ICTS

During these lectures I will cover the following topics:
(1) Motivation: total positivity
(2) Introduction to cluster algebras
(3) cluster structures on partial flag varieties
(4) realization of configuration spaces in quantum field theory as partial flag varieties
(5) applications of cluster structures to scattering amplitudes

Cluster algebras, introduced by Fomin and Zelevinsky, are commutative algebras characterised by intricate combinatorial structures and have applications across geometry and Lie theory, including examples such as Grassmannians, double Bruhat cells, and open Richardson varieties. In this talk, we will explore the Frobenius categorification of cluster algebras with coefficients. Using the additive categorification framework developed by Jensen--King--Su, we will explicitly determine the g-vectors of Plücker coordinates for the Grassmannian variety with respect to the triangular initial seed. This talk is based on a joint work with Bernhard Keller https://arxiv.org/pdf/2410.01037.

The theory of flag varieties attracts a lot of attention due to a very rich structure and a huge number of applications in various fields of mathematics. Algebraic geometry, representation theory and combinatorics play major role in the theory; merging together they provide powerful techniques and allow the derivation of deep and intriguing theorems. Flag varieties are defined as quotients of Lie groups by parabolic subgroups, they are the central objects in geometric representation theory and capture many properties of the corresponding Lie groups and Lie algebras. However, their importance is seen not only in the Lie theory, but also in many neighbouring fields. In particular, various degenerations provide connections with other classes of groups and varieties, such as toric varieties. We will review main definitions and constructions and give various examples.

In her pioneering works V.Lakshmibai constructed an explicit degeneration of Grassmannians and flag varieties to certain normal toric varieties. The construction admits numerous generalizations and has many deep and surprising applications including constructions of bases in highest weight irreducible representations. The degenerations in question allow one to connect seemingly unrelated objects and to use these connections to solve complicated problems of geometric, algebraic and combinatorial nature. One of the examples is provided by the theory of quiver Grassmannians -- a far reaching generalizations of the classical Grassmann varieties. We will explain how do quiver Grassmannians enter the story, why are they useful for the description of various degenerations and what is the outcome of the use of this powerful machinery.

The classical flags are finite-dimensional projective algebraic varieties defined via finite-dimensional Lie groups. The construction admits a natural generalization in the infinite-dimensional world using the theory of affine Kac-Moody Lie algebras and affine Kac-Moody Lie groups. This generalization is natural and important in various problems and constructions (e.g. in mathematical physics) and leads to the so-called flag ind-varieties. These are infinite-dimensional objects which can be obtained as direct limits of embedded finite-dimensional projective algebraic varieties. The finite-dimensional pieces showing up in this construction are known as affine Schubert varieties and play an important role in the theory. It turned out that quiver Grassmannians (for cyclic equioriented quivers) are also useful in this context. We will define the relevant quiver Grassmannians and describe the corresponding degenerations.

Quantum affine algebras are important examples of Drinfeld-Jimbo quantum groups. Their representation theory is very rich and it has been investigated intensively during the past thirty five years from different point of views. Relatively recently, it was discovered that the finite-dimensional representations can be studied from the point of view of cluster algebras (remarkable commutative algebras with distinguished set of generators obtained from inductive processes). The aim of these lectures will be to explain this connection, and some of the developments in this direction.

As a modern analogue and generalization of the representation-directed algebras, we introduce the notion of brick-directed algebras. We study some main properties of these algebras and show how this new family gives a vast generalization of the classical notion and also includes various types of algebras which are of infinite representation type. A key tool in our study is the notion of brick-splitting torsion pairs, which itself is a novel generalization of splitting torsion pairs. Through these generalizations, we obtain several interesting results on the modern and classical families of algebras. If time permits, we also discuss how brick-directed algebras give a full classification of those algebras whose lattice of torsion classes is trim. This is based on joint work with Sota Asai, Osamu Iyama, and Charles Paquette.

The theory of flag varieties attracts a lot of attention due to a very rich structure and a huge number of applications in various fields of mathematics. Algebraic geometry, representation theory and combinatorics play major role in the theory; merging together they provide powerful techniques and allow the derivation of deep and intriguing theorems. Flag varieties are defined as quotients of Lie groups by parabolic subgroups, they are the central objects in geometric representation theory and capture many properties of the corresponding Lie groups and Lie algebras. However, their importance is seen not only in the Lie theory, but also in many neighbouring fields. In particular, various degenerations provide connections with other classes of groups and varieties, such as toric varieties. We will review main definitions and constructions and give various examples.

