Related papers: Gauss-Bonnet for multi-linear valuations

Kempf [1976] studied proper, G-equivariant maps from equivariant vector bundles over flag manifolds to G-representations V, which he called _collapsings_. We give a simple formula for the G-equivariant cohomology class on V,…

We rederive in a simplified version the Lehmann-Sommers eigenvalue distribution for the Gaussian ensemble of asymmetric real matrices, invariant under real orthogonal transformations, as a basis for a detailed derivation of a Pfaffian…

The arithmetic of N, Z, Q, R can be extended to a graph arithmetic where N is the semiring of finite simple graphs and where Z and Q are integral domains, culminating in a Banach algebra R. A single network completes to the Wiener algebra.…

Let G be a connected bipartite graph with color classes E and V and root polytope Q. Regarding the hypergraph (V,E) induced by G, we prove that its interior polynomial is equivalent to the Ehrhart polynomial of Q, which in turn is…

We prove that any finite collection of quadratic forms (overlaps) of general deterministic matrices and eigenvectors of an $N\times N$ Wigner matrix has joint Gaussian fluctuations. This can be viewed as the random matrix analogue of the…

Counting homomorphisms from a graph $H$ into another graph $G$ is a fundamental problem of (parameterized) counting complexity theory. In this work, we study the case where \emph{both} graphs $H$ and $G$ stem from given classes of graphs:…

We prove two multivariate $q$-binomial identities conjectured by Bousseau, Brini and van Garrel [Geom. Topol. 28 (2024), 393-496, arXiv:2011.08830] which give generating series for Gromov-Witten invariants of two specific log Calabi-Yau…

We establish a generalization of Bourgain double recurrence theorem by proving that for any map $T$ acting on a probability space $(X,\mathcal{A},\mu)$, and for any non-constant polynomials $P, Q$ mapping natural numbers to themselves, for…

The correlation functions of supersymmetric gauge theories on a four-manifold X can sometimes be expressed in terms of topological invariants of X. We show how the existence of superconformal fixed points in the gauge theory can provide…

We show that for any integer $N$, there are only finitely many cuspidal algebraic automorphic representations of ${\rm GL}_n$ over $\mathbb{Q}$, with $n$ varying, whose conductor is $N$ and whose weights are in the interval…

We study character varieties arising as moduli of representations of an orientable surface group into a reductive group $G$. We first show that if $G/Z$ acts freely on the representation variety, then both the representation variety and the…

Let $G = (V,E)$ be a graph, and for each $e \in E(G)$, let $L_e$ be a list of real numbers. Let $w:E(G) \to \cup_{e \in E(G)}L_e$ be an edge weighting function such that $w(e) \in L_e$ for each $e \in E(G)$, and let $c_w$ be the vertex…

We give an explicit and simple construction of the incidence graph for the integral cohomology of real Grassmann manifold Gr(k,n) in terms of the Young diagrams filled with the letter q in checkered pattern. It turns out that there are two…

We describe methods for calculation of polytopes of quasiadjunction for plane curve singularities which are invariants giving a Hodge theoretical refinement of the zero sets of multivariable Alexander polynomials. In particular we identify…

Let {\cal G}=(G,w) be a positive-weighted simple finite graph, that is, let G be a simple finite graph endowed with a function w from the set of the edges of G to the set of the positive real numbers. For any subgraph G' of G, we define…

The long-standing Gaussian product inequality (GPI) conjecture states that $E [\prod_{j=1}^{n}X_j^{2m_j}]\geq\prod_{j=1}^{n}E[X_j^{2m_j}]$ for any centered Gaussian random vector $(X_1,\dots,X_n)$ and $m_1,\dots,m_n\in\mathbb{N}$. In this…

In this work, we introduce the boundary polynomial of a graph $G$ as the ordinary generating function in two variables $B(G;x,y):= \displaystyle\sum_{S\subseteq V(G)} x^{|B(S)|}y^{|S|}$, where $B(S)$ denotes the outer boundary of $S$. We…

We consider Weil sums of binomials of the form $W_{F,d}(a)=\sum_{x \in F} \psi(x^d-a x)$, where $F$ is a finite field, $\psi\colon F\to {\mathbb C}$ is the canonical additive character, $\gcd(d,|F^\times|)=1$, and $a \in F^\times$. If we…

Recently, Brouwer, Cioab\u{a}, Ihringer and McGinnis obtained some new results involving the eigenvalues of various graphs coming from association schemes and posed some conjectures related to the eigenvalues of Grassmann graphs, bilinear…

We investigate a hierarchy of semidefinite bounds $\vartheta^{(r)}(G)$ for the stability number $\alpha(G)$ of a graph $G$, based on its copositive programming formulation and introduced by de Klerk and Pasechnik [{\em SIAM J. Optim.} 12…