×

Hausel, Letellier and Rodriguez-Villegas 's conjectural formula predicts the mixed Hodge polynomials of parabolic character varieties of Riemann surfaces. Via the P=W theorem this formula also computes the perverse Poincare polynomials of the moduli spaces of Higgs bundles. On the one side we have a certain generating series of the mixed Hodge/perverse Poincare polynomials. On the other side is a certain partition function involving Macdonald polynomials. In the theory of Macdonald polynomials the are certain operators that want to be interpreted as the T-matrix and the S-matrix of a kind of TQFT, and one can write partition functions that want to be related to some invariants of moduli spaces associated with configurations of nodal curves. We make guesses about the general statement, confirmed by the existing conjectures (including conjectures about knot invariants and plain singularities) and explicit computations in rank 1. We conjecture that the partition function in general encodes two weight filtrations: one on the Betti (character variety) side, and the other one on the Dolbeault (Higgs bundles) side.

×

The Tutte polynomial of a graph is a two-variable polynomial, which is the universal polynomial satisfying the deletion contraction recursion. In this talk I will explain how this polynomial arises as a special case of a bicharacteristic polynomial defined for pairs of symplectic dual conical resolutions of singularities. More precisely, the Tutte polynomial records the dimensions of the graded pieces of the cohomology of hypertoric varieties (which I’ll introduce) along with the two filtrations by cohomological degree, coming from symplectic duality and Maulik-Okounkov stable envelopes (which I will also introduce).  These results produce new inequalities of coefficients of Tutte polynomials of matroids. This talk is based on joint work with Michael McBreen.

×

Chromatic quasisymmetric functions of labeled graphs were introduced by Shareshian and Wachs as a $q$-analogue of the chromatic symmetric functions defined by Stanley. For the class of labeled graphs called unit interval graphs, the chromatic quasisymmetric functions have interpretations in terms of the geometry of Hessenberg varieties and the representation theory of Hecke algebras. In this talk, I will present a probabilistic interpretation of these functions. For a given unit interval graph, I construct a stochastic process whose probability distribution describes the elementary symmetric function expansion of the chromatic quasisymmetric function. In particular, I prove the $e$-positivity of chromatic symmetric functions of unit interval graphs, which was conjectured by Stanley and Stembridge in 1993.

×

The coinvariant algebra, the quotient of the coordinate ring of affine space by the ideal generated by positive degree invariant polynomials, plays a basic role in algebraic combinatorics and the representation theory of the symmetric group, equipping itsregular representation with a graded algebra structure. Using the coordinate ring of (P^1)^n in its Segre embedding, I will introduce adegeneration of the coinvariant algebra, the projective coinvariant algebra, which gives a bigraded structure on the regular representation of the symmetric group with interesting Frobenius character that generalises a classical result of Lusztig and Stanley. I will also show how this algebra contains bigraded versions of partial coinvariant algebras, coming from coordinate rings of all possible Segre embeddings.Hilbert functions of the bigraded algebras are controlled by natural expressions known from algebraic combinatorics. Relations to Haiman's diagonal coinvariant algebra, and (quantum) cohomology of full and partial flag varieties, will also be discussed.

×

Hilbert schemes of points on planar curve singularities can be realized as generalized GL_N-affine Springer fibers attached to the direct sum of the adjoint and standard representations. In this talk, I will establish the cohomological purity of the Hilbert schemes associated with generic irreducible singularities by constructing an explicit affine paving, thereby confirming a new case of the purity conjecture for generalized affine Springer fibers.

The combinatorics of the paving - cell indices and dimensions - are governed by (nd,md)-Dyck paths, extending results of Gorsky-Mazin-Oblomkov on compactified Jacobians. Time permitting, I will also explain connections to the Oblomkov-Rasmussen-Shende and Cherednik-Danilenko-Philipp conjectures, which link the enumerative geometry of planar curve singularities to knot invariants and double affine Hecke algebras. Based on joint work with Taiwang Deng.

×

The quantum cohomology ring of a smooth variety is a deformation of usual cohomology ring, it is a commutative associative algebra. The (equivariant) intersection cohomology of a singular affine variety is a module over the cohomology of a point, it does not generally admit a commutative ring structure.

However, for conical symplectic resolution, the intersection cohomology of the singular cone is conjectured to be related with quantum cohomology of its resolution in the work of Mcbreen-Proudfoot. This gives a conjectural commutative ring structure on intersection cohomology. In this talk, I will explain the notion of quantum cohomology, and discuss recent progress in this direction. Examples of Springer resolution and Nakajima quiver varieties are included. This is joint work in preparation with Michael Mcbreen.

×

The homology of character stacks of 2-manifolds for type A gauge groups has been studied intensively in connection with the Langlands duality and the non-abelian Hodge correspondence. Building on recent developments in Donaldson-Thomas theory, the extension of these studies to 3-manifolds and to arbitrary reductive gauge groups is gradually becoming active, where we use the critical cohomology instead of the homology. I will survey recent progress and open problems in this direction.

×

Abelian fibrations are proper morphisms whose general fiber is an abelian variety (a complex torus), while singular fibers may be quite complicated. These geometric objects arise naturally in the study of mirror symmetry, integrable systems, hyper-Kähler geometry, and related areas. In this talk, I will discuss natural degenerations of the cohomology rings of the total spaces of many interesting abelian fibrations; such degenerations have already appeared in the study of DAHA and the (now proven) P = W conjecture. I will also present further new structural results and open questions.

×

Cohomological Hall algebras of quivers with potentials give a geometric way to realize quantum groups . In this talk, I will define a certain vertex coproduct called a Joyce vertex coalgebra on the Cohomological Hall algebras of a quiver with potential. This structure is compatible with the multiplication and upgrades the cohomological Hall algebra into a vertex bialgebra. I will then explain how to compare this structure to Deformed drinfeld coproducts on ADE Yangians.This is joint work with Shivang Jindal and Alexei Latyntsev.