one by Bazhanov-Lukyanov-Zamolodchikov and another by Litvinov). I want to discuss these constructions as well as their relation to W algebras and Yangian symmetry.
December 7 (online)
Rouven Frassek (University of Modena)
Title: Non-compact spin chains and integrable particle processes
Abstract: I will discuss the relation between non-compact spin chains studied first in the context of high energy physics following ideas of Lipatov, Faddeev and Korchemsky and the zero-range processes introduced by Sasamoto-Wadati, Povolotsky and Barraquand-Corwin. The main difference compared to the prime examples of integrable particle processes, namely the SSEP and the ASEP, is that for the models discussed in this talk several particles can occupy one and the same site. I will introduce integrable boundary conditions for these models that are obtained from the boundary Yang-Baxter equation and which allow to define analogues of the open SSEP and ASEP with boundary reservoirs. Finally, for the rational case with symmetric hopping rates, I will present an explicit mapping of these boundary driven models to equilibrium. This mapping allows us to obtain closed-form solutions of the probabilities in steady state and of k-point correlations functions.
Spring 2022
January 26 (in-person)
Matvei Libine (IU Bloomington)
Title: Quaternionic Analysis, Vacuum Polarization and Quaternionic Algebras
Abstract: I will give a quick colloquium-style introduction to quaternionic analysis. Then I will present recent developments in quaternionic analysis from the point of view of representation theory of the conformal group GL(2,H) (group locally isomorphic to O(5,1)). In particular, I will talk about connections with vacuum polarization diagrams from particle physics and then about a new algebra structure on quaternionic functions that commutes with the action of GL(2,H). I will not assume any knowledge of quaternionic analysis from the audience.
March 2 (in-person)
Sasha Tsymbaliuk (Purdue University)
Title: BGG-type relations for transfer matrices of rational spin chains and the shifted Yangians
Abstract: In this talk, I will discuss: (1) the new BGG-type resolutions of finite dimensional representations of simple Lie algebras that lead to BGG-relations expressing finite-dimensional transfer matrices via infinite-dimensional ones, (2) the factorization of infinite-dimensional ones into the product of two Q-operators, (3) the construction of a large family of rational Lax matrices from antidominantly shifted Yangians. This talk is based on the joint works with R.Frassek, I. Karpov, and V.Pestun.
March 9 (in-person)
Ilya Kachkovskiy (Michigan State University)
Title: Perturbative Diagonalisation for Quasiperiodic Operators with Monotone Potential
Abstract: We consider quasiperiodic operators on with unbounded monotone sampling functions ("Maryland-type”), and construct the Rayleigh—Schrodinger formal perturbation series for the eigenvalues and eigenvectors. We discuss the combinatorial structure and cancellations in such series and sufficient conditions for their convergence. This allows to establish Anderson localization for several classes of mononone quasiperiodic operators
by constructing explicit converging expansions for eigenvalues and eigenvectors. If time permits, we will discuss cases where the requirement of strict monotonicity can be relaxed (the talk is based on the joint work with S. Krymskii, L. Parnovskii, and R. Shterenberg, both published and in progress).
March 23 (online)
Andrea Appel (University of Parma)
Title: Continuum Quantum Groups
Abstract: Continuum Kac-Moody algebras are a natural generalisation of standard Kac-Moody algebras, where the role of simple roots is rather played by a “continuous" space of real roots. In this talk, I will provide an overview of such exotic Lie algebras and their quantum groups, describing both the combinatorics which underlies their definition and their “true” origin as Hall algebras of certain categories of geometric origin. This is based on joint work with F. Sala and O. Schiffmann.