In her pioneering works V.Lakshmibai constructed an explicit degeneration of Grassmannians and flag varieties to certain normal toric varieties. The construction admits numerous generalizations and has many deep and surprising applications including constructions of bases in highest weight irreducible representations. The degenerations in question allow one to connect seemingly unrelated objects and to use these connections to solve complicated problems of geometric, algebraic and combinatorial nature. One of the examples is provided by the theory of quiver Grassmannians -- a far reaching generalizations of the classical Grassmann varieties. We will explain how do quiver Grassmannians enter the story, why are they useful for the description of various degenerations and what is the outcome of the use of this powerful machinery.

The classical flags are finite-dimensional projective algebraic varieties defined via finite-dimensional Lie groups. The construction admits a natural generalization in the infinite-dimensional world using the theory of affine Kac-Moody Lie algebras and affine Kac-Moody Lie groups. This generalization is natural and important in various problems and constructions (e.g. in mathematical physics) and leads to the so-called flag ind-varieties. These are infinite-dimensional objects which can be obtained as direct limits of embedded finite-dimensional projective algebraic varieties. The finite-dimensional pieces showing up in this construction are known as affine Schubert varieties and play an important role in the theory. It turned out that quiver Grassmannians (for cyclic equioriented quivers) are also useful in this context. We will define the relevant quiver Grassmannians and describe the corresponding degenerations.

Quantum affine algebras are important examples of Drinfeld-Jimbo quantum groups. Their representation theory is very rich and it has been investigated intensively during the past thirty five years from different point of views. Relatively recently, it was discovered that the finite-dimensional representations can be studied from the point of view of cluster algebras (remarkable commutative algebras with distinguished set of generators obtained from inductive processes). The aim of these lectures will be to explain this connection, and some of the developments in this direction.

Let G=GLn(Fq) be the general linear group over a finite field. The Jacquet module is an important tool for understanding the structure of representations of G. For a cuspidal representation of G, the Jacquet module is always trivial, which motivates the study of its twisted versions. In this talk, we will discuss recent results about the structure of twisted Jacquet modules for cuspidal representations of G. This is based on joint work with Kumar Balasubramanian and Krishna Kaipa.

The theory of flag varieties attracts a lot of attention due to a very rich structure and a huge number of applications in various fields of mathematics. Algebraic geometry, representation theory and combinatorics play major role in the theory; merging together they provide powerful techniques and allow the derivation of deep and intriguing theorems. Flag varieties are defined as quotients of Lie groups by parabolic subgroups, they are the central objects in geometric representation theory and capture many properties of the corresponding Lie groups and Lie algebras. However, their importance is seen not only in the Lie theory, but also in many neighbouring fields. In particular, various
degenerations provide connections with other classes of groups and varieties, such as toric varieties. We will review main definitions and constructions and give various examples.

In her pioneering works V.Lakshmibai constructed an explicit degeneration of Grassmannians and flag varieties to certain normal toric varieties. The construction admits numerous generalizations and has many deep and surprising applications including constructions of bases in highest weight irreducible representations. The degenerations in question allow one to connect seemingly unrelated objects and to use these connections to solve complicated problems of geometric, algebraic and combinatorial nature. One of the examples is provided by the theory of quiver Grassmannians -- a far reaching generalizations of the classical Grassmann varieties. We will explain how do quiver Grassmannians enter the story, why are they useful for the description of various degenerations and what is the outcome of the use of this powerful machinery.

The classical flags are finite-dimensional projective algebraic varieties defined via finite-dimensional Lie groups. The construction admits a natural generalization in the infinite-dimensional world using the theory of affine Kac-Moody Lie algebras and affine Kac-Moody Lie groups. This generalization is natural and important in various problems and constructions (e.g. in mathematical physics) and leads to the so-called flag ind-varieties. These are infinite-dimensional objects which can be obtained as direct limits of embedded finite-dimensional projective algebraic varieties. The finite-dimensional pieces showing up in this construction are known as affine Schubert varieties and play an important role in the theory. It turned out that quiver Grassmannians (for cyclic equioriented quivers) are also useful in this context. We will define the relevant quiver Grassmannians and describe the corresponding degenerations.

Quantum affine algebras are important examples of Drinfeld-Jimbo quantum groups. Their representation theory is very rich and it has been investigated intensively during the past thirty five years from different point of views. Relatively recently, it was discovered that the finite-dimensional representations can be studied from the point of view of cluster algebras (remarkable commutative algebras with distinguished set of generators obtained from inductive processes). The aim of these lectures will be to explain this connection, and some of the developments in this direction.

