Related papers: Local formulae for combinatorial Pontrjagin classe…

A closed formula is obtained for the integral $\int_{\mathcal{\bar{H}}_g^1}\kappa_{1}\psi^{2g-2}$ of tautological classes over the locus of hyperelliptic Weierstra\ss{} points in the moduli space of curves. As a corollary, a relation…

In 1940s Steenrod asked if every homology class $z\in H_n(X,\mathbb{Z})$ of every topological space $X$ can be realised by an image of the fundamental class of an oriented closed smooth manifold. Thom found a non-realisable 7-dimensional…

We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincar\'e duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. As applications, we obtain proofs of Dowling and Wilson's Top-Heavy…

Let P(z) and Q(y) be polynomials of the same degree k>=1 in the complex variables z and y, respectively. In this extended abstract we study the non-linear functional equation P(z)=Q(y(z)), where y(z) is restricted to be analytic in a…

For any polynomial f with complex coefficients we find a remarkable subset of poles of the motivic zeta function. It is combinatorially determined by any log resolution and it admits an intrinsic interpretation in terms of contact loci of…

For any flag simplicial complex $K$, we describe the multigraded Poincare series, the minimal number of relations and the degrees of these relations in the Pontryagin algebra of the corresponding moment-angle complex $Z_K$. We compute the…

We establish an analogue of Pontryagin duality for modules over compact discrete valuation rings $R$. Namely, we define the dual of a topological $R$ module to be its continuous $R$-module homomorphisms into $K/R$, the quotient module of…

In this paper we study the mixed Poincar\'e polynomial of generic $\mathrm{PGL}_n(\mathbb{C})$-character stacks with coefficients in some local systems arising from the conjugacy classes of $\mathrm{PGL}_n(\mathbb{C})$ which have…

Let $G_{\mathbb{R}}$ be a simple real linear Lie group with maximal compact subgroup $K_{\mathbb{R}}$ and assume that ${\rm rank}(G_\mathbb{R})={\rm rank}(K_\mathbb{R})$. In \cite{MPVZ} we proved that for any representation $X$ of…

Kerov's polynomials give irreducible character values in term of the free cumulants of the associated Young diagram. We prove in this article a positivity result on their coefficients, which extends a conjecture of S. Kerov. Our method,…

Let $K$ be a finite field of characteristic $p$. We study a certain class of functions $K\rightarrow K$ that agree with an $\mathbb{F}_p$-affine function $K\rightarrow K$ on each coset of a given additive subgroup $W$ of $K$ - we call them…

The goal of this article is to try understand where Hodge cycles on a singular complex projective variety X come from. As a first step we consider Hodge cycles on the maximal pure quotient $H^{2p}(X)/W_{2p-1}$, and introduce a class of…

We describe a class of real Banach manifolds, which classify $K^{-1}$. These manifolds are Grassmannians of (hermitian) lagrangian subspaces in a complex Hilbert space. Certain finite codimensional real subvarieties described by incidence…

We prove that the rank of the cohomology of a closed symplectic manifold with coefficients in a field of characteristic $p$ is smaller than the number of periodic orbits of any non-degenerate Hamiltonian flow. Following Floer, the proof…

The main results of this article concern the definition of a compactly supported cohomology class for the congruence group $\Gamma_0(p^n)$ with values in the second Milnor $K$-group (modulo 2-torsion) of the ring of $p$-integers of the…

The dual Kontsevich cycles in the double dual of associative graph homology correspond to polynomials in the Miller-Morita-Mumford classes in the integral cohomology of mapping class groups. I explain how the coefficients of these…

Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F, and let P be a prime of F at which f is new. Let K be a quadratic extension of F, and L(f/K,s) the L-function of the base-change of…

We give a simple matrix-based proof of congruence equations modulo a prime $p$ involving sums of binomial coefficients appearing in Pascal's triangle. These equations can be used to construct some groups of exponent $p^n$. These groups, as…

Given a set of forms f={f_1,...,f_m} in R=k[x_1,...,x_n], where k is a field of characteristic zero, we focus on the first syzygy module Z of the transposed Jacobian module D(f), whose elements are called differential syzygies of f. There…

Bipartite calculus is a direct generalization of Kauffman planar expansion from $N=2$ to arbitrary $N$, applicable to the restricted class of knots which are entirely made of antiparallel lock tangles. Whenever applicable, it allows a…