March 30 (in-person)
Joshua Wen (Northeastern University)
Title: The quantum (deformed) Harish-Chandra isomorphism for GL_n
Abstract: The deformed Harish-Chandra isomorphism of Etingof-Ginzburg (and Gan-Ginzburg) identifies the spherical type A rational Cherednik algebra with a quantization of the Hilbert scheme of points on the plane. As a Nakajima quiver variety, the Hilbert scheme can be presented as a Hamiltonian reduction, and likewise its quantization is obtained as a quantum Hamiltonian reduction of the ring of differential operators on a vector space of matrices. I will present a "multiplicative" analogue of this isomorphism. The rational Cherednik algebra is replaced with the usual Cherednik DAHA while the Hilbert scheme is replaced with a multiplicative quiver variety that also happens to be a character variety for the torus. The quantized character variety has a collection of subalgebras indexed by relatively prime integers (a,b), each generated by quantized Wilson loops supported on the (a,b) torus knot. Under this isomorphism, this (a,b) subalgebra is sent to the slope a/b subalgebra of the spherical DAHA.
April 6 (online)
Jose Simental (MPIM Bonn)
Title: Braids, links and cluster algebras
Abstract: To a positive braid \beta on n strands we associate an affine algebraic variety X(\beta) as the solution space of an incidence problem in the flag variety of GL_n(C). This variety has a number of nice properties, for example, it is smooth and admits smooth compactifications which correspond to different braid words for \beta. Many interesting varieties appearing in Lie theory, such as open Richardson varieties in type A, appear as varieties of the form X(\beta) for special types of braids \beta. I will define these varieties and explain some of their combinatorial and geometric properties. Most importantly, I will describe an A-cluster structure on X(\beta) defined using the formalism of algebraic weaves, a graphical calculus that allows us to find open tori in X(\beta). In particular, this yields a cluster structure on type A Richardson varieties. Finally and time permitting, I will elaborate on the relationship that X(\beta) bears to the Legendrian link appearing as the Legendrian (-1)-closure of the braid \beta, and how a cluster structure on X(\beta) may help in the understanding of the symplectic geometry of this link. This talk is based on joint works with Roger Casals, Eugene Gorsky and Mikhail Gorsky.
April 13 (in-person)
Nathan Haouzi (IAS)
Title: What can little strings teach us about the geometric Langlands program?
Abstract: In its simplest incarnation, the geometric Langlands program was defined by Beilinson and Drinfeld in the late 90’s as relating, on one side, a flat connection on a Riemann surface, and on the other side, a more sophisticated structure known as a D-module. Since its inception, this conjectured correspondence has been a highly active and fruitful topic of research both for mathematicians and theoretical physicists. In this talk, we will review a generalization of the correspondence known as the quantum q-Langlands program, due to Aganagic-Frenkel-Okounkov, which establishes an isomorphism between q-deformed versions of conformal blocks for a W-algebra on one side, and for a Langlands dual affine Lie algebra on the other side. The physical motivation for this isomorphism is known as little string theory, a powerful framework for studying the correspondence. For instance, we will invoke little string motivations to give a precise mathematical formulation of ramification, or adding punctures on the Riemann surface, in this q-Langlands program. As an application, when the Lie algebra is specialized to be su(2), one obtains a new (dual) perspective on recent results of Nekrasov and Tsymbaliuk.
April 20 (in-person)
Gus Schrader (Northwestern University)
Title: Whittaker Functions for Quantum Groups
Abstract: In one of the first applications of representation theory to quantum integrability, Kostant showed in the 1970's that the classical Whittaker functions for split real Lie groups are eigenfunctions for quantum Toda chains. These Whittaker functions admit a q-deformation - now associated to split real quantum groups - which are eigenfunctions for the q-difference analogs of the Toda chains. These q-deformed Whittaker functions have turned out to have applications across different areas of mathematics and physics: they govern the decomposition of a tensor product of principal series representations of the split real quantum group into irreducibles, provide the key to the proof of the modular functor conjecture in quantum higher Teichmuller theory, and they encode the BPS spectrum of supersymmetric quiver gauge theories. I will survey these results, and explain how they arise from the cluster-algebraic construction of the Toda chain and its eigenfunctions.
