Related papers: The symplectic plactic monoid, crystals, and MV cy…

Moduli spaces of admissible covers and stable maps of target curves give rise to cycles on $\overline{M}_{g,n}$. We prove a formula relating these cycles. It recovers both the Ekedahl-Lando-Shapiro-Vainshtein formula and the…

This paper proves a conjecture of Fomin and Shapiro that their combinatorial model for any Bruhat interval is a regular CW complex which is homeomorphic to a ball. The model consists of a stratified space which may be regarded as the link…

We develop the deformation theory of symplectic foliations, i.e. regular foliations equipped with a leafwise symplectic form. The main result of this paper is that each symplectic foliation has an attached $L_\infty$-algebra controlling its…

Fundamental representations of real simple Poisson Lie groups are Poisson actions with a suitable choice of the Poisson structure on the underlying (real) vector space. We study these (mostly quadratic) Poisson structures and corresponding…

An embedding of a graph on a translation surface is said to be \emph{systolic} if each vertex of the graph corresponds to a singular point (or marked point) and each edge corresponds to a shortest saddle connection on the translation…

We consider canonical symplectic structure on the moduli space of flat ${\g}$-connections on a Riemann surface of genus $g$ with $n$ marked points. For ${\g}$ being a semisimple Lie algebra we obtain an explicit efficient formula for this…

We define a family of groups that include the mapping class group of a genus g surface with one boundary component and the integral symplectic group Sp(2g,Z). We then prove that these groups are finitely generated. These groups, which we…

Let M denote either Euclidean or hyperbolic n-space, and let G be a discrete group of isometries of M, with the property that G respects and acts tile-transitively on a convex-polyhedral tesselation of M. Given an arbitrary base point p in…

Let (M,w,L) be a symplectic manifold endowed with a lagrangian foliation L. Liberman and Weinstein have shown that the leaves of L are endowed with an affine structure. In this paper we provide links between the theories of affine manifolds…

The classifying space of a crossed complex generalises the construction of Eilenberg-Mac Lane spaces. We show how the theory of fibrations of crossed complexes allows the analysis of homotopy classes of maps from a free crossed complex to…

We develop a general framework for studying phases of mixed states with strong and weak symmetries, including non-invertible or categorical symmetries. The central idea is to consider a purification of the mixed state density matrix, which…

Let G be a connected reductive group. In this paper we are studying the invariant theory of symplectic G-modules. Our main result is that the invariant moment map is equidimensional. We deduce that the categorical quotient is a fibration…

Each point $x$ in Gr$(r,n)$ corresponds to an $r \times n$ matrix $A_x$ which gives rise to a matroid $M_x$ on its columns. Gel'fand, Goresky, MacPherson, and Serganova showed that the sets $\{y \in \mathrm{Gr}(r,n) | M_y = M_x\}$ form a…

We describe the structure of the Whittaker or Gelfand-Graev module on a $n$-fold metaplectic cover of a $p$-adic group $G$ at both the Iwahori and spherical level. We express our answer in terms of the representation theory of a quantum…

This survey paper begins with the description of the duality between arc systems and ribbon graphs embedded in a punctured surface. Then we explain how to cellularize the moduli space of curves in two different ways: using Jenkins-Strebel…

In this paper, we classify homomorphisms from the braid group of $n$ strands to the mapping class group of a genus $g$ surface. In particular, we show that when $g<n-2$, all representations are either cyclic or standard. Our result is sharp…

Motivated by the work of Salvetti and Settepanella we introduce certain total orderings of the faces of any shellable regular CW-complex (called `shelling-type orderings') that can be used to explicitly construct maximum acyclic matchings…

We define and study a class of finite topological spaces, which model the cell structure of a space obtained by gluing finitely many Euclidean convex polyhedral cells along congruent faces. We call these finite topological spaces,…

This paper is the first part in a 2 part study of an elementary functorial construction from the category of finite non-abelian groups to a category of singular compact, oriented 2-manifolds. After a desingularization process this…

We present a model for smectic elastomer membranes which includes elastic and liquid crystalline degrees of freedom. Based on our model, we determined the qualitative phase diagram of a smectic elastomer membrane using mean-field theory.…