In this talk, we recall the notion of Shen–Larsson modules over the Hamiltonian Lie algebra, also known as the Lie algebra of Hamiltonian vector fields on a torus and provide necessary and sufficient conditions for the irreducibility of these modules. If time permits, we shall then describe the Jordan-Hölder series of the reducible ones, which we call exceptional modules.

The theory of flag varieties attracts a lot of attention due to a very rich structure and a huge number of applications in various fields of mathematics. Algebraic geometry, representation theory and combinatorics play major role in the theory; merging together they provide powerful techniques and allow the derivation of deep and intriguing theorems. Flag varieties are defined as quotients of Lie groups by parabolic subgroups, they are the central objects in geometric representation theory and capture many properties of the corresponding Lie groups and Lie algebras. However, their importance is seen not only in the Lie theory, but also in many neighbouring fields. In particular, various degenerations provide connections with other classes of groups and varieties, such as toric varieties. We will review main definitions and constructions and give various examples.

In her pioneering works V.Lakshmibai constructed an explicit degeneration of Grassmannians and flag varieties to certain normal toric varieties. The construction admits numerous generalizations and has many deep and surprising applications including constructions of bases in highest weight irreducible representations. The degenerations in question allow one to connect seemingly unrelated objects and to use these connections to solve complicated problems of geometric, algebraic and combinatorial nature. One of the examples is provided by the theory of quiver Grassmannians -- a far reaching generalizations of the classical Grassmann varieties. We will explain how do quiver Grassmannians enter the story, why are they useful for the description of various degenerations and what is the outcome of the use of this powerful machinery.

The classical flags are finite-dimensional projective algebraic varieties defined via finite-dimensional Lie groups. The construction admits a natural generalization in the infinite-dimensional world using the theory of affine Kac-Moody Lie algebras and affine Kac-Moody Lie groups. This generalization is natural and important in various problems and constructions (e.g. in mathematical physics) and leads to the so-called flag ind-varieties. These are infinite-dimensional objects which can be obtained as direct limits of embedded finite-dimensional projective algebraic varieties. The finite-dimensional pieces showing up in this construction are known as affine Schubert varieties and play an important role in the theory. It turned out that quiver Grassmannians (for cyclic equioriented quivers) are also useful in this context. We will define the relevant quiver Grassmannians and describe the corresponding degenerations.

Quantum affine algebras are important examples of Drinfeld-Jimbo quantum groups. Their representation theory is very rich and it has been investigated intensively during the past thirty five years from different point of views. Relatively recently, it was discovered that the finite-dimensional representations can be studied from the point of view of cluster algebras (remarkable commutative algebras with distinguished set of generators obtained from inductive processes). The aim of these lectures will be to explain this connection, and some of the developments in this direction.

The Mather class of a variety X plays a central role in the theory of characteristic classes of singular varieties. In Sabbah and Ginzburg's `Lagrangian model' for MacPherson's theory of functorial Chern classes for singular varieties, the Mather class corresponds to the conormal space of X relative to some ambient manifold M. I will explain how one can use a desingularization of conormal spaces of Schubert varieties in cominuscule Grassmannians, due to R. Singh, to give a formula for Mather classes of Schubert varieties, uniform across all Lie types. I will also discuss Schubert positivity, and conjectural unimodality, and log concavity properties of these classes. This is based on joint work with R. Singh and with P. Aluffi, J. Schurmann, and C. Su.

'd like to talk about descriptions of 3-point $K$-theoretic Gromov-Witten invariants in terms of the quantum Bruhat graph. The main part of my talk is "the quantum $K$-theoretic divisor axiom for flag manifolds",which was conjectured by Buch and Mihalcea, and was proved in a joint work (arXiv:2505.16150) with C. Lenart, S. Naito, and W.Xu.

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 maximal parabolic subalgebras, certain divisibility conditions will appear.

Following the work of Hernandez and Leclerc, the connection between cluster algebras and the Grothendieck ring of certain subcategories of quantum affine algebra modules has made the classification of prime and real modules a central topic. A simple module is called real if its tensor square is simple; otherwise, it is imaginary. While cluster algebra methods can produce real modules via sequences of mutations, classifying real and imaginary modules via Drinfeld polynomials remains challenging. In type A, under regularity assumptions, Schur–Weyl duality and the work of Lapid and Minguez on representations of p-adic groups provide such a classification. In this talk, we present new families of imaginary modules appearing as subquotients of tensor products of snake modules, including examples beyond the regular case.

Given a finite group G acting on a simple Lie algebra and a positive integral level, I will define the notion of twisted conformal blocks and describe an analogue of the Verlinde formula to compute their dimensions. Our approach is based on studying the relationship of twisted conformal blocks with the notion of G-crossed modular fusion categories and a categorical Verlinde formula in this setting. The main goal of the talk will be to describe this relationship. This is based on joint work with S. Mukhopadhyay.

