Abstract:  The affine and periodic Temperley–Lieb algebras are families of infinite-dimensional algebras with a diagrammatic presentation. They have been studied in the last 30 years, mostly for their physical applications in statistical mechanics, where the diagrammatic presentation encodes the connectivity property of the models. Most of the relevant representations for physics are finite-dimensional. In this work, we define finite-dimensional quotients of these algebras that we name uncoiled algebras in reference to the diagrammatic interpretation of the quotient and construct a family of Jones–Wenzl idempotents, each of which projects onto one of the one-dimensional modules these algebras admit. We also prove that, most of the time, the uncoiled algebras are affine cellular and we define a notion of skew sandwich cellularity for the remaining cases. This is joint work with Alexi Morin-Duchesne.

Time permitting, I will also cover future generalisations on diagrammatic calculus on surfaces via cobordisms.

Abstract:  The Robinson--Schensted (RS) correspondence admits diagrammatic interpretations via Fomin's growth diagrams and Viennot's shadow line construction. Works of Spaltenstein, Springer, Steinberg, and van Leeuwen connected this combinatorial construction to the relative position map of Springer fibers. Motivated by Kazhdan--Lusztig cell theory, various generalizations of the Robinson--Schensted correspondence into the affine type A setting have been studied. Prominent examples include Shi's insertion algorithm and the affine matrix ball construction by Chmutov--Pylyavskyy--Yudovina. In particular, using the affine matrix ball construction, Boixeda--Ying--Yue showed that the two-sided cells and S-cells agree when the nilpotent is of rectangular type. 

In this talk, we introduce a new combinatorial construction of the affine RS correspondence via growth diagrams and shadow lines that is in a sense dual to Shi's insertion and the affine matrix ball construction, and geometrically natural in terms of relative positions of affine flags. 

Ongoing joint work with Sylvester Zhang.

Abstract:  Kleinian singularities are quotients of C^2 by finite subgroups of SL_2(C). They are in bijection with the ADE Dynkin diagrams via the McKay correspondence. The filtered Poisson deformations of Kleinian singularities are parametrized by h/W, where h is the Cartan space and W is the Weyl group. In this talk, I will introduce certain singular Lagrangian subvarieties in the deformations and resolutions of Kleinian singularities that are related to the geometric classification of Harish-Chandra modules over quantizations of nilpotent orbits. The irreducible components of these singular Lagrangian subvarieties are projective lines and affine curves. I will describe how they deform along smooth curves in the parameter spaces.

Abstract:  These lectures will provide an elementary introduction into cluster algebras and cluster varieties from a combinatorial standpoint, loosely following our book draft (https://people.math.harvard.edu/~williams/book.html). Examples and computations will be emphasized throughout. No prior familiarity with the subject will be assumed.

We will start by briefly reviewing two key motivations for cluster algebras: total positivity and canonical bases. We will introduce the machinery of mutations of quivers and skew-symmetrizable matrices, and discuss several examples in which these mutations naturally appear, including pseudoline arrangements, triangulations of surfaces, and plabic graphs. We will then define cluster algebras, formulate their key properties, and present the finite type classification.

Many cluster algebras arise in the context of representation theory, invariant theory, etc., in the form of coordinate rings of affine algebraic varieties. Typical examples include varieties of matrices of various kinds, Grassmannians, and basic affine spaces. We will discuss several examples of this nature, together with relevant general theory.

If time permits, we will review some recent results on the combinatorics of quiver mutations. 

Abstract:  A longstanding goal of Schubert calculus is to give a positive formula for the structure constants for the Schubert basis in the cohomology ring of the d-step flag variety. This goal can be generalized by replacing "cohomology ring" with "torus-equivariant cohomology ring", "K-ring", and "torus-equivariant K-ring". It can also be generalized in an orthogonal direction by replacing "d-step flag variety" with "cotangent bundle of the d-step flag variety".

Recently, Allen Knutson and Paul Zinn-Justin proved a positive formula in terms of Knutson-Tao puzzles for the structure constants in the basis of motivic Segre classes of Schubert cells in (a localization of) the torus-equivariant K-ring of the cotangent bundle of the Grassmannian. Their proof heavily uses the theory of quantum integrable systems.

