Related papers: Quantitative $(p,q)$ theorems in combinatorial geo…
We prove inverse theorems for the size of sumsets and the $L^q$ norms of convolutions in the discretized setting, extending to arbitrary dimension an earlier result of the author in the line. These results have applications to the…
The classical Busemann-Petty problem (1956) asks, whether origin-symmetric convex bodies in $\mathbb {R}^n$ with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if $n\le 4$…
What use can there be for a function from the $p$-adic numbers to the $q$-adic numbers, where $p$ and $q$ are distinct primes? The traditional answer, courtesy of the half-century old theory of non-archimedean functional analysis: not much.…
This paper studies projections of uniform random elements of (co)adjoint orbits of compact Lie groups. Such projections generalize several widely studied ensembles in random matrix theory, including the randomized Horn's problem, the…
This paper transfers a randomized algorithm, originally used in geometric optimization, to computational problems in commutative algebra. We show that Clarkson's sampling algorithm can be applied to two problems in computational algebra:…
Set partitions and permutations with restrictions on the size of the blocks and cycles are important combinatorial sequences. Counting these objects lead to the sequences generalizing the classical Stirling and Bell numbers. The main focus…
Using a recent result of Orponen (Invent. math. '21), we show that sets with plenty of big projections (PBP) admit an Analyst's Travelling Salesman Theorem. We then show that sets with PBP which are uniformly non-flat (or wiggly) have large…
We announce two breakthrough results concerning important questions in the Theory of Computational Complexity. In this expository paper, a systematic and comprehensive geometric characterization of the Subset Sum Problem is presented. We…
The generalized volume conjecture relates asymptotic behavior of the colored Jones polynomials to objects naturally defined on an algebraic curve, the zero locus of the A-polynomial $A(x,y)$. Another "family version" of the volume…
The classical Busemann-Petty problem asks whether smaller central hyperplane sections of origin-symmetric convex bodies necessarily imply smaller total volume. Zvavitch studied this question for arbitrary measures with continuous even…
We introduce two explicit examples of polynomials orthogonal on the unit circle. Moments and the reflection coefficients are expressed in terms of Jacobi elliptic functions. We find explicit expression for these polynomials in terms of a…
We establish a novel connection between the well-known chromatic threshold problem in extremal combinatorics and the celebrated $(p,q)$-theorem in discrete geometry. In particular, for a graph $G$ with bounded clique number and a natural…
In this article we discuss classical theorems from Convex Geometry in the context of topological drawings and beyond. In a simple topological drawing of the complete graph $K_n$, any two edges share at most one point: either a common vertex…
We prove two results on convex subsets of Euclidean spaces invariant under an orthogonal group action. First, we show that invariant spectrahedra admit an equivariant spectrahedral description, i.e., can be described by an equivariant…
We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures (POVMs), as opposed to the restricted class of orthogonal projection-valued measures used in the original…
In the general context of computable metric spaces and computable measures we prove a kind of constructive Borel-Cantelli lemma: given a sequence (constructive in some way) of sets $A_{i}$ with effectively summable measures, there are…
Assume that $k \le d$ is a positive integer and $\C$ is a finite collection of convex bodies in $\R^d$. We prove a Helly type theorem: If for every subfamily $\C^*\subset \C$ of size at most $\max \{d+1,2(d-k+1)\}$ the set $\bigcap \C^*$…
This is an overview of Erlangen Programme at Large. Study of objects and properties, which are invariant under a group action, is very fruitful far beyond the traditional geometry. In this paper we demonstrate this on the example of the…
We deduce some growth properties of composite entire functions in the light of their relative $(p,q)$\ th order by extending some results of J. Tu, Z. X. Chen and X. M. Zheng.
We develop geometry-of-numbers methods to count orbits in coregular vector spaces having bounded invariants over any global field. We apply these techniques to bound the average ranks and determine average Selmer group sizes of elliptic…