Associated to two given sequences of eigenvalues is a natural polytope, the polytope of augmented hives with the specified boundary data, which is associated to sums of random Hermitian matrices with these eigenvalues. As a first step towards the asymptotic analysis of random hives, we show that if the eigenvalues are drawn from the GUE ensemble, then the associated augmented hives exhibit concentration as the number of eigenvalues tends to infinity.

Our main ingredients include a representation due to Speyer of augmented hives involving a supremum of linear functions applied to a product of Gelfand--Tsetlin polytopes; known results by Klartag on the KLS conjecture in order to handle the aforementioned supremum; covariance bounds of Cipolloni--Erdős--Schröder of eigenvalue gaps of GUE; and the use of the theory of determinantal processes to analyze the GUE minor process. This is a joint work with Scott Sheffield and Terence Tao.

Given dominant integral weights λ, μ, ν of a finite-dimensional simple Lie algebra g and an element w of its Weyl group, the refined tensor product multiplicity cν λμ(w)
is the multiplicity of the irreducible g-module V (ν) in the so-called Kostant–Kumar submodule K(λ, w, μ) of the tensor product V (λ)⊗V (μ). In type A, we obtain a hive model for the cν λμ(w) and prove that the saturation and strong semigroup properties hold if the permutation w is 312-avoiding, 231-avoiding, or a commuting product of such elements. We also remark that, in the case of g is other than type A, saturation and strong semigroup properties hold for many suitable Weyl group elements. We also discuss some formulations and variations of the saturation conjecture. This talk is based on joint work with K. N. Raghavan and Sankaran Viswanath.

The Gromov width of a symplectic manifold is a fundamental symplectic invariant that measures the size of the largest standard symplectic ball that can be symplectically embedded into the manifold. In this talk, we will first review known results on the Gromov width for specific classes of manifolds and the methods used to estimate it. Finally, we will focus on generalized Bott–Samelson manifolds, which extend both Bott–Samelson and flag manifolds.

Let $G$ be a semisimple algebraic group over $C$, the field of complex numbers. Let $T$ be a maximal torus of $G$ and $B$ be a Borel subgroup of $G$ containing $T$. Let $R\subset X(T)$ be the root system of $G$ relative to $T$ and $R^{+}\subset R$ be the set of positive roots relative to $(B, T)$. Let $G/B$ be the full flag variety of all Borel subgroups of $G$. Let $W:=N_{G}(T)$ be the Weyl group of $G$ relative to $T$. For $w\in W$, let $X(w)\subset G/B$ be the Schubert variety corresponding to $w$. For $\lambda in X(B)$ be a character of $B$, let $L_{\lambda}$ be the line bundle on $G/B$ associated to $\lambda$. Let $\rho$ denotes the half sum of positive roots.
By Borel-Weil-Bott’s theorem states:

1. If $\lambda + \rho$ is singular, then all cohomologies vanish.

2. If $\lambda+ \rho$ is non singular, then $H^{i}(G/B, L_{\lambda})=0$ for all $i\neq l(w)$ and $H^{l(w)}(G/B, L_{\lambda})=H^{0}(G/B, L_{w(\lambda+\rho)-rho})$, where $w\in W$ is the unique element for which $w(\lambda+\rho)-\rho$ is dominant. In this talk we give a description of Horospherical Schubert varieties $X(w)$ in terms of the combinatoric properties of $w$.

Further assume that $G$ is simply laced. Then, given a Horospherical Schubert variety $X(w)$, and we give a description of $w$ and $\phi\in W$ for which a generic character $\lambda$ of $B$ in chamber corresponding to an element $\phi\in W$, there is only one cohomology is non zero. This is a joint work with Mahir Bilen Can and Pinakinath Saha.

It is known that centerless Virasoro algebra (Witt algebra) acts on Affine Kac-Moody algebra and its semi direct product is an interesting object of study. We generalize this for several variables. 

We first define Toroidal algebra which is generalization of Affine Kac-Moody algebra and state Classification of irreducible integrable modules. We then define Witt algebra in several variables. It is denoted by DerA which are derivations on A(=Laurent polynomials in several variables). Now we define Shen-Laesson modules for DerA and state two important results. It is also A module and as (A DerA) module it is always irreducible. We then describe all submodules of Shen-Larsson module as a DerA module. In fact, these are all the irreducible modules for DerA proved by Futorny and Billig. If time permits we will state results on Full Toroidal algebra which is an extension of Toroidal algebra and Witt algebra.