Related papers: Face vectors of flag complexes

This note provides a detailed proof of the fact that a linear vector field on a vector bundle has a flow by vector bundle isomorphisms. It implies then easily the existence of global solutions to linear non-autonomous ODE's, with a standard…

The paper is motivated by the study of graded representations of Takiff algebras, cominuscule parabolics, and their generalizations. We study certain special subsets of the set of weights (and of their convex hull) of the generalized Verma…

A conjecture regarding the structure of expander graphs is discussed.

Matroids are ubiquitous in modern combinatorics. As discovered by Gelfand, Goresky, MacPherson and Serganova there is a beautiful connection between matroid theory and the geometry of Grassmannians: realizable matroids correspond to torus…

We compare various viewpoints on down-sets (simplicial complexes), illustrating how the combinatorial inclusion-exclusion principle may serve as an alternative to more advanced methods of studying their face numbers.

A relative simplicial complex is a collection of sets of the form $\Delta \setminus \Gamma$, where $\Gamma \subset \Delta$ are simplicial complexes. Relative complexes played key roles in recent advances in algebraic, geometric, and…

We propose a formulation of the Equivariant Tamagawa Number Conjecture for modular motives with coefficients in universal deformation rings and Hecke algebras; something which seems to have been heretofore missing because the complexes of…

Iosevich and Senger (2008) showed that if a subset of the d-dimensional vector space over a finite field is large enough, then it contains many k-tuples of mutually orthogonal vectors. In this note, we provide a graph theoretic proof of…

We formulate some conjectures about the K-theory of symplectic manifolds and their Fukaya categories, and prove some of them in very special cases.

In this paper we prove that a Finsler metrics has constant flag curvature if and only if the curvature of the induced nonlinear connection satisfies an algebraic identity with respect to some arbitrary second rank tensors. Such algebraic…

A complete classification of isotropic vector equations of the geometric type that possess higher symmetries is proposed. New examples of integrable multi-component systems of the geometric type and their auto-Backlund transformations are…

Given two arbitrary vector bundles on the Fargues-Fontaine curve, we completely classify all vector bundles which arise as their extensions.

Eberhard-type theorems are statements about the realizability of a polytope (or more general polyhedral maps) given the valency of its vertices and sizes of its polygonal faces up to a linear linear degree of freedom. We present new…

The half-open hypersimplex $\Delta'_{n,k}$ consists of those $x = (x_{1}, \ldots, x_{n}) \in[0,1]^n$ with $k-1<x_1+\cdots+x_n\le k$, where $0 < k \leq n$. The $f$-vector of a half-open hypersimplex and related generating functions are…

We prove a new family of total positivity criteria for partial flag varieties for simply-connected complex algebraic group in the simply laced case.

One describes, using a detailed analysis of Atiyah--Hirzebruch spectral sequence, the tuples of cohomology classes on a compact, complex manifold, corresponding to the Chern classes of a complex vector bundle of stable rank. This…

We give a new definition of lattice-face polytopes by removing an unnecessary restriction in the paper "Ehrhart polynomials of lattice-face polytopes", and show that with the new definition, the Ehrhart polynomial of a lattice-face polytope…

Lurie's representability theorem gives necessary and sufficient conditions for a functor to be an almost finitely presented derived geometric stack. We establish several variants of Lurie's theorem, making the hypotheses easier to verify…

The edge group of a simplicial complex is a well-known, combinatorial version of the fundamental group. It is a group associated to a simplicial complex that consists of equivalence classes of edge loops and that is isomorphic to the…

Under the assumption of the Hodge, Tate and Fontaine-Mazur conjectures we give a criterion for a compatible system of l-adic representations to be isomorphic to the second cohomology of a K3 surface.