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.
Surveying connections between graph combinatorics and algebraic right-angled Artin groups.
New model for links uses meander diagrams and combinatorics.
Paper studies geometric and combinatorial properties of circular snakes.
Graph dynamics link combinatorics to geometry, revealing manifold intersections and stability.
New geometric object for polynomials simplifies complex data.
We show that any two geometric triangulations of a closed hyperbolic, spherical or Euclidean manifold are related by a sequence of Pachner moves and barycentric subdivisions of bounded length. This bound is in terms of the dimension of the manifold, the number of top dimensional simplexes and bound on the lengths of ed…
State surfaces are spanning surfaces of links that are obtained from link diagrams guided by the combinatorics underlying Kauffman's construction of the Jones polynomial via state models. Geometric properties of such surfaces are often dictated by simple link diagrammatic criteria, and the surfaces themselves carry imp…
We give a geometric realization of the polyhedra governed by the structure of associative algebras with co-inner products, or more precisely, governed by directed planar trees. Our explicit realization of these polyhedra, which include the associahedra in a special case, shows in particular that these polyhedra are hom…
Survey on GKM theory in low dimensions, highlighting combinatorics-geometry interplay.
In this note we study numerically the combinatorics of curves and geodesics on the torus with one boundary component. A potential computational difficulty is avoided by counting inside specific orbits of the mapping class group up to a certain length, either geometric or combinatorial. Some cases are rigurolosly determ…
The quantum cohomology algebra of the (full) flag manifold is a fundamental example in quantum cohomology theory, with connections to combinatorics, algebraic geometry, and integrable systems. Using a differential geometric approach, we give an algorithm for computing the multiplicative structure constants of this alge…
Constructs geometric models for moduli spaces of Higgs bundles over Riemann sphere.
In previous work, the authors have developed a geometric theory of fundamental strata to study connections on the projective line with irregular singularities of parahoric formal type. In this paper, the moduli space of connections that contain regular fundamental strata with fixed combinatorics at each singular point …
We give a method for computing upper and lower bounds for the volume of a non-obtuse hyperbolic polyhedron in terms of the combinatorics of the 1-skeleton. We introduce an algorithm that detects the geometric decomposition of good 3-orbifolds with planar singular locus and underlying manifold the 3-sphere. The volume b…
New method determines arrangement combinatorics from Milnor fiber boundary.
New discrete cmc surfaces defined from sphere packings and combinatorics.
For the pants graph, there is little known about the behaviour of geodesics, as opposed to quasigeodesics. Brock-Masur-Minsky showed that geodesics or geodesic segments connecting endpoints satisfying a bounded combinatorics condition, such as the stable/unstable laminations of a pseudo-Anosov, all have bounded combina…
Formulae for special almost-complex structures on Vogan diagrams.
Study restricts line arrangements with odd points using topological arguments.
The maximum hyperbolic polyhedron volume is found to be the rectification of its skeleton.
We present a simplified exposition of some classical and modern results on graph drawings in the plane. These results are chosen so that they illustrate some spectacular recent higher-dimensional results on the border of topology and combinatorics. We define a mod2-valued self-intersection invariant (i.e. the van Kampe…
Paper introduces a new geometric homology theory and applies it to Gromov-Witten theory.
