A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Graph products inherit Morse local-to-global property from their components.
Local-to-global principle for Morse actions on symmetric spaces.
New group with non-loxodromic Morse element found.
Investigates maps and properties in spaces with negative dimensions and curvature.
This paper improves a local-to-global principle for Morse quasigeodesics.
We show the mapping class group, CAT(0) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgro…
Expands Bredon's trick for applications in geometry and topology.
Exponential growth of stable subgroups in Morse geodesics.
This thesis explores GNNs, categorizing them into local and global approaches.
A quasi-geodesic is Morse if and only if it is strongly contracting in injective spaces.
This paper illustrates the themes of the title in terms of: van Kampen type theorems for the fundamental groupoid; holonomy and monodromy groupoids; and higher homotopy groupoids. Interaction with work of the writer is explored.
Novel proof technique for Gelfand-Fuks cohomology.
We give an explicit construction of a deformation quantization of the algebra of functions on a Poisson manifolds, based on Kontsevich's local formula. The deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection.
For an oriented manifold whose dimension is less than , we use the contractibility of certain complexes associated to its submanifolds to cut into simpler pieces in order to do local to global arguments. In particular, in these dimensions, we give a different proof of a deep theorem of Thurston in foliation …
In this work we present a new local to global criterion for proving a form of high dimensional expansion, which we term cosystolic expansion. Applying this criterion on Ramanujan complexes, yields for every dimension, an infinite family of bounded degree complexes with the topological overlapping property. This answer …
New findings show different cost functions yield equivalent curvature bounds.
Develops a logifold structure for understanding datasets.
We define the notion of whiskered categories and groupoids, showing that whiskered groupoids have a commutator theory. So also do whiskered -categories, thus answering questions of what might be `commutative versions' of these theories. We relate these ideas to the theory of Leibniz algebras, but the commutator theo…
L2G2G improves graph autoencoder accuracy without sacrificing scalability.
Given any K and N we show that there exists a compact geodesic metric measure space satisfying locally the CD(0,4) condition but failing CD(K,N) globally. The space with this property is a suitable non convex subset of R^2 equipped with the l^\infty-norm and the Lebesgue measure. Combining many such spaces gives a (non…
Logifold improves ensemble machine learning by identifying fuzzy domains.
We study Linial-Meshulam random 2-complexes, which are two-dimensional analogues of Erdős-Rényi random graphs. We find the threshold for simple connectivity to be p = n^{-1/2}. This is in contrast to the threshold for vanishing of the first homology group, which was shown earlier by Linial and Meshulam to be p = 2 log(…
We study local, global and local-to-global properties of threefolds with certain singularities. We prove criteria for these threefolds to be rational homology manifolds and conditions for threefolds to satisfy rational Poincaré duality. We relate the topological Euler characteristic of elliptic Calabi-Yau threefolds wi…
The study examines the systole of 3-manifolds with positive scalar curvature.
We study Linial-Meshulam random 2-complexes, which are two-dimensional analogues of Erdős-Rényi random graphs. We find the threshold for simple connectivity to be p = n^{-1/2}. This is in contrast to the threshold for vanishing of the first homology group, which was shown earlier by Linial and Meshulam to be p = 2 log(…
This paper is devoted to the analysis of metric measure spaces satisfying locally the curvature-dimension condition CD(K,N) introduced by the second author and also studied by Lott & Villani. We prove that the local version of CD(K,N) is equivalent to a global condition CD*(K,N), slightly weaker than the (usual, global…
We solve the integration problem for generalized complex manifolds, obtaining as the natural integrating object a weakly holomorphic symplectic groupoid, which is a real symplectic groupoid with a compatible complex structure defined only on the associated stack, i.e., only up to Morita equivalence. We explain how such…
We give a new proof for the local existence of a smooth isometric embedding of a smooth -dimensional Riemannian manifold with nonzero Riemannian curvature tensor into -dimensional Euclidean space. Our proof avoids the sophisticated arguments via microlocal analysis used in earlier proofs. In Part 1, we introduce …
Richberg technique adapted for nonlinear subequations.
Bredon's trick helps extend local properties to global topological spaces.
Paper resolves Chern conjecture for 4D minimal hypersurfaces in S5.
Research shows that certain metric spaces cannot contain rigid structures and provides evidence for loose embeddings into Euclidean spaces.
Describes the relationship between two spectral sequences and their joint refinement.
In the wake of recent advances in experimental methods in neuroscience, the ability to record in-vivo neuronal activity from awake animals has become feasible. The availability of such rich and detailed physiological measurements calls for the development of advanced data analysis tools, as commonly used techniques do …
The paper conjectures and proves fixed points for certain group actions on nonpositively curved spaces.
New curvature measure for causal sets derived from optimal transport.
We consider the problem of embedding unweighted, directed k-nearest neighbor graphs in low-dimensional Euclidean space. The k-nearest neighbors of each vertex provides ordinal information on the distances between points, but not the distances themselves. We use this ordinal information along with the low-dimensionality…
Extends Loday-Quillen-Tsygan theorem to bornological Lie algebra homology.
Study finds eigenvalue bounds for non-convex domains using cohomology.
Framework for optimizing portfolios under model uncertainty.
The goal of this paper is twofold: we study metric measure spaces with variable lower bounds for the Ricci curvature and we study pathwise coupling of Brownian motions. Given any lower semicontinuous function we introduce the curvature-dimension condition which canonically ex…
This paper studies transformer learning dynamics and initialization.
New action-angle coordinates found for singular symplectic manifolds.
In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X,d,m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm and Villani geodesic convexity condition for t…
A new method boosts graph neural networks by preventing over-smoothing and over-squashing.
Convolutional Neural Networks (CNN) have been pivotal to the success of many state-of-the-art classification problems, in a wide variety of domains (for e.g. vision, speech, graphs and medical imaging). A commonality within those domains is the presence of hierarchical, spatially agglomerative local-to-global interacti…
Paper studies Transformer learning theory for Euclidean and Riemannian domains.
Study introduces new curvature conditions for Lorentzian spaces using Rényi entropy.