In this talk, we will describe a one-parameter deformation of the motivic Segre classes of Schubert cells in the Grassmannian which comes from the so-called "connective formal group law", and we give a positive formula for the structure constants in the basis of deformed classes in terms of Knutson-Tao puzzles. The proof of the puzzle formula involves the representation theory of the multi-parameter quantum group of affine type A.

Abstract:  A semicanonical basis is a basis of the negative half of the universal enveloping algebra of a symmetric generalized Kac-Moody algebra.  

It is realized via constructible functions on Lusztig’s nilpotent quiver varieties. In this talk, I’ll provide an explicit description of this basis in the case of the Jordan quiver, namely,  the quiver with one vertex and one loop. 

Abstract:  M. Kontsevich conjectured and T. Bitoun proved that if M is a nonzero holonomic D-module then the p-support of a generic reduction of M to characteristic p>0 is Lagrangian. I will discuss a new elementary proof of this theorem and also generalize it to q-D-modules. The proofs are based on Bernstein's theorem that any holonomic D-module can be transformed by an element of the symplectic group into a vector bundle with a flat connection, and a q-analog of this theorem. I will also discuss potential applications to quantizations of symplectic singularities and to quantum cluster algebras.

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Abstract:  Higgs and Coulomb branches of quiver gauge theories form two important families of Poisson varieties that are expected to be exchanged under so-called 3D mirror symmetry. The representation theory of quantized Coulomb branches is deeply connected with the enumerative geometry of Higgs branches. One important approach to studying modules over quantized Coulomb branches is by analyzing their graded traces. Graded traces generalize the notion of characters and are closely related to the q-characters introduced by Frenkel and Reshetikhin. Any graded trace defines a solution of the D-module of graded traces introduced by Kamnitzer, McBreen, and Proudfoot.

In this talk, I will discuss techniques that allow us to explicitly compute characters and graded traces of certain modules over quantized Coulomb branches. Time permitting, I will explain how some of these results naturally appear on the Higgs side, leading to an explicit description of the D-module of graded traces for a quantized Coulomb branch via the geometry of the Higgs branch. We prove these results for ADE quivers and formulate explicit conjectures in the general case. Talk is based on joint works with Dinkins, Karpov, Klyuev, and Lance.

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Abstract:  I will discuss various aspects of the geometry of the joint moduli space and nonabelian Hodge correspondence for Higgs bundles

on Riemann surfaces with varying complex structures.  Specifically, there are four objects that are related in a surprising way: the isomonodromic

distribution, the degeneracy of the hermitian pairing arising from the Atiyah-Bott-Goldman form, the "Kodaira-Spencer" form, and the energy functional

for equivariant harmonic maps.  I will also sketch proofs of the existence of pseudo-Kaehler metrics in a number

of cases of Higgs bundles with special holonomy.  This recovers and extends several recent constructions of various authors.

This work is part of a collaboration with Brian Collier and Jeremy Toulisse.

Abstract:  Let $G$ be a connected simply-connected simple algebraic group over $\mathbb{C}$ and let $T$ be a maximal torus, $B \supset T$ a Borel subgroup

and $K$ a maximal compact subgroup. Then, the product in the (algebraic) based loop group $\Omega(K)$ gives rise to a comultiplication in the topological

$T$-equivariant $K$-ring $K^{\mathrm{top}}_T(\Omega(K))$. Recall that $\Omega(K)$ is identified with the affine Grassmannian $X$ (of $G$) and hence we get a comultiplication in

$K^{\mathrm{top}}_T(X)$. Dualizing, one gets the Pontryagin product in the $T$-equivariant $K$-homology $K^T_0(X)$, which in-turn gets identified with the convolution

product (due to S.~Kato).

Now, $K^{\mathrm{top}}_T(X)$ has a basis $\{\xi_w\}$ over the representation ring $R(T)$ given by the ideal sheaves corresponding to the finite

codimension Schubert varieties $X_w$ in $X$. We make a positivity conjecture on the comultiplication structure constants in the above basis.

Using some results of Kato, this conjecture gives rise to an equivalent conjecture on the positivity of the multiplicative structure constants in $T$-equivariant

quantum $K$-theory $QK_T(G/B)$ in the Schubert basis. 

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