arXiv research

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

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

Quantum dilogarithm function proven from a linear difference equation.

Wave trace singularity formula for fibre bundles generalizes Poisson summation.

Paper extends trigonometric summation formula with weights.

New findings on cusped Borel Anosov representations and their properties.

In recent work, Cheu et al. (Eurocrypt 2019) proposed a protocol for $n$-party real summation in the shuffle model of differential privacy with $O_{ε, δ}(1)$ error and $Θ(ε\sqrt{n})$ one-bit messages per party. In contrast, every local model protocol for real summation must incur error $Ω(1/\sqrt{n})$, and there exist …

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

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.

New theorem shows every integer can be represented by knot summation.

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 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.

The paper is concerned with the Kontsevich-Zagier formal power series $f(q)=\sum_{n=0}^\infty (1-q)... (1-q^n)$ and its analytic properties. To begin with, we give an explicit formula for the Borel transform of the associated formal power series $F(x)=e^{-1/(24x)}f(e^{-1/x})$ from which its analytic continuation, i…

Improved greedy 2-coordinate updates for optimization problems with constraints.

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.

Stochastic gradient descent updates parameters with summation gradient computed from a random data batch. This summation will lead to unbalanced training process if the data we obtained is unbalanced. To address this issue, this paper takes the error variance and error mean both into consideration. The adaptively adjus…

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.

Kernel density estimation (KDE) is a popular statistical technique for estimating the underlying density distribution with minimal assumptions. Although they can be shown to achieve asymptotic estimation optimality for any input distribution, cross-validating for an optimal parameter requires significant computation do…

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.

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

A new method slices and sums radial kernels faster.

We provide faster algorithms for the problem of Gaussian summation, which occurs in many machine learning methods. We develop two new extensions - an O(Dp) Taylor expansion for the Gaussian kernel with rigorous error bounds and a new error control scheme integrating any arbitrary approximation method - within the best …

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 consider fast kernel summations in high dimensions: given a large set of points in $d$ dimensions (with $d \gg 3$) and a pair-potential function (the {\em kernel} function), we compute a weighted sum of all pairwise kernel interactions for each point in the set. Direct summation is equivalent to a (dense) matrix-vec…

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}…

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

Complex equivalence classes found in graph homotopy.

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.

New 4D shapes found that defy smoothness rules.

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

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…

Dirichlet-Neumann duality for Riemannian submersions

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

Let $G_0$ be a connected, simply connected real simple Lie group. Suppose that $G_0$ has a compact Cartan subgroup $T_0$, so it has discrete series representations. Relative to $T_0$ there is a distinguished positive root system $Δ^+$ for which there is a unique noncompact simple root $ν$, the "Borel -- de Siebenthal s…

Proves resurgence properties for Habiro elements from radial limits of theta series.

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

Cantor Riemannium is a new type of space from holomorphic germs.

Let S be a non-exceptional oriented surface of finite type. We discuss the action of subgroups of the mapping class group of S on the CAT(0)-boundary of the completion of Teichmueller space with respect to the Weil-Petersson metric. We show that the set of invariant Borel probability measures for the Weil-Petersson flo…