Related papers: The mixing conjecture under GRH

Mixture of experts (MoE) models are widely applied for conditional probability density estimation problems. We demonstrate the richness of the class of MoE models by proving denseness results in Lebesgue spaces, when inputs and outputs…

We obtain a combinatorial formula for the Miller-Morita-Mumford classes for the mapping class group of punctured surfaces and prove Witten's conjecture that they are proportional to the dual to the Witten cycles. The proportionality…

We conjecture the existence of special elements in odd degree higher algebraic K-groups of number fields that are related in a precise way to the values at strictly negative integers of the derivatives of Artin L-functions of finite…

Let $\textbf{G}$ be a simply connected semisimple algebraic group over a field of characteristic greater than the Coxeter number. We construct a monoidal action of the diagrammatic Hecke category on the principal block…

In 2019, P. Higgins formulated [1] a question about bipartite graphs (see Conjecture 1 below); this question arises in the study of regular finite semigroups. F. V. Petrov formulated [2] another combinatorial conjecture (Conjecture 3);…

Let G be a connected reductive real Lie group, and H a compact connected subgroup. Harish-Chandra associates to a regular coadjoint admissible orbit M of G some unitary representations of G. Using the character formula for these…

Let $(R, \mathfrak m)$ be a Noetherian local ring. In this work we extend the notion of mixed multiplicities of modules, given in \cite{Kleiman-Thorup2} and \cite{Kirby-Rees1} (see also \cite{Bedregal-Perez}), to an arbitrary family…

In [7], Higashitani, Kummer, and Micha{\l}ek pose a conjecture about the symmetric edge polytopes of complete multipartite graphs and confirm it for a number of families in the bipartite case. We confirm that conjecture for a number of new…

The Ghahramani-Lau conjecture is established; in other words, the measure algebra of every locally compact group is strongly Arens irregular. To this end, we introduce and study certain new classes of measures (called approximately…

We prove an equidistribution result for Hecke operators acting on the basic stratum of certain Shimura varieties. We relate the rate of convergence to the bounds from the Ramanujan conjecture of certain cuspidal automorphic representations…

We prove a version of weakly functorial big Cohen-Macaulay algebras that suffices to establish Hochster-Huneke's vanishing conjecture for maps of Tor in mixed characteristic. As a corollary, we prove an analog of Boutot's theorem that…

We study families of partitions with gap conditions that were introduced by Schur and Andrews, and describe their fundamental connections to combinatorial q-series and automorphic forms. In particular, we show that the generating functions…

While examples of Ramanujan-type congruences are amply available via their relation to Hecke operators, it remains unclear which of them should be considered of combinatorial origin and which of them are mere artifacts of the connection…

We prove two new cases of the Brou\'e-Malle-Rouquier freeness conjecture for the Hecke algebras associated to complex reflection groups. These two cases are the complex reflection groups of rank 2 called $G_{20}$ and $G_{21}$ in the…

We prove joint universality theorems on the half plane of absolute convergence for general classes of Dirichlet series with an Euler-product, where in addition to vertical shifts we also allow scaling. This generalizes our recent joint…

In this paper we extend some classical results of Convex Analysis to the sub-Riemannian setting of the Heisenberg group. In particular, we provide a horizontal version of Minty's theorem concerning maximal H-monotone operators defined in…

Let $K$ be the function field of a $p$-adic curve, $G$ a semisimple simply connected group over $K$ and $X$ a $G$-torsor over $K$. A conjecture of Colliot-Th\'el\`ene, Parimala and Suresh predicts that if for every discrete valuation $v$ of…

We prove the Hodge-Riemann bilinear relations, the hard Lefschetz theorem and the Lefschetz decomposition for compact Kahler manifolds in the mixed situation.

The Mishchenko-Fomenko conjecture says that for each real or complex finite-dimensional Lie algebra $\goth g$ there exists a complete set of commuting polynomials on its dual space $\goth g^*$. In terms of the theory of integrable…

We prove that the complement of a very general pair of hypersurfaces of total degree $2n$ in $\mathbb{P}^n$ is algebraically hyperbolic modulo a proper closed subvariety. This provides evidence towards conjectures of Lang-Vojta and…