arXiv research

Properties of pairs of product conjugate connections are stated with a special view towards the integrability of the given almost product structure. We define the analogous in product geometry of the structural and the virtual tensors from the Hermitian geometry and express the product conjugate connections in terms of…

Survey on conjugate surfaces in product spaces.

Generalizes Collins' theorem to products of locally indicable groups.

Study dispersive estimates for Schrödinger and wave equations on a cone with specific metric.

Geodesic flows on certain surfaces are shown to be semi-conjugate to expansive flows.

The paper classifies 3-manifold groups with specific torsion elements.

New method for pricing financial products without no-arbitrage condition.

Let $G$ be a compact, simply connected Lie group. If $\mathcal{C}_1,\mathcal{C}_2$ are two $G$-conjugacy classes, then the set of elements in $G$ that can be written as products $g=g_1g_2$ of elements $g_i\in \mathcal{C}_i$ is invariant under conjugation, and its image under the quotient map $G\to G/\operatorname{Ad}(G…

Let $G=*_λG_λ$ be a free product of torsion-free groups, and let $g\in[G,G]$ be any element not conjugate into a $G_λ$. Then scl$_G(g)\ge1/2$. This generalizes, and gives a new proof of a theorem of Duncan-Howie.

Let $f\colon M\to M$ be an expansive homeomorphism with dense topologically hyperbolic periodic points, $M$ a compact manifold. Then there is a local product structure in an open and dense subset of $M$. Moreover, if some topologically hyperbolic periodic point has codimension one, then this local product structure is …

A nontrivial element in a group is a generalized torsion element if some nonempty finite product of its conjugates is the identity. We prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental…

The paper studies conjugate points on Lie groups with specific metrics.

We introduce the notion of a conjugation-free geometric presentation for a fundamental group of a line arrangement's complement, and we show that the fundamental groups of the following family of arrangements have a conjugation-free geometric presentation: A real arrangement L, whose graph of multiple points is a union…

Geodesic flows on compact manifolds without conjugate points are shown to have a unique measure of maximal entropy.

In this paper, we study the Ricci flow on closed manifolds equipped with warped product metric $(N\times F,g_{N}+f^2 g_{F})$ with $(F,g_{F})$ Ricci flat. Using the framework of monotone formulas, we derive several estimates for the adapted heat conjugate fundamental solution which include an analog of G. Perelman's dif…

Study on geodesics in Cartan group sub-Riemannian problem, proving conjugate time relation to Maxwell time.

Random groups prove length constraints on product of conjugates.

The study restricts normal subgroups of Kähler groups, proving specific cases and general restrictions.

New algebraic fundamental groups identified for fake projective planes.

We explain a connection between the algebraic and geometric properties of groups of contact transformations, open book decompositions, and flexible Legendrian embeddings. The main result is that, if a closed contact manifold $(V, ξ)$ has a supporting open book whose pages are flexible Weinstein manifolds, then the conn…

This paper classifies reversible and strongly reversible elements in affine groups.

If $G$ is a free product of finite groups, let $ΣAut_1(G)$ denote all (necessarily symmetric) automorphisms of $G$ that do not permute factors in the free product. We show that a McCullough-Miller [D. McCullough and A. Miller, {\em Symmetric Automorphisms of Free Products}, Mem. Amer. Math. Soc. 122 (1996), no. 582] an…

The class of chain event graph models is a generalisation of the class of discrete Bayesian networks, retaining most of the structural advantages of the Bayesian network for model interrogation, propagation and learning, while more naturally encoding asymmetric state spaces and the order in which events happen. In this…

We prove certain localized and global differential Harnack inequality for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace-Beltrami operator is replaced with "$…

New generalized torsion found in 3-manifolds.

This paper introduces Jensen, an easily extensible and scalable toolkit for production-level machine learning and convex optimization. Jensen implements a framework of convex (or loss) functions, convex optimization algorithms (including Gradient Descent, L-BFGS, Stochastic Gradient Descent, Conjugate Gradient, etc.), …

Classifies reversible and strongly reversible elements in quaternionic groups.

