Related papers: Computing Kazhdan constants by semidefinite progra…
The discrete logarithm problem in Jacobians of curves of high genus $g$ over finite fields $\FF_q$ is known to be computable with subexponential complexity $L_{q^g}(1/2, O(1))$. We present an algorithm for a family of plane curves whose…
Let G be a locally compact abelian group (LCA group) and U be an open, 0-symmetric set. Let F:=F(U) be the set of all real valued continuous functions from G to R which are supported in U and are positive definite. The Turan constant T(U)…
In 2010, Invent. Math., Ershov and Jaikin-Zapirain proved Kazhdan's property (T) for elementary groups. This expository article focuses on presenting an alternative simpler proof of that. Unlike the original one, our proof supplies no…
Notions of higher Kazhdan property can be defined in terms of vanishing of unitary group cohomology in higher degrees. Garland's theorem for simple groups over non-archimedean fields provides the first examples of a higher Kazhdan property.…
We establish a new spectral criterion for Kazhdan's property $(T)$ which is applicable to a large class of discrete groups defined by generators and relations. As the main application, we prove property $(T)$ for the groups $EL_n(R)$, where…
Let $\Lambda$ be a subgroup of an arithmetic lattice in SO(n+1,1). The quotient $\mathbb{H}^{n+1} / \Lambda$ has a natural family of congruence covers corresponding to primes in some ring of integers. We establish a super-strong…
We establish several fine boundary regularity results of weak solutions to non-homogeneous $s$-fractional Laplacian type equations. In particular, we prove sharp Calder\'on-Zygmund type estimates of $u/d^s$ depending on the regularity…
As an indicator of the stability of spectral clustering of an undirected weighted graph into $k$ clusters, the $k$th spectral gap of the graph Laplacian is often considered. The $k$th spectral gap is characterized in this paper as an…
The Lipschitz constant is an important quantity that arises in analysing the convergence of gradient-based optimization methods. It is generally unclear how to estimate the Lipschitz constant of a complex model. Thus, this paper studies an…
We present an algorithm for solving the discrete logarithm problem in Jacobians of families of plane curves whose degrees in $X$ and $Y$ are low with respect to their genera. The finite base fields $\FF_q$ are arbitrary, but their sizes…
The word stable is used to describe a situation when mathematical objects that almost satisfy an equation are close to objects satisfying it exactly. We study operator-algebraic forms of stability for unitary representations of groups and…
We study lattices in a product $G = G_1 \times \dots \times G_n$ of non-discrete, compactly generated, totally disconnected locally compact (tdlc) groups. We assume that each factor is quasi just-non-compact, meaning that $G_i$ is…
A group may be considered $C^*$-stable if almost representations of the group in a $C^*$-algebra are always close to actual representations. We initiate a systematic study of which discrete groups are $C^*$-stable or only stable with…
We reformulate and extend the geometric method for proving Kazhdan property T developed by Dymara and Januszkiewicz and used by Ershov and Jaikin. The main result says that a group G, generated by finite subgroups G_i, has property T if the…
In this paper, we provide several instances in which interesting approximation and stability properties are inherited by quotients with respect to finitely generated normal subgroups or, more strongly, normal subgroups with Kazhdan's…
We have compared numerical simulations to observations for the nearby (< 40 Mpc) groups of galaxies (Huchra & Geller 1982 and Ramella et al. 2002). The group identification is carried out using a group-finding algorithm developed by Huchra…
A much-studied problem posed by Motzkin asks to determine, given a finite set $D$ of integers, the so-called Motzkin density for $D$, i.e., the supremum of upper densities of sets of integers whose difference set avoids $D$. We study the…
Class groups of real quadratic fields represent fundamental structures in algebraic number theory with significant computational implications. While Stark's conjecture establishes theoretical connections between special units and class…
The long-standing problem of minimal projections is addressed from a computational point of view. Techniques to determine bounds on the projection constants of univariate polynomial spaces are presented. The upper bound, produced by a…
For $q,n,d \in \mathbb{N}$, let $A_q^L(n,d)$ denote the maximum cardinality of a code $C \subseteq \mathbb{Z}_q^n$ with minimum Lee distance at least $d$, where $\mathbb{Z}_q$ denotes the cyclic group of order $q$. We consider a…