Mathematics

We show that every stable UCT Kirchberg algebra has a principal \'etale groupoid model, and thus contains a C$^*$-diagonal. Every unital UCT Kirchberg algebra $A$ for which $[1_A]_0$ has infinite order in $K_0(A)$ is also covered by our…

We generalize the classical Ceva's and Menelaus's theorems to curvilinear triangles bounded by circular arcs. We introduce trilinear coordinates associated with such triangles and develop several geometric constructions. In particular, for…

The term Gibbons conjecture is widely used in connection with symmetry results for the Allen-Cahn equation. However, its origin is less transparent than its frequent citation suggests. In this note, we revisit its emergence, tracing it to a…

In this paper we prove the WALA conjecture.

We consider a smooth compact manifold with boundary, $M$, embedded in a smooth manifold of the same dimension on which an amenable group $\Gamma$ acts by isometries. We do not assume $M$ to be invariant under $\Gamma$. This results in a…

In this paper, we study the dual Minkowski problem under group symmetry. For $0<q\le n$, we give a complete existence characterization in the framework of $G$-invariant convex bodies when the group $G\subset O(n)$ has no nonzero fixed…

We obtain a Galois correspondence between the lattice of intermediate C*-discrete subalgebras intermediate to a given irreducible C*-discrete inclusion, and characterize these as targets of compatible expectations under a traciality…

We investigate some genuine Poisson geometric objects in the modular theory of an arbitrary von Neumann algebra $\mathfrak{M}$. Specifically, for any standard form realization $(\mathfrak{M},\mathcal{H},J,\mathcal{P})$, we find a canonical…

This paper gives a complete description of the solutions of the global positioning problem, emphasizing the under-determined case. We show that the solutions form a quadric, which may degenerate in various ways. Perhaps more surprisingly,…

We introduce the notion of $k$-regular factorizations for contractions into $k$ factors, generalizing the classical notion of regular factorization due to Sz.-Nagy and Foia\c{s}, and develop a systematic framework for their analysis. Using…

Let $G$ be a finitely generated virtually abelian group and $[\sigma]\in H^2(G;\mathbb{T})$ such that $\sigma(x,y)$ is always a root of unity. We show that the nuclear dimension of the twisted group $C^*$-algebra $C^*(G,\sigma)$ is equal to…

In one of his posthumous papers, conserved in G\"ottingen, Riemann considers the derivatives of $\log\zeta(s)$ at the point $1/2$, giving explicit values for them. Around 2010 we shared Riemann's value of the second derivative with some…

We define a distance analogous to the Gromov-Hausdorff distance that enables the comparison of arbitrary quasi-isometric spaces. We also investigate properties preserved under limits with respect to this distance, as well as properties of…

We extend a property of Mazzola's theory of cadential sets in relation to the modulation between minor and major tonalities from triadic to tetradic harmony, using the PLRQ group of Cannas et al. (2017) as the analogue of the classical PLR…

In this paper, we explore the connections between Christiaan Huygens and Niels Henrik Abel through the tautochrone problem. The problem -- determining the curve along which a particle descends under gravity in the same time, regardless of…

We prove a generalization of the orthogonality relations of Duflo and Moore for ergodic, trace-preserving group actions on von Neumann algebras that are integrable in a suitable sense. We also obtain convolution inequalities that generalize…

We study the bi-Lipschitz embedding problem for a class of metric spaces called slit carpets. First we show that the $n$th stage $\mathbb{M}_n$ of the standard slit carpet of Merenkov admits a bi-Lipschitz embedding into Euclidean space…

We study the Fredholm solvability for a new class of nonlocal boundary value problems associated with group actions on smooth manifolds. Namely, we consider the case in which the group action is defined on an ambient manifold without…

The chord-power integrals $I_k$ are classical integral-geometric functionals of a planar convex body, obtained by integrating powers of the chord length against the kinematic measure on the space of lines meeting the body. We establish a…

Semiclassical analysis and noncommutative geometry are two pillars of quantum theory. It is only recently that bridges between them have been emerging. In this monograph, we combine various techniques from functional analysis and spectral…