April 27 (online)
Curtis Wendlandt (University of Saskatchewan)
Title: The restricted quantum double of the Yangian
Abstract: Many advances in the representation theory of quantum groups have been inspired by a desire to understand, and produce, solutions of the quantum Yang-Baxter equation (QYBE), called R-matrices. One of the most basic tools in this regard is Drinfeld's quantum double method which takes as input a Hopf algebra (subject to certain constraints) and outputs a second Hopf algebra roughly twice as large which, crucially, comes equipped with a universal R-matrix. A large number of important quantum groups can be realized this way, including the Drinfeld-Jimbo and quantum affine algebras. A main exception to this pattern is the Yangian associated to a simple Lie algebra. This is a remarkable quantum group which admits a universal R-matrix R(z) satisfying a parameter dependent QYBE, but which does not appear to arise from the quantum double construction. The goal of this talk is to address this phenomenon by explaining that, in a certain sense, a Yangian is nearly equal to its own quantum double, and R(z) can in fact be obtained as a byproduct of this construction. Making this rigorous involves a proof of a conjecture which appeared in the 1990's in the work of Khoroshkin and Tolstoy.
Fall 2021
September 2 (in-person)
Arun Debray (Purdue University)
Title: Topological phases and topological field theories
Abstract: It is generally believed that the low-energy effective theory of a topological phase of matter is a topological field theory (TFT), providing an avenue for mathematical work in TFT to address questions in condensed-matter physics. However, making this belief precise is a difficult open problem. In this talk, I'll describe what we do know about this correspondence between topological phases and TFTs, delving into the easier invertible case as well as my work on a particular example in the noninvertible case. With the remaining time, I’ll discuss a potential next step of understanding this correspondence for phases with spatial symmetries.
September 16 (online)
Peter Koroteev (UC Berkeley)
Title: q-Opers, QQ-Systems and Bethe Ansatz
Abstract: We introduce the notions of (G,q)-opers and Miura (G,q)-opers, where G is a simply-connected complex simple Lie group, and prove some general results about their structure. We then establish a one-to-one correspondence between the set of (G,q)-opers of a certain kind and the set of nondegenerate solutions of a system of Bethe Ansatz equations. Additionally we associate to a (G,q)-oper a class of meromorphic sections of a G-bundle, satisfying certain difference equations, which we refer to as generalized q-Wronskians. We show that the QQ-systems and their extensions emerge as the relations between generalized minors, thereby putting the Bethe Ansatz equations in the framework of cluster mutations known in the theory of double Bruhat cells.
September 23 (online)
HyunKyu Kim (Ewha Womans University)
Title: Three dimensional construction of the Virasoro-Bott group
Abstract: I will present a three-dimensional geometric construction of the Virasoro-Bott group, which is a central extension of the group of diffeomorphisms of the circle. I will first review an analogous construction in 1980's of a central extension of the loop group, which is the group of smooth maps from the circle to a Lie group, where the three dimensional geometry is used in the Wess-Zumino term. I will discuss the corresponding Lie algebra constructions and possible generalizations. Most of the talk will be made accessible to graduate students. Based on the joint work 2107.11693 with Igor B. Frenkel.
September 30 (online)
Miroslav Rapcak (UC Berkeley)
Title: M5-branes in toric 3-folds and vertex operator algebras
Abstract: M-theory admits six-dimensional objects called M5-branes. Let us place an M5-brane on a product of a Riemann surface with a four-dimensional manifold. The low energy behavior of the M5-brane can be described by a quantum field theory living on its support. Compactifying such a theory on the four-manifold leads to an effective description in terms of a theory on the Riemann surface containing local operators forming a vertex operator algebra. Compactifying on the Riemann surface leads to a description in terms of a 4d theory living on the four-manifold with sectors of different instanton numbers. This physical picture provides an explanation for the famous work of Nakajima, who showed a rather close correspondence between moduli spaces of instantons on four-manifolds and the representation theory of vertex operator algebras. Recently, the correspondence was extensively enlarged by allowing more general configurations of intersecting M5-branes relating spiked instantons of Nekrasov with a larger class of vertex operator algebras introduced in my work with Gaiotto and Prochazka.
October 7 (online)
Rouven Frassek (University of Modena)
Title: Baxter Q-operators and functional relations for the rational so(2r) spin chain
Abstract: I will review the QQ-system and oscillator construction of Baxter Q-operators for su(r+1) spin chains and discuss its generalisation to so(2r) spin chains.
