arXiv research

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

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

Geometrically proves WKB solutions of Schrödinger equations are resurgent.

Complex equivalence classes found in graph homotopy.

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

Extends holomorphic functions on complex manifolds to larger spaces.

Resurgence of Joyce structures gauged to a standard form using gauge transformations.

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

Quantum dilogarithm function proven from a linear difference equation.

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

New findings on cusped Borel Anosov representations and their properties.

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

For a hyperbolic link complement with a triangulation, there are hyperbolicity equations of the triangulation, which guarantee the hyperbolic structure of the link complement. In this paper, we explain that the number of the essential solutions of the equations is equal to or bigger than the extension degree of the inv…

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

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

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…

If $Γ<\mathrm{PSL}(2,\mathbb{C})$ is a lattice, we define an invariant of a representation $Γ\rightarrow \mathrm{PSL}(n,\mathbb{C})$ using the Borel class $β(n)\in \mathrm{H}^3_\mathrm{c}(\mathrm{PSL}(n,\mathbb{C}),\mathbb{R})$. We show that the invariant is bounded and its maximal value is attained by conjugation of 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.

Classifies actions of groups on hyperbolic spaces, proving dichotomy.

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.

Let G be a finite group. The unit sphere in a finite-dimensional orthogonal G-representation motivates the definition of homotopy representations, due to tom Dieck. We introduce an algebraic analogue, and establish its basic properties including the Borel-Smith conditions and realization by finite G-CW-complexes.

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.

We continue our study of the Complex Monge-Ampère Operator on the Weighted Pluricomplex energy classes. We give more characterizations of the range of the classes $\mathcal E_ χ$ by the Complex Monge-Ampère Operator. In particular, we prove that a non-negative Borel measure $μ$ is the Monge-Ampère of a unique function …

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

Proves summability of state integrals for specific hyperbolic knots.

Let G be a connected complex semi-simple group, B a Borel subgroup of G, and T a maximal torus in B. We construct a class of smooth T-stable subvarieties inside the flag variety G/B, each of which is an embedding of a product of projective lines.

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.

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…

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

Study shows how nonstandard hulls can contain metric completions.

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

In this note we prove that the Borel class of representations of 3-manifold groups to PGL(n,C) is preserved under Cartan involution up to sign. For representations to PGL(3,C) this is implied by a more general result of E. Falbel and Q. Wang, however our proof appears to be much shorter for that special case.

We solve the Dirichlet problem for the complex Monge-Ampère equation on a strictly pseudoconvex with the right hand side being a positive Borel measure which is dominated by the Monge-Ampère measure of a Hölder continuous plurisubharmonic function. If the boundary data is continuous, then the solution is continuous. If…

We prove that a word hyperbolic group which admits a $P_{2q+1}$-Anosov representation into $\mathsf{PGL}(4q+2, \mathbb{R})$ contains a finite-index subgroup which is either free or a surface group. As a consequence, we give an affirmative answer to Sambarino's question for Borel Anosov representations into $\mathsf{SL}…

Heat kernel resurgent structure from Picard-Lefschetz theory

New representations of surface groups into higher-dimensional PSL generalize pleated surfaces.

For a compact Lie group acting on a smooth manifold, we define the differential cohomology of a certain quotient stack involving principal bundles with connection. This produces differential equivariant cohomology groups that map to the Cartan-Weil equivariant forms and to Borel's equivariant integral cohomology. We sh…

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

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…

New 4D shapes found that defy smoothness rules.

Novel mathematical approach using resurgent analysis reveals new structures in complex Chern-Simons theory.

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…

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

We prove a generalization of a theorem of Borel-Harish-Chandra on closed orbits of linear actions of reductive groups. Consider a real reductive algebraic group $G$ acting linearly and rationally on a real vector space $V$. $G$ can be viewed as the real points of a complex reductive group $G^\mathbb C$ which acts on $V…