The paper improves sparse Gaussian processes by optimizing predictive loss.

A long-standing conjecture asserts that any Anosov diffeomorphism of a closed manifold is finitely covered by a diffeomorphism which is topologically conjugate to a hyperbolic automorphism of a nilpotent manifold. In this paper, we show that any closed 4-manifold that carries a Thurston geometry and is not finitely cov…

Global rigidity theorem for metrics on Σ×S1 without conjugate points.

Unified Kantorovich duality for multimarginal optimal transport on Polish spaces.

The energy of any $C^1$ representative of a homotopy class of maps from a compact and connected Riemannian manifold with nonnegative Ricci curvature into a complete Riemannian manifold with no conjugate points is bounded below by a constant determined by the asymptotic geometry of the target, with equality if and only …

The reflection length of an element of a Coxeter group is the minimal number of conjugates of the standard generators whose product is equal to that element. In this paper we prove the conjecture of McCammond and Petersen that reflection length is unbounded in any non-affine Coxeter group. Among the tools used, the con…

New method simplifies Bayesian analysis for categorical data.

New manifolds found without interior conjugate points.

We obtain a result about the existence of only a finite number of geodesics between two fixed non-conjugate points in a Finsler manifold endowed with a convex function. We apply it to Randers and Zermelo metrics. As a by-product, we also get a result about the finiteness of the number of lightlike and timelike geodesic…

Study ergodic properties of geodesic flows on specific manifolds without conjugate points.

We prove that closed surfaces of all topological types, except for the non-orientable odd-genus ones, can be minimally embedded in the Riemannian product of a sphere and a circle of arbitrary radius. We illustrate it by obtaining some periodic minimal surfaces in $\mathbb{S}^2\times\mathbb{R}$ via conjugate constructio…

Given a group $G$ and a subset $X \subset G$, an element $g \in G$ is called quasi-positive if it is equal to a product of conjugates of elements in the semigroup generated by $X$. This notion is important in the context of braid groups, where it has been shown that the closure of quasi-positive braids coincides with t…

We show that the bar version of the $\mathrm{Pin}(2)$-monopole Floer homology of a three-manifold $Y$ equipped with a self-conjugate spin$^c$ structure $\mathfrak{s}$ is determined by the triple cup product of $Y$ together with the Rokhlin invariants of the spin structures inducing $\mathfrak{s}$. This is a manifestati…

Study on homeomorphism groups of telescoping 2-manifolds showing strong distortion.

A condition for a statistical manifold to have an equiaffine structure is studied. The facts that dual flatness and conjugate symmetry of a statistical manifold are sufficient conditions for a statistical manifold to have an equiaffine structure were obtained in [2] and [3]. In this paper, a fact that a statistical man…

We define a new differential geometric structure, called Lie rackoid. It relates to Leibniz algebroids exactly as Lie groupoids relate to Lie algebroids. Its main ingredient is a selfdistributive product on the manifold of bisections of a smooth precategory. We show that the tangent algebroid of a Lie rackoid is a Leib…

In this article, we will prove that the subsectors of $α$-induced sectors for $M \rtimes \hat{G} \supset M$ forms a modular category, where $M \rtimes \hat{G}$ is the crossed product of $M$ by the group dual $\hat{G}$ of a finite group $G$. In fact, we will prove that it is equivalent to Müger's crossed product. By usi…

The paper tackles efficient computation of optimal transport by approximating conjugates with amortized optimization.

Estimates geodesics on surfaces without conjugate points.

New family of measurable pseudo-Anosov maps on spheres.

For each $k\geq2$, we construct two families of surfaces with constant mean curvature $H$ for $H\in[0,1/2]$ in $Σ(κ)\times\R$ where $κ+4H^2\leq0$. The surfaces are invariant under $2π/k$-rotations about a vertical fiber of $Σ(κ)\times\R$, have genus zero, and a finite number of ends. The first family generalizes the no…