October 21 (online)
Luan Bezerra (University of Sao Paulo)
Title: Partitions with parity and representations of quantum toroidal superalgebras.
Abstract: The representation theory of quantum toroidal (super)algebras is a very technical and difficult subject. On the other hand, a large class of modules where the central element C acts by 1 have an easy description through the combinatorial framework of partitions with parity. This combinatorics is not only interesting in its own right, but it is expected to be related to other concepts such as crystal bases, fixed points of the moduli spaces of BPS states, and equivariant K-theory of moduli spaces of maps. In this talk, I will explain how to construct these modules. This is a joint project with Evgeny Mukhin.
October 28 (in-person)
Colleen Delaney (IU Bloomington)
Title: Knots and modular isotopes
Abstract: The algebraic theory of quasiparticles called anyons in certain 2-dimensional topological phases of matter is given by a unitary modular tensor category (UMTC). Until recently all known examples of UMTCs were determined by two invariants called the modular data: the framed link invariants that a UMTC assigns to a once-twisted unknot and the Hopf link. These invariants can be thought of as topological Feynman diagrams that capture the self and mutual statistics of the anyons in question. However, there now exists examples of different UMTCs with the same modular data, called modular isotopes, which require stronger link invariants to detect and new theory to interpret.
November 4 (in-person)
Eugene Rabinovich (University of Notre Dame)
Title: Factorization Algebras for Bulk-Boundary Systems
Abstract: A factorization algebra is a cosheaf-like object that is meant to model the observables of a
quantum field theory. Indeed, Costello and Gwilliam have shown how to construct a factorization algebra of observables for any perturbative quantum gauge theory. In my dissertation, I have extended their results to bulk-boundary systems on manifolds with boundary. In this talk, I will provide an introduction to factorization algebras and a survey of some of the low-dimensional bulk-boundary systems to which the techniques of my dissertation apply.
November 11 (in-person)
Javier Zuniga (Universidad del Pacifico)
Title: A combinatorial model for bordered stable Riemann surfaces
Abstract: Moduli spaces of Riemann surfaces are ubiquitous in many branches of mathematics and physics. Different applications require the surfaces considered to be marked, stable, bordered, etc. One way to study these surfaces and their moduli is to consider combinatorial models known as ribbon (or fat) graphs. We will review these models and propose an extension to bordered stable surfaces using the double construction.
December 2 (online)
Yu-An Chen (The Joint Center for Quantum Informationand Computer Science)
Title: Classification of invertible fermionic topological phases by G-crossed braided tensor category
Abstract: The integer quantum Hall states, the quantum spin Hall insulator, and the p-wave topological
superconductor each have an important place in condensed matter physics due to their quantized symmetry-protected topological invariants. These systems have a unique ground state on any closed manifold in (2+1) dimensions and are examples of 'invertible' topological phases of fermions. Here I will describe a general theory describing the universal properties of invertible phases, and classifies them based on their symmetries. This approach is 'categorical': it does not depend on microscopic models. Our theories can be considered as the symmetry-enriched Kitaev's 16-fold way. Some new applications of the theory include an interacting version of the 'tenfold way' classification of topological insulators and superconductors, and also the prediction of an interesting invertible phase.
December 9 (in-person)
Sachin Gautam (Ohio State University)
Title: R-matrices and Yangians
Abstract: An R-matrix is a solution to the Yang-Baxter equation (YBE), a central object in Statistical Mechanics, discovered in 1970's. The R-matrix also features prominently in the theory of quantum groups formulated in the eighties. In recent years, many areas of mathematics and physics have found methods to construct R-matrices and solve the associated integrable system. In this talk I will present one such method, which produces meromorphic solutions to (YBE) starting from the representation theory of a family of quantum groups called Yangians. Our techniques give (i) a constructive proof of the existence of the universal R-matrix of Yangians, which was obtained via cohomological methods by Drinfeld in 1983, and (ii) prove that Drinfeld's universal R-matrix is analytically well behaved. This talk is based on joint works with Valerio Toledano Laredo and Curtis Wendlandt.