arXiv research

An algebra $A$ with identity $(a\circ b)\circ c-a\circ(b\circ c)=(a\circ c)\circ b-a\circ(c\circ b),$ is called right-symmetric. Cohomology and deformation theory for right-symmetric algebras are developed. Cohomologies of $gl_n$ and half-Witt algebras $W_n^{rsym}, p=0,$ $W_n^{rsym}({\bf m}), p>0,$ are calculated. In p…

We obtain the curvature form $F^\nabla=P\circ d^\nabla\circ\nabla-d^\nabla\circ P\circ\nabla+d^\nabla\circ\nabla\circ P$ for a vector bundle pseudoconnection $\nabla$, where $d^\nabla$ is the exterior derivative associated to $\nabla$. We use $F^\nabla$ to obtain the curvature of $\nabla$. We also prove that $F^\nabla=…

The paper classifies actions of a specific group on certain manifolds.

Study proves rigidity of critical points in hydrophobic capillary systems.

Connected boundaries of strata of differentials are always connected in various compactifications.

New algorithm estimates edge density of random graphs robustly, achieving optimal breakdown point.

The circle transfer $QΣ(LX_{hS^1})_+ \to QLX_+$ has appeared in several contexts in topology. In this note we observe that this map admits a geometric re-interpretation as a morphism of cobordism categories of 0-manifolds and 1-cobordisms. Let $C_1(X)$ denote the 1-dimensional cobordism category and let $Circ(X) \subse…

In this paper we use 3-manifold techniques to illuminate the structure of the category of tangles. In particular, we show that every idempotent morphism $A$ in such a category naturally splits as $A=B\circ C$ such that $C\circ B$ is an identity morphism.

Proves a generalization of a multiplicity one theorem for specific groups.

Let π: V \rightarrow M be a (real or holomorphic) vector bundle whose base has an almost Frobenius structure (\circ_{M},e_{M}, g_{M}) and typical fiber has the structure of a Frobenius algebra (\circ_{V},e_{V},g_{V}). Using a connection D on the bundle V and a morphism α: V \rightarrow TM, we construct an almost Froben…

Deforms complex pair of pants to tropical hyperplane.

An elementary family of local Hamiltonians $H_{\c ,\ell}, \ell = 1,2,3, ldots$, is described for a $2-$dimensional quantum mechanical system of spin $={1/2}$ particles. On the torus, the ground state space $G_{\circ,\ell}$ is $(\log)$ extensively degenerate but should collapse under $ł$perturbation" to an anyonic syste…

The paper studies horofunction compactifications of symmetric cones under Finsler distances.

New algorithm for decentralized matching markets without prior preference rankings.

We prove that a monomorphic functor $F:Comp\to Comp$ with finite supports is epimorphic, continuous, and its maximal $\emptyset$-modification $F^\circ$ preserves intersections. This implies that a monomorphic functor $F:Comp\to Comp$ of finite degree $deg F\le n$ preserves (finite-dimensional) compact ANR's if the spac…

This work continues the study of $F$--manifolds $(M,\circ)$, first defined by Hertling and Manin and investigated in [He]. The notion of a compatible flat structure $\nabla$ is introduced, and it is shown that many constructions known for Frobenius manifolds do not in fact require invariant metrics and can be developed…

A regular F-manifold is an F-manifold (with Euler field) (M, \circ, e, E), such that the endomorphism {\mathcal U}(X) := E \circ X of TM is regular at any p\in M. We prove that the germ ((M,p), \circ, e, E) is uniquely determined (up to isomorphism) by the conjugacy class of {\mathcal U}_{p} : T_{p}M \rightarrow T_{p}M…

New method uses Kernel Flows to improve neural network training without changing structure.

Deep neural networks can approximate complex functions through repeated compositions of a fixed-size ReLU network.

A computational technique for calculating nullity vectors and kernel vectors, using the new Finsler package, is introduced. As an application, three interesting counterexamples are given. The first counterexample shows that the two distributions $\mathrm{Ker}_R$ and $\N_R$ do not coincide. The second shows that the nul…

Given a result of Herman, we provide a new elementary proof of the fact that the connected component of the group of compactly supported diffeomorphisms is perfect and hence simple. Moreover, we show that every diffeomorphism $g$, which is sufficiently close to the identity, can be represented as a product of four comm…

The ideal of relations in the (small) quantum cohomology ring of the generalized flag manifold $G/B$ has been determined by B. Kim. We are going to point out a limited number of properties that, if they are satisfied by an $R[q_1,...,q_l]$-bilinear product $\circ$ on $H^*(G/B)\otimes R[q_1,...,q_l$, then the ring $(H^*…

CIRCE measures conditional independence for learning invariant features.

