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Let $G$ be a simple algebraic group of type $F_{4}$, $E_{6}$, $E_{7}$ or $E_{8}$, and let $\mathfrak{g}$ be its Lie algebra. The adjoint variety $X_{ad} \subseteq \mathbb{P} \mathfrak{g}$ is defined as the unique closed orbit of the adjoint…

We study the nonlinear supersymmetric hyperbolic sigma model introduced by Zirnbauer in 1991. This model can be related to the mixing measure of a vertex- reinforced jump process. We prove that the two-point correlation function has a…

We prove most of Lusztig's conjectures from the paper "Bases in equivariant K-theory II", including the existence of a canonical basis in the Grothendieck group of a Springer fiber. The conjectures also predict that this basis controls…

We begin with proving a formula relating the Hilbert series of a graded algebra $A$ and the Poincar\'{e} series for $A$ in two variables. This gives the Fr\"oberg formula in the case where the bigraded $Tor^A(k,k)$ is concentrated on the…

It is known that characters of BPS representations of extended superconformal algebras do not have good modular properties due to extra singular vectors coming from the BPS condition. In order to improve their modular properties we apply…

We construct a categorification of the maximal commutative subalgebra of the type $A$ Hecke algebra. Specifically, we propose a monoidal functor from the (symmetric) monoidal category of coherent sheaves on the flag Hilbert scheme to the…

In this paper we consider certain closed subvarieties of the flag variety, known as Hessenberg varieties. We give a connectedness criterion for semisimple Hessenberg varieties generalizing a criterion given by Anderson and Tymoczko. We show…

We construct embeddings $\mathcal{Y}_{n,\tau} \to {Hilb}^n (\Sigma_{\tau})$ for each of the classical Lie algebras $\ger{sp}_{2m}(\Cc)$, $\ger{so}_{2m}(\Cc)$, and $\ger{so}_{2m+1}(\Cc)$. The space $\mathcal{Y}_{n,\tau}$ is the fiber over a…

Nilpotence in the homotopy of $\mathbb{E}_\infty$-ring spectra is detected by the classical $H\mathbb{Z}$-Hurewicz homomorphism. Inspired by questions of Mathew, Noel, and Naumann, we investigate the extent to which this criterion holds in…

The principal goal of the paper is to apply the approach inspired by the theory of integrable systems to construct explicit sections of line bundles over the combinatorial model of the moduli space of pointed Riemann surfaces based on…

Let K be an infinite field and denote by H(n,K) the family of pairs (A,B) of commuting nilpotent n by n matrices with entries in K. There has been substantial recent study of the connection between H(n,K) and the fibre H[n] of the punctual…

Let $R$ be a root datum with affine Weyl group $W^e$, and let $H = H (R,q)$ be an affine Hecke algebra with positive, possibly unequal, parameters $q$. Then $H$ is a deformation of the group algebra $\mathbb C [W^e]$, so it is natural to…

In the first part of the paper we defined and studied a binary operation on the set of irreducible components of Lusztig's nilpotent varieties of a quiver. For type $A$ we conjecture, following Geiss and Schr\"oer, that this operation is…

This work pursues a circle of Lie-theoretic ideas involving Hessenberg varieties, Poisson geometry, and wonderful compactifications. In more detail, one may associate a symplectic Hamiltonian $G$-variety…

We study the biholomorphic action of the Heisenberg group $\mathbb{H}_n$ on the Siegel domain $D_{n+1}$ ($n \geq 1$). Such $\mathbb{H}_n$-action allows us to obtain decompositions of both $D_{n+1}$ and the weighted Bergman spaces…

We construct a canonical sesquilinear pairing on the relative crystalline cohomology of a smooth proper family of varieties over a complete discretely valued $p$-adic field. Motivated by the role of Saito's higher residue pairing in the…

In this paper we establish $L^p$ boundedness properties for maximal operators, Littlewood-Paley functions and variation operators involving Poisson semigroups and resolvent operators associated with nonsymmetric Ornstein-Uhlenbeck…

Hessenberg varieties are subvarieties of the flag variety parametrized by a linear operator $X$ and a nondecreasing function $h$. The family of Hessenberg varieties for regular $X$ is particularly important: they are used in quantum…

We study combinatorial aspects of the Schubert calculus of the affine Grassmannian Gr associated with SL(n,C). Our main results are: 1) Pieri rules for the Schubert bases of H^*(Gr) and H_*(Gr), which expresses the product of a special…

This paper studies a flat degeneration P_n of the classical coinvariant algebra R_n, a bigraded Artinian Gorenstein algebra that arises from the coordinate ring of the Segre embedding of the n-fold self-product of the projective line. The…