arXiv research

Study shows Roller compactification's median graph has limited asymptotic dimension.

The work discusses equivariant asymptotic dimension (also known as "wide equivariant covers", "$N$-$\mathcal F$-amenability" or "amenability dimension", and "$d$-BLR condition") and its generalisation, transfer reducibility, which are versions of asymptotic dimension invented for the proofs of the Farrell--Jones and Bo…

In this project we further investigate the idea of reducing the dimensionality of datasets using a Borel isomorphism with the purpose of subsequently applying supervised learning algorithms, as originally suggested by my supervisor V. Pestov (in 2011 Dagstuhl preprint). Any consistent learning algorithm, for example kN…

The study shows how certain ODEs and integrals are regular under Borel summation.

Equivariant homotopy methods developed over the last 20 years lead to recent breakthroughs in the Borel isomorphism conjectures for Loday assembly maps in K- and L-theories. An important consequence of these algebraic conjectures is the topological rigidity of compact aspherical manifolds. Our goal is to strip the basi…

Study proves hyperbolic groups have specific subgroup properties.

We show that the Borel sums of the Voros symbols considered in the theory of exact WKB analysis arise naturally as Fock-Goncharov coordinates of framed $PGL_2(\mathbb{C})$-local systems on a marked bordered surface. Using this result, we show that these Borel sums can be meromorphically continued to any point of $\math…

Let $\mathcal M_g$ denote the moduli space of compact Riemann surfaces of genus $g$ and let $\mathcal A_g$ be the space of principally polarized abelian varieties of (complex) dimension $g$. Let $J:\mathcal M_g\longrightarrow \mathcal A_g$ be the map which associates to a Riemann surface its Jacobian. The map $J$ is in…

New 4D shapes found that defy smoothness rules.

Constructs projective moduli spaces for Calabi-Yau pairs.

Study on Chern-Simons theory at generic levels, revealing universal resurgent structure.

The paper studies graded manifolds and their functorial relationship.

New findings on null measurability in symmetrization interface of VC learning.

Mathematical structures link Gromov-Witten to Donaldson-Thomas invariants.

Heat kernel resurgent structure from Picard-Lefschetz theory

Explicitly found generators of cohomology for SL_n(Z) using sharbly cycles and cosharbly cocycles.

The Borel Conjecture predicts that closed aspherical manifolds are topological rigid. We want to investigate when a non-aspherical oriented connected closed manifold M is topological rigid in the following sense. If f: N --> M is an orientation preserving homotopy equivalence with a closed oriented manifold as target, …

We study spaces obtained from a complete finite volume complex hyperbolic n-manifold M by removing a compact totally geodesic complex (n-1)-submanifold. The main result is that the fundamental group of M-S is relatively hyperbolic, relative to fundamental groups of the ends of M-S, and M-S admits a complete finite volu…

We show that in all dimensions >7 there are closed aspherical manifolds whose fundamental groups have nontrivial center but do not possess any topological circle actions. This disproves a conjectured converse (proposed by Conner and Raymond) to a classical theorem of Borel.

New findings on cusped Borel Anosov representations and their properties.

The group of simplicial automorphisms of a Tits-Kac-Moody ininite building of thickness q associated to a cocompact reflexion group with fundamental domain a simplex, is Kazhdan for q sufficiently large. Thus we obtain families of new Kazhdan groups: two in dimension 3 and one in dimension 4. The proof uses continuos c…

We will prove that Ruelle L-function for a cuspidal local system on an odd dimensional hyperbolic manifold with finite volume satisfies a functional equation and an analog of the Riemann hypothesis. We will also compute its Laurent expansion at the origin and will prove that the second coefficient coincides with a rati…

We study resurgence properties of partition function of SU(2) Chern-Simons theory (WRT invariant) on closed three-manifolds. We check explicitly that in various examples Borel transforms of asymptotic expansions posses expected analytic properties. In examples that we study we observe that contribution of irreducible f…

We establish the submaximal symmetry dimension for Riemannian and Lorentzian conformal structures. The proof is based on enumerating all subalgebras of orthogonal Lie algebras of sufficiently large dimension and verifying if they stabilize a non-zero Weyl tensor up to scale. Our main technical tools include Dynkin's cl…

Let $Γ$ be the fundamental group of a complete hyperbolic $3$-manifold $M$ with toric cusps. We define the $ω$-Borel invariant $β_n^ω(ρ_ω)$ associated to a representation $ρ_ω: Γ\rightarrow SL(n,\mathbb{C}_ω)$, where $\mathbb{C}_ω$ is a field which can be constructed as a quotient of a suitable subset of $\mathbb{C}^\m…

Study rigidity of real moment-angle manifolds using cubical geometry.

