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We study highest weight vectors for symmetric and alternating spaces of tensors, whose dimensions are given by generalized Kronecker coefficients. We describe the algebraic relations for classical constructions of corresponding spanning…
In this article, we prove a weighted version of Saitoh's conjecture. As an application, we prove a weighted version of Saitoh's conjecture for higher derivatives.
We prove refined generating series formulae for characters of (virtual) cohomology representations of external products of suitable coefficients, e.g., (complexes of) constructible or coherent sheaves, or (complexes of) mixed Hodge modules…
We prove an effective version of the Oppenheim conjecture with a polynomial error rate. The proof is based on an effective equidistribution theorem which in turn relies on recent progress towards restricted projection problem.
The existence of a "Plastikstufe" for a contact structure implies the Weinstein conjecture for all supporting contact forms.
We construct the geometric Langlands functor in one direction (from the automorphic to the spectral side) in characteristic zero settings (i.e., de Rham and Betti). We prove that various forms of the conjecture (de Rham vs Betti, restricted…
The Wehrl entropy conjecture for coherent (highest weight) states in representations of the Heisenberg group, which was proved in 1978 and recently extended by us to the group $SU(2)$, is further extended here to symmetric representations…
In this short note, we prove Hadwiger's conjecture for strongly monotypic polytopes.
We introduce the $\omega$-Vaught's conjecture, a strengthening of the infinitary Vaught's conjecture. We believe that if one were to prove the infinitary Vaught's conjecture in a structural way without using techniques from higher recursion…
We give direct and inverse theorems for the weighted approximation of functions with inner singularities by combinations of Bernstein polynomials.
We isolate a combinatorial property of capacities leading to a construction of proper forcings. Then we show that many classical capacities such as the Newtonian capacity satisfy the property.
We study the properties of the multiplicative structure on valuations on convex sets. We prove a new version of the hard Lefschetz theorem for even translation invariant continuous valuations, and discuss related problems of integral…
We prove some constructive results that on first and maybe even on second glance seem impossible.
According to the similarity theorem on the distributions of the effective prime factors and by using two-part method, Goldbach theorem and, consequently, Goldbach conjecture was proved.
We state and verify up to weight 172 a conjecture on the existence of a certain generating set for spaces of classical Siegel modular forms. This conjecture is particularly useful for calculations involving Fourier expansions. Using this…
We prove a closed formula for the derivative, of any order, of a implicit function, in terms of some binomial building blocks, and explain the combinatorics behind the coefficients appearing in the formula.
In this paper we introduce a new method to produce lower bounds for the Waring rank of symmetric tensors. We also introduce the notion of $e$-computability and we use it to prove that Strassen's Conjecture holds in infinitely many new…
We prove the constructive version of Birkhoff's ergodic theorem following Vyugin but trying to separate and state explicitly the combinatorial statement on which this proof is based. We pose some questions related to this statement (and the…
In this article we consider the exterior power and the symmetric tensors of the polynomial ring in one variable. The structure of an associative semigraded algebra of this polynomial ring induces on the symmetric tensors the structure of an…
By analogy with Weinstein's neighbourhood theorem, we prove a uniqueness result for symplectic neighbourhoods of a large family of stratified subspaces. This result generalizes existing constructions, e.g., in the search for exotic…