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In the framework of the semiclassical approach the universal spectral correlations in the Hamiltonian systems with classical chaotic dynamics can be attributed to the systematic correlations between actions of periodic orbits which (up to…

We present a nonstandard hull construction for locally uniform groups in a spirit similar to Luxembourg's construction of the nonstandard hull of a uniform space. Our nonstandard hull is a local group rather than a global group. We…

Given an extension $0\to V\to G\to Q\to1$ of locally compact groups, with $V$ abelian, and a compatible essentially bijective $1$-cocycle $\eta\colon Q\to\hat V$, we define a dual unitary $2$-cocycle on $G$ and show that the associated…

Partially ordered sets (posets) play a universal role as an abstract structure in many areas of mathematics. For finite posets, an explicit enumeration of distinct partial orders on a set of unlabelled elements is known only up to a…

Following Arthur's study of the representations of the orthogonal and symplectic groups, we prove many cases of both the local and global Arthur conjectures for tempered representations of the unitary group. This completes the proof of…

A compact manifold $M$ together with a Riemannian metric $h$ on its universal cover $\tilde M$ for which $\pi_1(M)$ acts by similarities is called a similarity structure. In the case where $\pi_1(M) \not\subset \mathrm{Isom}(\tilde M, h)$…

Strictly proper kernel scores are well-known tool in probabilistic forecasting, while characteristic kernels have been extensively investigated in the machine learning literature. We first show that both notions coincide, so that insights…

Uncertainty quantification is essential in decision-making, especially when joint distributions of random variables are involved. While conformal prediction provides distribution-free prediction sets with valid coverage guarantees, it…

On unitary compact groups the decomposition of a generic element into product of reflections induces a decomposition of the characteristic polynomial into a product of factors. When the group is equipped with the Haar probability measure,…

We introduce and study a class of discrete particle ensembles that naturally arise in connection with classical random matrix ensembles, log-gases and Jack polynomials. Under technical assumptions on a general analytic potential we prove…

We provide the detailed asymptotic behavior for first-order aggregation models of heterogeneous oscillators. Due to the dissimilarity of natural frequencies, one could expect that all relative distances converge to definite positive value…

Using the theory of group action, we first introduce the concept of the automorphism group of an exponential family or a graphical model, thus formalizing the general notion of symmetry of a probabilistic model. This automorphism group…

We define a class of random measures, spatially independent martingales, which we view as a natural generalisation of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian…

Rational discrete cohomology and homology for a totally disconnected locally compact group $G$ is introduced and studied. The $\mathrm{Hom}$-$\otimes$ identities associated to the rational discrete bimodule $\mathrm{Bi}(G)$ allow to…

The Continuous Spontaneous Localization (CSL) model is the best known and studied among collapse models, which modify quantum mechanics and identify the fundamental reasons behind the unobservability of quantum superpositions at the…

The coadjoint orbits of compact Lie groups each carry a canonical (positive definite) K\"ahler structure, famously used to realize the group's irreducible representations in holomorphic sections of appropriate line bundles (Borel-Weil…

In one of their seminal articles on allowable sequences, Goodman and Pollack gave combinatorial generalizations for three problems in discrete geometry, one of which being the Dirac conjecture. According to this conjecture, any set of $n$…

We study bipartite spin-singlet correlations when rotational symmetry is described by a quantum group rather than an ordinary Lie group. We show that, even though the single-spin observables act as in the undeformed theory, the non-trivial…

This work is motivated by the problem of finding locally compact group topologies for piecewise full groups (a.k.a.~ topological full groups). We determine that any piecewise full group that is locally compact in the compact-open topology…

The main goal of this article is to bring together the theories of holomorphic iteration in the unit disc and semigroups of holomorphic functions. We develop a technique that allows us to partially embed the orbit of a holomorphic self-map…