Lecture title: Quantum Toroidal Symmetries in Gauge and String Theory 

Lecture abstract: The following topics will be covered.

1. Quantum toroidal symmetries: What is a quantum toroidal algebra? Applications in mathematics and physics.

2. Quantum toroidal gl(1): Definition, representations (vector, Fock H&V, MacMahon), vertex operators. 

3. (p,q)-brane web and algebraic engineering: 5d N=1 gauge theories, (p,q)-brane realization, correspondence with webs of representations. 

4. 5d AGT correspondence: From free field representations to q-deformed W-algebras, role of Miki's automorphism. 

5. Orbifolds and Bethe/gauge correspondence: Surface defect partition functions, quantum toroidal gl(N) with a cyclic shift

Seminar title: Free field representations of quantum groups and q-deformed W-algebras through cluster algebras (slides)

Seminar abstract: Following the development of the AGT correspondence, new relations between free field representations of quantum groups and W-algebras were obtained. The simplest one is the homomorphism between the level $(N,0)$ horizontal representation of the quantum toroidal gl(1) algebra and (dressed) q-deformed $W_N$ algebras. In this talk, I will explain how to extend this type of relations to the Wakimoto representations of quantum affine sl(N) algebras using the 'surface defect' deformation of the quantum toroidal sl(N) algebra.

Lecture title: Hall algebras and Hopf Algebras (slides)

Lecture abstract: The goal of this two-hour talk is to provide a concise introduction to the theory of (Cohomological) Hall Algebras (COHAs) from the perspective of Hopf algebras. In the first hour, I will focus on Hall algebras associated with abelian categories, particularly the Hall algebra of nilpotent representations of a quiver. 

In the second hour, I will turn to the example of the quantum toroidal algebra of gl_1, first presenting it via the Hall algebra of coherent sheaves on an elliptic curve, and then explaining its realization as a K-theoretic Hall algebra of the one-loop quiver. In the final part of the talk, I will place this example in a broader context, discussing the theory of two-dimensional Cohomological Hall algebras of quivers and the associated Maulik–Okounkov Yangians, as well as the two-dimensional K-theoretic Hall algebras of quivers and the Okounkov–Smirnov quantum loop algebras.

No prior knowledge of Hall algebras or their refined versions (cohomological, K-theoretic, etc.) will be assumed. However, some familiarity with the basic notions of Hopf algebras may be helpful.

Seminar title: COHAs of coherent sheaves on smooth surfaces and affine Yangians (slides)

Seminar abstract: The aim of this talk is to present a (conjecturally) new half of the affine Yangian of type ADE, which arises naturally from the theory of Cohomological Hall Algebras (COHAs) associated to coherent sheaves on a minimal resolution X of a Kleinian singularity. The construction uses the derived McKay correspondence and the variation of t-structures to establish a connection between the COHA of the affine ADE quiver and the COHA of the surface X.

Seminar title: Infinitesimal braidings and quasitriangular bialgebras

Seminar abstract: In this talk we present an infinitesimal version of a braided monoidal category, that we call braided pre-Cartier, whose corresponding infinitesimal braiding is a first-order deformation of the given braiding. This extends the (symmetric) Cartier categories, where an additional commutativity of the infinitesimal braiding with the involutive braiding is required. 

It is known that the monoidal category of modules over a bialgebra is braided precisely if the bialgebra is quasitriangular, i.e. it is equipped with a so-called universal R-matrix which is a solution of the quantum Yang-Baxter equation. We consider the pre-Cartier structure for the categories of modules over a quasitriangular bialgebra and we characterize the resulting infinitesimal R-matrices on the bialgebra. Among their main properties, infinitesimal R-matrices are 2-cocycles in Hochschild cohomology and they satisfy an infinitesimal quantum Yang-Baxter equation. We provide explicit examples of infinitesimal R-matrices on some known Hopf algebras, such as on the Hopf algebras E(n) which are a generalization of the Sweedler’s Hopf algebra E(1). This talk is based on the following papers.

[1] Ardizzoni A., Bottegoni L., Sciandra A.,Weber T., Infinitesimal braidings and pre-Cartier bialgebras, Commun. Contemp. Math. 27, no. 5 (2025) 2450029.

[2] Bottegoni L., Renda F., Sciandra A., Infinitesimal R-matrices for some families of Hopf algebras, preprint arXiv:2412.02350.

Seminar title: Spiralling branes, R-matrices and elliptic integrable systems

Seminar abstract: I will explain how R-matrices of quantum toroidal algebras arise from a certain brane setup in Type IIB string theory. String theory implies relations fo the R-matrices and eventually determines their explicit form. I will then show that such R-matrices lead to a new compact formulation of quantum trigonometric and elliptic Ruijsenaars-Schneider type integrable systems. They will also provide a link to some new elliptic integrable systems recently proposed by Koroteev and Shakirov.