Wreath Macdonald polynomials are generalizations of Macdonald polynomials wherein the symmetric groups are replaced with their wreath products with a cyclic group of order $\ell$. They were defined by Haiman, and mirroring the usual Macdonald theory, it is not obvious that they exist. Haiman also conjectured for them a generalization of his celebrated proof of Macdonald positivity where the Hilbert scheme of points on the plane is replaced with certain cyclic Nakajima quiver varieties. This conjecture was proven by Bezrukavnikov and Finkelberg, which also implies the existence of the polynomials. Analogues of standard formulas and results of usual Macdonald theory remain to be explored. I will present an approach to the study of the wreath variants via the quantum toroidal algebra of $\mathfrak{sl}_\ell$, generalizing the fruitful interactions between the usual Macdonald theory and the quantum toroidal algebra of $\mathfrak{gl}_1$. As applications, I'll present an analogue of the norm formula and a conjectural path towards "wreath Macdonald operators" that makes contact with the spin Ruijsenaars-Schneider integrable system.

In 2015, Brosnan and Chow, and independently Guay-Paquet, proved the Shareshian--Wachs conjecture, which links the combinatorics of chromatic symmetric functions to the geometry of Hessenberg varieties via a permutation group action on the cohomology ring of regular semisimple Hessenberg varieties. This talk will give a brief overview of that story and discuss how the dot action can be computed in all Lie types using the Betti numbers of certain nilpotent Hessenberg varieties. As an application, we obtain new geometric insight into certain linear relations satisfied by chromatic symmetric functions, known as the modular law. This is joint work with Eric Sommers.

A 2-group is a categorified version of a group. In this talk, we will study the structure of moduli stacks and spaces of principal bundles for 2-groups. In a special case where the isomorphism classes of objects in our 2-group form a finite (ordinary) group $G$, we show that the moduli stack provides a higher-categorical enhancement of the Freed–Quinn line bundle appearing in Chern–Simons theory for the finite group $G$. This is joint work with Eric Berry, Dan Berwick-Evans, Laura Murray, Apurva Nakade, and Emma Phillips.

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Let $S$ be a K3 surface and $C$ a smooth curve in $S$. We consider the moduli space $M$ of coherent sheaves on $S$ which are supported on a curve rational equivalent to $nC$ and have fixed Euler characteristic (coprime to $n$). Then $M$ is an irreducible holomorphic symplectic manifold equipped with a Lagrangian fibration given by taking supports. This is the beautiful Mukai system.

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One source of interest in the Mukai system is, that it deforms to the Hitchin system on $C$. And there is a notion of the nilpotent cone in the Mukai system deforming to the nilpotent cone in the Hitchin system. In my talk, I present some results about the nilpotent cone on the Mukai side (in the lowest dimensional case), which can then be transferred to the Hitchin side.

3d mirror symmetry is a conjectured duality among symplectic varieties that expects deep relationships between enumerative invariants of varieties that may appear to be unrelated. In this talk, I will describe the general setup of 3d mirror symmetry and will then explain its nontrivial combinatorial implications in the example of the cotangent bundle of the Grassmannian and its mirror variety. In this case, the 3d mirror relationship is governed by a new family of difference operators which characterize the Macdonald polynomials.

Let $G$ be a complex reductive group, set $\mathfrak g={\mathrm{Lie\,}}G$. The algebra ${\mathcal S}(\mathfrak g)^{\mathfrak g}$ of symmetric $\mathfrak g$-invariants and the centre ${\mathcal Z}(\mathfrak g)$ of the enveloping algebra ${\mathcal U}(\mathfrak g)$ are polynomial rings in ${\mathrm{rk\,}}\mathfrak g$ generators. There are several isomorphisms between them, including the symmetrisation map $\varpi$, which exists also for the Lie algebras $\mathfrak q$ with $\dim\mathfrak q=\infty$.

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However, in the infinite dimensional case, one may need to complete ${\mathcal U}(\mathfrak q)$ in order to replace ${\mathcal Z}(\mathfrak q)$ with an interesting related object. Roughly speaking, the Feigin-Frenkel centre arises as a result of such completion in case of an affine Kac-Moody algebra. From 1982 until 2006, this algebra existed as an intriguing black box with many applications. Then explicit formulas for its elements appeared first in type ${\sf A}$, later in all other classical types, and it was discovered that the FF-centre is the centraliser of the quadratic Casimir element.

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We will discuss the type-free role of the symmetrisation map in the description of the FF-centre and present new explicit formulas for its generators in types ${\sf B}$, ${\sf C}$, ${\sf D}$, and ${\sf G}_2$. One of our main technical tools is a certain map from ${\mathcal S}^{k}(\mathfrak g)$ to $\Lambda^2\mathfrak g \otimes {\mathcal S}^{k-3}(\mathfrak g)$.

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The aim of this joint work in progress with Roman Bezrukavnikov is to unite different approaches to Khovanov-Rozansky triply graded link homology. The original definition is completely algebraic in terms of Soergel bimodules. It has been conjectured by Gorsky, Negut and Rasmussen that it can also be calculated geometrically in terms of cohomology of sheaves on Hilbert schemes. Motivated by string theory Oblomkov and Rozansky constructed a link invariant in terms of matrix factorizations on related spaces and later proved that it coincides with Khovanov-Rozansky homology. In this talk I’ll discuss a direct relation between the different constructions and how one might invent these spaces starting directly from definitions.