If K is a rationally null-homologous knot in a 3-manifold M, the rational genus of K is the infimum of -χ(S)/2p over all embedded orientable surfaces S in the complement of K whose boundary wraps p times around K for some p (hereafter: S is a p-Seifert surface for K). Knots with very small rational genus can be constru…
A finite subset S of a closed hyperbolic surface F canonically determines a "centered dual decomposition" of F: a cell structure with vertex set S, geodesic edges, and 2-cells that are unions of the corresponding Delaunay polygons. Unlike a Delaunay polygon, a centered dual 2-cell Q is not determined by its collection …
Considering the Teichmüller space of a surface equipped with Thurston's Lipschitz metric, we study geodesic segments whose endpoints have bounded combinatorics. We show that these geodesics are cobounded, and that the closest-point projection to these geodesics is strongly contracting. Consequently, these geodesics are…
We use ending laminations for Weil-Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for Weil-Petersson geodesic segments, rays, and lines. Further, a more general notion of non-annular bounded combinatorics, which allows arbitrarily large Dehn-twisting, corresponds to an equ…
This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the combinatorics of such a polyhedron, while keeping it within the class of right-angl…
We prove the filling area conjecture in the hyperelliptic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu's result in genus 0. We translate the problem into a question about closed ovalless real surfaces. The conjecture then results from a combination of two ingr…
We introduce a coarse combinatorial description of the Weil-Petersson distance d_WP(X,Y) between two finite area hyperbolic Riemann surfaces X and Y. The combinatorics reveal a connection between Riemann surfaces and hyperbolic 3-manifolds conjectured by Thurston: the volume of the convex core of the quasi-Fuchsian man…
We suggest a new definition for discrete minimal surfaces in terms of sphere packings with orthogonally intersecting circles. These discrete minimal surfaces can be constructed from Schramm's circle patterns. We present a variational principle which allows us to construct discrete analogues of some classical minimal su…
Following the general strategy proposed by G.Rybnikov, we present a proof of his well-known result, that is, the existence of two arrangements of lines having the same combinatorial type, but non-isomorphic fundamental groups. To do so, the Alexander Invariant and certain invariants of combinatorial line arrangements a…
We introduce a program aimed to studying problems arising from the theory of complex networks with differential geometric means. We study the propagation of influences on manifolds assuming that at each point only a finite number of propagation velocities are allowed. This leads to the computation of the volume of the …
We introduce Quintessence: a family of burr puzzles based on the geometry and combinatorics of the 120-cell. We discuss the regular polytopes, their symmetries, the dodecahedron as an important special case, the three-sphere, and the quaternions. We then construct the 120-cell, giving an illustrated survey of its geome…
A projective mirror polyhedron is a projective polyhedron endowed with reflections across its faces. We construct an explicit diffeomorphism between the moduli space of a mirror projective polyhedron with fixed dihedral angles in , and the union of copies of , when the polyhedron has the combin…
New combinatorial framework for geometric realizations of subword complexes.
Gordon-Litherland pairing connects combinatorics and topology.
The problem of defining correctly geometric objects such as the curvature is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs. He named it coarse Ricci curvature because it coincides, up to some given factor…
New method detects dangerous states in gridworlds using geometric defects.
Bounded-type 3-manifolds arise as combinatorially bounded gluings of irreducible 3-manifolds chosen from a finite list. We prove effective hyperbolization and effective rigidity for a broad class of 3-manifolds of bounded type and large gluing heights. Specifically, we show the existence and uniqueness of hyperbolic me…
A topological version of a longstanding conjecture of H. Hopf, originally proposed by W. Thurston, states that the sign of the Euler characteristic of a closed aspherical manifold of dimension depends only on the parity of . Gromov defined several hyperbolization functors which produce an aspherical manifold …
This paper proves exponential mixing for frame flows on hyperbolic manifolds with cusps.
In the present paper, we introduce -braids and, more generally, -braids for an arbitrary group . They form a natural group-theoretic counterpart of -knots, see \cite{reidmoves}. The underlying idea, used in the construction of these objects --- decoration of crossings with some additional informa…
In this paper, we review the problem of matrix completion and expose its intimate relations with algebraic geometry, combinatorics and graph theory. We present the first necessary and sufficient combinatorial conditions for matrices of arbitrary rank to be identifiable from a set of matrix entries, yielding theoretical…
The article studies embeddings of edge-colored graphs related to balanced 3- and 4-manifolds.
New polynomial connects knot genus to 3-manifold geometry.
Study conic line arrangements of degree 7, finding their topology and connected components.
We utilize ideal bipyramids to obtain new upper bounds on volume for hyperbolic link complements in terms of the combinatorics of their projections.
One can realize higher laminations as positive configurations of points in the affine building. The duality pairings of Fock and Goncharov give pairings between higher laminations for two Langlands dual groups and . These pairings are a generalization of the intersection pairing between measured laminatio…