arXiv research

Lower bounds set for infinite-precision transformers.

Innovative PGMs match neural networks, revealing precise approximations during forward propagation.

We construct a infinite-dimensional manifold structure adapted to analytic Lie pseudogroups of infinite type. More precisely, we prove that any isotropy subgroup of an analytic Lie pseudogroup of infinite type is a regular infinite-dimensional Lie group, modelled on a locally convex strict inductive limit of Banach spa…

The paper examines how deep linear neural networks behave as they become infinitely wide.

Study on infinite energy maps from surfaces to CAT(0) spaces.

VC dimensions of group CNNs are infinite for certain kernels and groups.

This paper concerns a study of three families of non-compact type symmetric spaces of infinite dimension. Although they have infinite dimension they have finite rank. More precisely, we show they have finite telescopic dimension. We also show the existence of Furstenberg maps for some group actions on these spaces. Suc…

We construct convergent and divergent lattices in negative curvature and give a precise asymptotic description of the behavior of their counting function.

The aim of this paper is the study of the geodesic distance in operator groups with several Riemannian metrics. More precisely we study the geodesic distance in self-adjoint operator groups with the left invariant Riemannian metric induced by the infinite trace and extend known results about the completeness of some cl…

The topological type of a non-compact Riemann surface is determined by its ends space and the ends having infinite genus. In this paper for a non-compact Riemann Surface $S_{m,s}$ with $s$ ends and exactly $m$ of them with infinite genus, such that $m,s\in \mathbb{N}$ and $1<m\leq s$, we give a precise description of t…

In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infinite number of closed immersed minimal surfaces. We use min-max theory for the area functional to prove this conjecture in the positive Ricci curvature setting. More precisely, we show that every compact Riemannian manifo…

Study infinite superelliptic curves and their Veech groups, providing geometric and algebraic insights.

Paper approximates solutions for complex decision processes with limited precision.

We construct an infinite family of topologically slice knots that are not smoothly concordant to their reverses. More precisely, if T denotes the concordance group of topologically slice knots and R is the involution of T induced by string reversal, then T/Fix(R) contains an infinitely generated free subgroup. The resu…

We introduce an asymptotic small noise expansion, a so called vol-of-vol expansion, for potentially infinite dimensional and rough stochastic volatility models. Thereby we extend the scope of existing results for finite dimensional models and validate claims for infinite dimensional models. Furthermore we provide new, …

Graph convolutional deep kernel machine learns representations for graph tasks.

New knots bound multiple non-isotopic ribbon disks.

While Recurrent Neural Networks (RNNs) are famously known to be Turing complete, this relies on infinite precision in the states and unbounded computation time. We consider the case of RNNs with finite precision whose computation time is linear in the input length. Under these limitations, we show that different RNN va…

New knots not rationally concordant to their reverses found.

Maximal cusps are not dense on Teichmüller space for infinite-type surfaces.

We present results connecting crossratios, representations of surface groups in $SL(n,\mathbb R)$ and in an infinite dimensional group related to the group of diffeomorphisms of the circle. More precisely, we show that representations of a surface group in $SL(n,\mathbb R)$ can be interpreted as crossratios satisfying …

New pseudomodular groups constructed from jigsaw construction.

The paper constructs solutions with infinite-time singularities in Lagrangian mean curvature flow.

Continuous metrics on ample bundles lie in infinite-dimensional cones.

In infinite dimensional Heisenberg group, degenerate distances linked to unbounded curvature.

For an orientable surface of finite type equipped with a flat metric with holonomy of finite order q, the set of maximal embedded cylinders can be empty, non-empty, finite, or infinite. The case when q < 3 is well-studied as such surfaces are (semi-)translation surfaces. Not only is the set always infinite, the core cu…

Study examines risk premium convergence rates in risk sharing contracts.

A classical result by Marston Morse asserts that on some ellipsoids of ${\mathbb R}^3$ there exists exactly 3 closed and simple geodesics. The goal of this presentation is to prove that this rigidity result does not extend to higher dimensions and, more precisely, on any smooth closed riemannian manifod of arbitrary di…

Proves unique continuation for area minimizing currents.

We study the learnability of a class of compact operators known as Schatten--von Neumann operators. These operators between infinite-dimensional function spaces play a central role in a variety of applications in learning theory and inverse problems. We address the question of sample complexity of learning Schatten-von…

We prove the Conley conjecture for a closed symplectically aspherical symplectic manifold: a Hamiltonian diffeomorphism of a such a manifold has infinitely many periodic points. More precisely, we show that a Hamiltonian diffeomorphism with finitely many fixed points has simple periodic points of arbitrarily large peri…

Löbell polyhedra have small systoles and are quasi-arithmetic.

Study on neural network initialization with shaped infinite depth-and-width networks.

Proposes an algorithm for infinite-dimensional sparse learning in system identification.

Study on four-dimensional Dehn twists and Milnor fibrations, revealing new phenomena.

This paper is about the geometry of flip-graphs associated to triangulations of surfaces. More precisely, we consider a topological surface with a privileged boundary curve and study the spaces of its triangulations with n vertices on the boundary curve. The surfaces we consider topologically fill this boundary curve s…

We develop a scalable method for Bayesian neural networks with stochastic differential equations.

We analyze the possibility of defining infinite-dimensional manifolds as ringed spaces. More precisely, we consider three definitions of manifolds modeled on locally convex spaces: in terms of charts and atlases, in terms of ringed spaces, and in terms of functored spaces, as introduced by Douady in his thesis. It is s…

We establish that, over certain ground fields, the set of osculating tangents of Cayley's ruled cubic surface gives rise to a (maximal partial) spread which is also a dual (maximal partial) spread. It is precisely the Betten-Walker spreads that allow for this construction. Every infinite Betten-Walker spread is not an …

I construct "fake algebraic curves" in $Cp^2$. More precisely, for any k>2, I construct infinitely many pairwise smoothly non-isotopic (and moreover not ambient diffeomorphic) smooth surfaces $F\subset Cp^2$ homeomorphic to a non-singular algebraic curve of degree 2k, realizing the same homology class as such a curve a…

Transformers with CoT don't enhance reasoning power across all tasks.

Study classifies mapping class groups with hyperbolic actions on infinite-type surfaces.

Study graded coverings for supermanifolds, proving their universal properties.

Classifies 3-manifolds with uniformly positive scalar curvature.

This paper is the last paper in a series of five papers. Building on earlier papers in this series, we prove an analogue of Kuratowski's characterisation of graph planarity for three dimensions. More precisely, a simply connected 2-dimensional simplicial complex embeds in 3-space if and only if it has no obstruction fr…

Study on the $L^p$ behavior of helix curves' energy density.

Derives integral representations for a Lévy process and its extremum, hitting time, with fast evaluation.

A simple surface amalgam is the union of a finite collection of surfaces with precisely one boundary component each and which have their boundary curves identified. We prove if two fundamental groups of simple surface amalgams act properly and cocompactly by isometries on the same proper geodesic metric space, then the…