Related papers: A Hadwiger Theorem for Simplicial Maps

The notion of a simplicial set originated in algebraic topology, and has also been utilized extensively in category theory, but until relatively recently was not used outside of those fields. However, with the increasing prominence of…

To a function with values in the power set of a pre-ordered, separated locally convex space a family of scalarizations is given which completely characterizes the original function. A concept of a Legendre-Fenchel conjugate for set-valued…

All continuous, SL$(n)$ and translation invariant valuations on the space of convex functions on ${\mathbb R}^n$ are completely classified.

The Minkowski tensors are valuations on the space of convex bodies in ${\mathbb R}^n$ with values in a space of symmetric tensors, having additional covariance and continuity properties. They are extensions of the intrinsic volumes, and as…

According to the basic idea of category theory, any Einstein algebra, essentially an algebraic formulation of general relativity, can be considered from the point of view of any object of the category of smooth algebras; such an object is…

A direct relation between the enumeration of ordinary maps and that of fully simple maps first appeared in the work of the first and last authors. The relation is via monotone Hurwitz numbers and was originally proved using Weingarten…

The nonlinear concepts of mixed summable families and maps for the spaces that only non-void sets are developed. Several characterizations of the corresponding concepts are achieved and the proof for a general Pietsch Domination-type…

This note deals with a simultaneous approximation of several matrices by a finite family of diagonalizable matrices satisfying an additional condition for the spectrum of a matrix product. That is the simplicity of all eigenvalues.

This article contains a review of categorifications of semisimple representations of various rings via abelian categories and exact endofunctors on them. A simple definition of an abelian categorification is presented and illustrated with…

We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log…

Let X be an algebraic variety of characteristic zero. Terminal valuations are defined in the sense of the minimal model program, as those valuations given by the exceptional divisors on a minimal model over X. We prove that every terminal…

All measurable and $\operatorname{SL}(n)$-covariant vector valued valuations on convex polytopes containing the origin in their interiors are completely classified. The moment vector is shown to be essentially the only such valuation.

We provide upper bounds for the cardinality of the value set of a polynomial map in several variables over a finite field. These bounds generalize earlier bounds for univariate polynomials.

We study the space of generalized translation invariant valuations on a finite-dimensional vector space and construct a partial convolution which extends the convolution of smooth translation invariant valuations. Our main theorem is that…

We lift the Lefschetz number from an algebraic invariant of maps between spaces to an invariant of morphisms of data over the spaces.

We work over strata of meromorphic differentials with poles of order 1, and on affine subspaces defined by linear conditions on the residues. We propose a definition of the volume of these objects as the integral of a tautological class on…

In this note, we state and give the main ideas of the proof of a real convexity theorem for group-valued momentum maps. This result is a quasi-Hamiltonian analogue of the O'Shea-Sjamaar theorem in the usual Hamiltonian setting. We prove…

We observe that the category of topological space, uniform spaces, and simplicial sets are all, in a natural way, full subcategories of the same larger category, namely the simplicial category of filters; this is, moreover, implicit in the…

We consider questions related to quantizing complex valued functions defined on a locally compact topological group. In the case of bounded functions, we generalize R. Werner's approach to prove the characterization of the associated normal…

A weighted simplicial complex is a simplicial complex with values (called weights) on the vertices. In this paper, we consider weighted simplicial complexes with $\mathbb{R}^2$-valued weights. We study the weighted homology and the weighted…