Proves resurgent nature of a series solution to deformed Painlevé I equation.

Geometrically proves WKB solutions of Schrödinger equations are resurgent.

We characterize groups admitting Anosov representations into $\mathsf{SL}(3,\mathbb R)$, projective Anosov representations into $\mathsf{SL}(4,\mathbb R)$, and Borel Anosov representations into $\mathsf{SL}(4,\mathbb R)$. More generally, we obtain bounds on the cohomological dimension of groups admitting $P_k$-Anosov r…

We describe an explicit semi-algebraic partition for the complement of a real hyperplane arrangement such that each piece is contractible and so that the pieces form a basis of Borel-Moore homology. We also give an explicit correspondence between the de Rham cohomology and the Borel-Moore homology.

The infinitesimal symmetry algebra of any Cartan geometry has maximum dimension realized by the flat model, but often this dimension drops significantly when considering non-flat geometries, so a gap phenomenon arises. For general (regular, normal) parabolic geometries of type (G,P), we use Tanaka theory to derive a un…

We call a group FJ if it satisfies the $K$- and $L$-theoretic Farrell-Jones conjecture with coefficients in $\mathbb Z$. We show that if $G$ is FJ, then the simple Borel conjecture (in dimensions $\ge 5$) holds for every group of the form $G\rtimes\mathbb Z$. If in addition $Wh(G\times \mathbb Z)=0$, which is true for …

We show that the bounded Borel class of any dense representation $ρ: G\to \PSL_n\bC$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group $G$. When $n=2$, the Borel class is equal to the $3$-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, …

The paper discusses rigidity results for inequalities on weighted Riemannian manifolds.

We introduce the notion of pullback along a measurable cocycle and we use it to extend the Borel invariant studied by Bucher, Burger and Iozzi to the world of measurable cocycles. The Borel invariant is constant along cohomology classes and has bounded absolute value. This allows to define maximal cocycles. We conclude…

We prove the Borel Conjecture for a class of groups containing word-hyperbolic groups and groups acting properly, isometrically and cocompactly on a finite dimensional CAT(0)-space.

New rigidity results for complex and quaternionic moment-angle manifolds.

The paper studies the dimension of limit sets using variational principles and stationary measures.

We show, up to h-cobordism, that the existence and uniqueness of connected sum decompositions of oriented 4-dimensional manifolds is an invariant of homotopy equivalence, assuming that the fundamental group of each summand is "good" in the sense of Freedman and Quinn. On a separate note, we observe that the Borel Conje…

Study on Borel Anosov subgroups in SL(d,R) for d≠5,8k±1.

Maximal and Borel Anosov representations in $Sp(4,\mathbb{R})$ are proven to be Hitchin.

Proves summability of state integrals for specific hyperbolic knots.

Let $X$ be an algebraic variety over $\mathbb{C}$. We say that $X$ is Borel hyperbolic if, for every finite type reduced scheme $S$ over $\mathbb{C}$, every holomorphic map $S^{an}\to X^{an}$ is algebraic. We use a transcendental specialization technique to prove that $X$ is Borel hyperbolic if and only if, for every s…

Let P be a locally finite circle packing in the plane invariant under a non-elementary Kleinian group Gamma and with finitely many Gamma-orbits. When Gamma is geometrically finite, we construct an explicit Borel measure on the plane which describes the asymptotic distribution of small circles in P, assuming that either…

Develops measures for non-Borel Anosov groups on Furstenberg boundary.

Study shows how nonstandard hulls can contain metric completions.

Extends equivariant contact structure results to mod p L-spaces.

Classifies manifolds and discrete subgroups of Lie groups using descriptive set theory.