arXiv subjects

Elyasheev Leibtag

Publications and source records attributed to Elyasheev Leibtag.

6 recordsLinked to original sources

arXiv2026

Semi-group compactifications of Algebraic Groups

We show that for connected affine algebraic groups over local fields of characteristic zero, the following are equivalent: every continuous homomorphism into a Hausdorff topological group has closed image, every unitary representation decomposes into a direct sum of finite-dimensional representations and representations that are mixing modulo their kernels, and the matrix coefficients are uniformly dense in the algebra of weakly almost periodic functions on the group. In our proof, we employ methods from semigroup theory. We establish that these groups are \emph{compactification-centric}, meaning $sG=Gs$ for every element $s$ in the weakly almost periodic compactification $w(G)$ of the group $G$.

math.GR↗

arXiv2026

Selflessness, MIF and opposition in groups acting on exotic buildings

We prove that groups acting freely and cocompactly on (possibly exotic) affine buildings of type $\tilde{A}_2$ and $\tilde C_2$ are mixed-identity free and have selfless reduced $C^*$-algebras. These results follow from a strong form of ping-pong dynamics that we call 'transversal contractivity'. Our main geometric result regards domesticity properties of elements in the associated polygons at infinity: we prove that, in our context, the opposite geometry of a hyperbolic element is topologically large.

math.GR↗

arXiv2026

On Conservative Matrix Fields: Continuous Asymptotics and Arithmetic

We present the Conservative Matrix Field (CMF) as a tool for the analysis and computation of D-finite functions. We use conservative matrix fields to establish asymptotic properties of families of linear forms in periods, including (but not limited to) multivariate Mellin integrals, via a discrete Levinson-type framework due to Benzaid and Lutz. Finally, we present an experimental analysis of the families of linear forms generated by these objects and formalize the resulting observations as conjectures on their continuous asymptotic and arithmetic properties.

math.NT↗

arXiv2026

The Ramanujan Challenge For AI

To help evaluate the mathematical skills of current AI systems, we present a set of formulas for fundamental mathematical constants. These problems are attractive for AI evaluation because they are concrete and can be checked numerically to arbitrary precision, yet proving them may require non-obvious mathematics. Mathematical constants such as $π$, $e$, Catalan's constant, and special values of the Riemann zeta function have fascinated mathematicians for centuries. The search for formulas evaluating mathematical constants has produced some of the most beautiful mathematics in the field, especially in cases that yield irrationality proofs or fast convergence rates. Ramanujan's legacy is emblematic of this tradition. The list we provide contains two types of problems: formulas whose proofs are known to the authors but will remain encrypted for a short initial period; and formulas that are not yet proven. We are curious to see the achievements of AI in both cases.

math.HO↗

arXiv2026

From Euler to AI: Unifying Formulas for Mathematical Constants

The constant $π$ has fascinated scholars throughout the centuries, inspiring numerous formulas for its evaluation, such as infinite sums and continued fractions. Despite their individual significance, many of the underlying connections among formulas remain unknown, missing unifying theories that could unveil deeper understanding. The absence of a unifying theory reflects a broader challenge across math and science: knowledge is typically accumulated through isolated discoveries, while deeper connections often remain hidden. In this work, we present an automated framework for the unification of mathematical formulas. Our system combines Large Language Models (LLMs) for systematic formula harvesting, an LLM-code feedback loop for validation, and a novel symbolic algorithm for clustering and eventual unification. We demonstrate this methodology on the hallmark case of $π$, an ideal testing ground for symbolic unification. Applying this approach to 455,050 arXiv papers, we validate 385 distinct formulas for $π$ and prove relations between 360 (94%) of them, of which 166 (43%) can be derived from a single mathematical object - linking canonical formulas by Euler, Gauss, Brouncker, and newer ones from algorithmic discoveries by the Ramanujan Machine. Our method generalizes to other constants, including $e$, $ζ(3)$, and Catalan's constant, demonstrating the potential of AI-assisted mathematics to uncover hidden structures and unify knowledge across domains.

math.HO↗

arXiv2023

Homomorphic images of algebraic groups

We study topological group theoretic properties of algebraic groups over local fields. In particular, we find conditions under which such groups have closed images under arbitrary continuous homomorphisms into arbitrary topological groups.

math.GR↗