This is a survey of the current state of the theory of $F$--(super)manifolds $(M,\circ)$, first defined in [HeMa] and further developed in [He], [Ma2], [Me1]. Here $\circ$ is an $\Cal{O}_M$--bilinear multiplication on the tangent sheaf $\Cal{T}_M$, satisfying an integrability condition. $F$--manifolds and compatible fl…

The paper links set cuspidality to function regularity and flatness.

Let $\widetilde{S}$ be a closed (compact without boundary) oriented surface with genus $g$, and $G$ be a group isomorphic to $$, where $p$ is a prime integer. An action of $G$ on $S$ is a pair $(\widetilde{S},f)$, where $f$ is a representation of $G$ in the group of orientation preserving autohomeom…

The paper classifies helicoidal surfaces with specific curvature functions.

This paper studies the relationship between fundamental groups of manifolds and their effective regular sets.

Let $Q$ be a smooth compact orientable 3--manifold with smooth boundary $\partial Q$. Let $\mathcal{B}$ be the set of exact 2--forms $B\inΩ^2(Q)$ such that $j_{\partial Q}^*B=0$, where $j_{\partial Q}:{\partial Q}\to Q$ is the inclusion map. The group $\mathcal{D}=\mathrm{Diff}_0(Q)$ of self-diffeomorphisms of $Q$ isot…

The paper finds the optimal wealth growth rate in betting games.

We consider four-dimensional Riemannian manifolds with commuting higher order Jacobi operators defined on two-dimensional orthogonal subspaces (polygons) and on their orthogonal subspaces. More precisely, we discuss higher order Jacobi operator $\mathcal{J}(X)$ and its commuting associated operator $\mathcal{J}(X^{\per…

Survey article analyzes pseudoholomorphic curves on symplectization via contact instantons.

The study proves conditions for rigidity of Kleinian groups using measure theory and ergodic theory.

Let $(M,ω)$ be a compact Kähler manifold with negative holomorphic sectional curvature. It was proved by Wu-Yau and Tosatti-Yang that $M$ is necessarily projective and has ample canonical bundle. In this paper, we show that any irreducible subvariety of $M$ is of general type. Moreover, we can extend the theorem to the…

We prove the existence of a complete, embedded, singly periodic minimal surface, whose quotient by vertical translations has genus one and two ends. The existence of this surface was announced in our paper in {\it Bulletin of the AMS}, 29(1):77--84, 1993. Its ends in the quotient are asymptotic to one full turn of the …

Kernel methods outperform neural nets in operator learning tasks.

Let $S$ be a closed Riemann surface of genus $g(\geqq 2)$ and set $\dot{S}=S \setminus \{\hat{z}_0 \}$. Then we have the composed map $\varphi\circ r$ of a map $r: T(S) \times U \rightarrow F(S)$ and the Bers isomorphism $\varphi: F(S) \rightarrow T(\dot{S})$, where $F(S)$ is the Bers fiber space of $S$, $T(X)$ is the …

The study finds continuous solutions to complex Hessian equations on compact Hermitian manifolds.

Extends neural network approximations to guarantee continuity of real-world learning tasks.

Study shows $L^2$-temperedness and absence of positive eigenfunctions in certain groups.

Let $S$ be a compact oriented surface with boundary together with finitely many marked points on the boundary, and let $S^\circ$ be the same surface equipped with the opposite orientation. We consider the double $S_\mathcal{D}$ obtained by gluing the surfaces $S$ and $S^\circ$ along corresponding boundary components. W…

Let $(M^\circ, g)$ be an asymptotically conic manifold, in the sense that $M^\circ$ compactifies to a manifold with boundary $M$ in such a way that $g$ becomes a scattering metric on $M$. A special case of particular interest is that of asymptotically Euclidean manifolds, where $\partial M = S^{n-1}$ and the induced me…

Harmonic maps pull convex functions on metric spaces to subharmonic ones.

The paper studies the topological properties of convex sets and their polar mappings.

Let F be a closed non-orientable surface. We classify all finite order invariants of immersions of F into R^3, with values in any Abelian group. We show they are all functions of the universal order 1 invariant that we construct as T \oplus P \oplus Q where T is a Z valued invariant reflecting the number of triple poin…

The study simplifies complex functions on surfaces using a special transformation.

It is known that there exists a natural functor $Φ$ from Lie supergroups to super Harish-Chandra pairs. A functor going backwards, that associates a Lie supergroup with each super Harish-Chandra pair, yielding an equivalence of categories, was found by Koszul [18]; this result was later extended by other authors, to di…

We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on the product $v(K) = (\Vol K)(\Vol K^\circ)$ of the volume of a symmetric convex body $K \in \R^n$ and its polar body $K^\circ$. The Mahler conjecture asserts that the Mahler volume $v(K)$ is minimiz…