arXiv research

Complex equivalence classes found in graph homotopy.

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

New tools analyze the complexity of left-ordering equivalence relations in groups and 3-manifolds.

Classifies actions of groups on hyperbolic spaces, proving dichotomy.

Study convex embeddability in linear and circular orders, applying to knots.

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

We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a compact orientable surface with higher complexity. We prove similar rigidity results for a finite direct product of mapping cl…

Geometrically connects Laplace eigenfunctions to Borel-Weil theory on symmetric spaces.

A Hilbert space embedding for probability measures has recently been proposed, wherein any probability measure is represented as a mean element in a reproducing kernel Hilbert space (RKHS). Such an embedding has found applications in homogeneity testing, independence testing, dimensionality reduction, etc., with the re…

Survey of recent measures of association, including a new coefficient.

Let X be a path-connected topological space admitting a universal cover. Let Homeo(X,a) denote the group of homeomorphisms of X preserving degree one cohomology class a. We investigate the distortion in Homeo(X,a). Let g be an element of Homeo(X,a). We define a Nielsen-type equivalence relation on the space of g-invari…

Two 4-manifolds are stably diffeomorphic if they become diffeomorphic after connected sum with S^2 x S^2's. This paper shows that two closed, orientable, homotopy equivalent, smooth 4-manifolds are stably diffeomorphic, provided a certain map from the second homology of the fundamental group with coefficients in Z/2 to…

The paper studies graded manifolds and their functorial relationship.

In the present paper, we prove that a lower bound on the $1$-weighted Ricci curvature is equivalent to a convexity of entropies on the Wasserstein space. Based on such characterization, we provide some interpolation inequalities such as the Pr'ekopa-Leindler inequality, the Borel-Branscamp-Lieb inequality, and the Brun…

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

New findings on cusped Borel Anosov representations and their properties.

Let $G$, $B$, and $H$ denote a complex semi-simple algebraic group, a Borel subgroup of $G$, and a maximal complex torus in $B$, respectively. Choose a compact real form $K$ of $G$ such that $T=K\cap H$ is a maximal torus in $T$. Then there are two models for the flag space of $G$: the complex quotient $X=G/B$ and the …

Paper solves quantum differential equations for projective bundles using Borel multitransforms.

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…

In this note we prove the Borel Conjecture for closed, irreducible and sufficiently collapsed three-dimensional Alexandrov spaces. We also pose several questions related to characterization of fundamental groups of three-dimensional Alexandrov spaces, finite groups acting on them and rigidity results.

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

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…

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

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…

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, …

A one-to-one correspondence is drawn between law invariant risk measures and divergences, which we define as functionals of pairs of probability measures on arbitrary standard Borel spaces satisfying a few natural properties. Divergences include many classical information divergence measures, such as relative entropy a…

We use the reflection group trick to glue manifolds with corners that are Borel-Serre compactifications of locally symmetric spaces of noncompact type and obtain aspherical manifolds. We call these \emph{piecewise locally symmetric} manifolds. This class of spaces provide new examples of aspherical manifolds whose fund…

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…

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

Geometrically proves WKB solutions of Schrödinger equations are resurgent.

We give a criterion of (micro-)kroneckerity of the linear Poisson pencil on $\frak{g}^*$ related to an algebraic Nijenhuis operator $N:\frak{g}\to \frak{g}$ on a finite-dimensional Lie algebra $\frak{g}$. As an application we get a series of examples of completely integrable systems on semisimple Lie algebras related t…

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.

On an odd-dimensional oriented hyperbolic manifold of finite volume with strongly acyclic coefficient systems, we derive a formula relating analytic torsion with the Reidemeister torsion of the Borel-Serre compactification of the manifold. In a companion paper, this formula is used to derive exponential growth of torsi…

Study shows fast rates for inverse reinforcement learning with linear rewards.

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

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…

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…

Study non-perturbative quantum geometry of string theories using finite difference equations and resurgence analysis.

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.

Free group automorphisms group rigidity proven.

This work establishes properties on diffeological structures for set-valued maps and measures.

New invariant recovers known contact element and considers finite coverings.

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…

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

The 2-rank of a compact Lie group $G$ is the maximal possible rank of the elementary 2-subgroup ${\mathbb Z}_{2}\times... {\mathbb Z}_{2}$ of $G$. The study of 2-ranks (and $p$-rank for any prime $p$) of compact Lie groups was initiated in 1953 by A. Borel and J.-P. Serre. Since then the 2-ranks of compact Lie groups h…