The quantum toroidal algebra is the affinization of the quantum Heisenberg algebra. Schiffmann-Vasserot, Feigin-Tsymbaliuk and Negut studied the quantum toroidal algebra action on the Grothendieck group of Hilbert schemes of points on surfaces, which generalized the action by Nakajima and Grojnowski in cohomology. In this talk, we will categorify the above quantum toroidal algebra action. Our main technical tool is a detailed geometric study of certain nested Hilbert schemes of triples and quadruples, through the lens of the minimal model program, by showing that these nested Hilbert schemes are either canonical or semi-divisorial log terminal singularities.

Spherical subalgebra of Cherednik's double affine Hecke algebra of type A admits a polynomial representation in which its generators act via elementary symmetric functions and Macdonald operators. Recognizing the elementary symmetric functions as eigenfunctions of quantum Toda Hamiltonians, and applying (the inverse of) the Toda spectral transform, one obtains a new representation of spherical DAHA. In this talk, I will discuss how this new representation gives rise to an injective homomorphism from the spherical DAHA into a quantum cluster algebra in such a way that the action of the modular group on the former is realized via cluster transformations. The talk is based on a joint work in progress with Philippe Di Francesco, Rinat Kedem, and Gus Schrader.

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Gillespie, Levinson and Purbhoo introduced a crystal-like structure for shifted tableaux, called the shifted tableau crystal. Following a similar approach as Halacheva, we exhibit a natural internal action of the cactus group on this structure, realized by the restrictions of the shifted Schützenberger involution to all primed intervals of the underlying crystal alphabet. This includes the shifted crystal reflection operators, which agree with the restrictions of the shifted Schützenberger involution to single-coloured connected components, but unlike the case for type A crystals, these do not need to satisfy the braid relations of the symmetric group.

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In addition, we define a shifted version of the Berenstein-Kirillov group, by considering shifted Bender-Knuth involutions. Paralleling the works of Halacheva and Chmutov, Glick and Pylyavskyy for type A semistandard tableaux of straight shape, we exhibit another occurrence of the cactus group action on shifted tableau crystals of straight shape via the action of the shifted Berenstein-Kirillov group. We conclude that the shifted Berenstein-Kirillov group is isomorphic to a quotient of the cactus group. Not all known relations that hold in the classic Berenstein-Kirillov group need to be satisfied by the shifted Bender-Knuth involutions, namely the one equivalent to the braid relations of the type A crystal reflection operators, but the ones implying the relations of the cactus group are verified, thus we have another presentation for the cactus group in terms of shifted Bender-Knuth involutions.

We will describe a theory of root of unity quantum cluster algebras, which cover as special cases the big quantum groups of De Concini, Kac and Process. All such algebras will be shown to be polynomial identity (PI) algebras. Inside each of them, we will construct a canonical central subalgebra which is proved to be isomorphic to the underlying cluster algebra. It is a far-reaching generalization of the De Concini-Kac-Procesi central subalgebras in big quantum groups and presents a general framework for studying the representation theory of quantum algebras at roots of unity by means of cluster algebras as the relevant data becomes (PI algebra, canonical central subalgebra)=(root of unity quantum cluster algebra, underlying cluster algebra). We will also present an explicit formula for the corresponding discriminants in this general setting that can be applied in many concrete situations of interest, such as the discriminants of all root of unity quantum unipotent cells for symmetrizable Kac-Moody algebras. This is a joint work with Bach Nguyen (Xavier Univ) and Kurt Trampel (Notre Dame Univ).

There are many different methods to compactly moduli spaces of varieties with a rich source of examples from compactifying moduli spaces of curves. In this talk, I will explain a relatively new compactification coming from K-stability and how it connects to serval other compactifications, focusing on the case of plane curves of degree $d$. In particular, we regard a plane curve as a log Fano pair $(\mathbb{P}^2, aC)$ and study the K-moduli compactifications and establish a wall crossing framework as a varies. We will describe all wall crossings for low degree plane curves and discuss the picture for general $\mathbb{Q}$-Gorenstein smoothable log Fano pairs. This is joint work with Kenneth Ascher and Yuchen Liu.

The rank of a matrix can be generalized to tensors. In fact, there are many different rank notions for tensors that all coincide for matrices, such as the tensor rank, border rank, subrank and slice rank (and asymptotic versions of each of these). In this talk I will discuss two notions of rank that are closely related to Geometric Invariant Theory, the non-commutative rank and the G-stable rank. The non-commutative rank can be used for giving lower bounds for tensor rank and border rank. The G-stable rank was recently used by my graduate student Zhi Jiang to improve the asymptotic upper bounds of Ellenberg and Gijswijt for the Cap Set Problem.

The loop O(n) model is a model for random configurations of non-overlapping loops on the hexagonal lattice, which contains many models of interest (such as the Ising model, self-avoiding walks, and random Lipshitz functions) as special cases. The physics literature conjectures that the model undergoes several different phase transitions, leading to a dazzling phase diagram; over the last several years, several features of the phase diagram have been proven rigorously. In this talk, I will describe the predicted behavior of the model and show some recent progress towards proving that typical samples of perturbations of the uniform measure on loop configurations have long loops. This is joint work with Nick Crawford, Alexander Glazman, and Ron Peled.