arXiv research

The study estimates curvature on graphs with large girth.

New construction for surfaces with logarithmically large systoles.

This paper defines girth for knots and links, linking it to Khovanov homology.

Paper constructs graphs with girth four and distinct properties.

The study finds conditions for groups acting on CAT(0) cube complexes to have infinite girth.

Proves Girth Alternative for some HNN extensions, finds counterexamples.

The study shows that certain Artin groups cannot contain hyperbolic manifold groups.

Research shows finiteness in triangulations with girth constraints.

The girth of a finitely generated group G is the supremum of the girth of Cayley graphs for G over all finite generating sets. Let G be a finitely generated subgroup of the mapping class group Mod(S), where S is a compact orientable surface. Then, either G is virtually abelian or it has infinite girth; moreover, if we …

We construct simple curves from immersed curves in the setting of handlebodies and Heegaard splittings. We define a measure of complexity we call girth for closed curves in a handlebody. We extend this complexity to Heegaard splittings and pose a conjecture about all Heegaard splittings. We prove a test case of this co…

Ancient Ricci flows with bounded girth found in 3D and higher.

Study on Lin-Lu-Yau curvature and diameter of amply regular graphs.

In this note we introduce a natural Finsler structure on convex surfaces, referred to as the projective Finsler structure, which is dual in a sense to the obvious inclusion of a convex surface in a normed space. It has an associated projective girth, which is similar to the notion of girth defined by Schäffer. We prove…

We introduce a new way to tabulate knots by representing knot diagrams using a pair of planar trees. This pair of trees have their edges labeled by integers, they have no valence 2 vertices, and they have the same number of valence 1 vertices. The number of valence 1 vertices of the trees is called the girth of the kno…

Ricci curvature was proposed by Ollivier in a general framework of metric measure spaces, and it has been studied extensively in the context of graphs in recent years. In this paper we prove upper bounds for Ollivier's Ricci curvature for bipartite graphs and for the graphs with girth at least 5. We also prove a genera…

We examine the large systole problem, which concerns compact hyperbolic Riemannian surfaces whose systole, the length of the shortest noncontractible loops, grows logarithmically in genus. The generalization of a construction of Buser and Sarnak by Katz, Schaps, and Vishne, which uses principal "congruence" subgroups o…

Study on modified Ricci curvature on graphs, proving rigidity and deriving formulas.

The paper solves curvature problems on graphs using a special flow.

Recent research in coarse geometry revealed similarities between certain concepts of analysis, large scale geometry, and topology. Property A of G.Yu is the coarse analog of amenability for groups and its generalization (exact spaces) was later strengthened to be the large scale analog of paracompact spaces using parti…

Khovanov and chromatic homologies show similar patterns, improving chromatic bounds and computing Jones polynomial.

In this paper, the second of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs has girth tending to infinity, then the maximal coarse Baum-Connes assembly map is an isomorphism for the associated metric space $X$. As discussed in the first paper in this s…

In this paper, the first of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs is an expander and the girth of the graphs tends to infinity, then the coarse Baum-Connes assembly map is injective, but not surjective, for the associated metric space $X$. Exp…

We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum-Connes conjecture. In particula…

The Ollivier Ricci flow with prescribed curvature on infinite graphs.

New binary matrices improve compressed sensing with faster and less storage requirements.

We prove that a dense subgroup of $\mathrm{Homeo}_{+}(I)$ is not elementary amenable. We also show that the topological group $\mathrm{Homeo}_{+}(I)$ does not satisfy the Stability of the Generators Property, moreover, any finitely generated subgroup of $\mathrm{Homeo}_{+}(I)$ admits a faithful discrete representation …

In [13], it is proved that any subgroup of $\mathrm{Diff}_{+}^{ω}(I)$ (the group of orientation preserving analytic diffeomorphisms of the interval) is either metaabelian or does not satisfy a law. A stronger question is asked whether or not the Girth Alternative holds for subgroups of $\mathrm{Diff}_{+}^{ω}(I)$. In th…

Curvature formulas on regular graphs identified bone idle edges and graphs.

New graphs with maximum degree 4 found to be Ricci-flat.

The paper studies Ricci flow on graphs with prescribed curvature.

The paper introduces and studies a new type of submersion in Riemannian geometry.

Sharp bounds on diameter and eigenvalues for amply regular graphs.

The paper explores curvatures on graphs and their implications for Ricci flatness.

Study on finiteness property of right-angled Artin groups actions on extension graphs.

We propose a new yet natural algorithm for learning the graph structure of general discrete graphical models (a.k.a. Markov random fields) from samples. Our algorithm finds the neighborhood of a node by sequentially adding nodes that produce the largest reduction in empirical conditional entropy; it is greedy in the se…

The paper investigates the distribution of systoles on arithmetic Riemann surfaces.

The Funk metric connects billiards, projective geometry, and convex geometry.

Introduces resemblance structure for large scale geometry.

Minimal surfaces with negative curvature found in large spheres.

A new law of large numbers for single catastrophic risks.

Ranky solves SVD for large sparse matrices in distributed systems.

Study shows HFT benefits large traders under certain conditions.

Study large deviations in life insurance portfolios without identical distributions.

Large knots have very varied boundary slopes.

Study minimizes large losses in financial portfolios.

Large batch training improves deep learning performance without needing warmup.

IVON optimizes large neural networks, matching or outperforming Adam.

Large deviations for fat tailed distributions, i.e. those that decay slower than exponential, are not only relatively likely, but they also occur in a rather peculiar way where a finite fraction of the whole sample deviation is concentrated on a single variable. The regime of large deviations is separated from the regi…