The traditional Riemann Mapping Theorem can be proved with circle packing techniques. We prove the Combinatorial Riemann Mapping Theorem for tilings of bounded size using circle packings.
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New combinatorial structure for hierarchically hyperbolic spaces.
problem Constructing new hierarchically hyperbolic spaces.
method Combinatorial hierarchical hyperbolicity criterion to construct and clarify HHS structures.
result HHSs admit a combinatorial structure, clarifying the application of the combinatorial HHS criterion.
Pure combinatorial models for BPL_n and Gauss map of a combinatorial manifold are described.
New technique shows any group can be a map's automorphism.
problem Finding maps with any given automorphism group.
method Universal technique for showing any finite automorphism group is possible.
result Any finite automorphism group can be realized by many non-isomorphic maps.
Bayesian optimization adapted for discrete spaces using random mappings.
problem Global optimization of expensive black-box functions with discrete variables.
method Embeds discrete space into a convex polytope, performs optimization in continuous space.
result Method outperforms existing methods in large combinatorial spaces.
In a previous paper, Sarkar and the third author gave a combinatorial description of the hat version of Heegaard Floer homology for three-manifolds. Given a cobordism between two connected three-manifolds, there is an induced map between their Heegaard Floer homologies. Assume that the first homology group of each boun…
Computing uniformization maps for surfaces has been a challenging problem and has many practical applications. In this paper, we provide a theoretically rigorous algorithm to compute such maps via combinatorial Calabi flow for vertex scaling of polyhedral metrics on surfaces, which is an analogue of the combinatorial Y…
We use the combinatorial harmonic map theory to study the isometric actions of discrete groups on Hadamard spaces. Given a finitely generated group acting by automorphisms, properly discontinuously and cofinitely on a simplicial complex and its isometric action on a Hadamard space, we formulate criterions for the actio…
Using bordered Floer theory, we give a combinatorial construction and proof of invariance for the hat version of Heegaard Floer homology. As a part of the proof, we also establish combinatorially the invariance of the linear-categorical representation of the strongly-based mapping class groupoid given by the same theor…
Alexander method extended to infinite-type surfaces.
problem Determining equality of elements in mapping class groups.
method Combinatorial tool extended to infinite-type surfaces.
result Verification of relations and triviality of centers in mapping class groups.
A combinatorial formula for Earle's twisted 1-cocycle on the mapping class group \mathcal{M}_{g,*}math.GT
We present a formula expressing Earle's twisted 1-cocycle on the mapping class group of a closed oriented surface of genus >=2 relative to a fixed base point, with coefficients in the first homology group of the surface. For this purpose we compare it with Morita's twisted 1-cocycle which is combinatorial. The key is t…
The hyperbolic structure of equilateral pentagons is mapped to a tiling of the hyperbolic plane.
problem Mapping the realization space of equilateral pentagons to a hyperbolic plane.
method Combining combinatorial correspondence, Riemann mapping theorem, and normalization procedure.
result A full conformal parameterization of the space of equilateral pentagons.
The study shows how quotients of mapping class groups are hierarchically hyperbolic.
problem Understanding the hierarchical hyperbolicity of mapping class groups and their quotients.
method A combinatorial criterion for hierarchical hyperbolicity applied to mapping class groups.
result Quotients of mapping class groups by large powers of Dehn twists are hierarchically hyperbolic.
We introduce a combinatorial energy for maps of triangulated surfaces with simplicial metrics and analyze the existence and uniqueness properties of the corresponding harmonic maps. We show that some important applications of smooth harmonic maps can be obtained in this setting.
The paper studies circle patterns on surfaces with specific angles and curvature maps.
problem Investigating circle patterns with obtuse angles on surfaces of finite type.
method Characterizing curvature maps and establishing combinatorial Ricci flow conditions.
result Generalizations of circle pattern theorem and a computational method to find patterns.
We study the space of linear difference equations with periodic coefficients and (anti)periodic solutions. We show that this space is isomorphic to the space of tame frieze patterns and closely related to the moduli space of configurations of points in the projective space. We define the notion of combinatorial Gale tr…
A connected combinatorial 2-manifold is called degree-regular if each of its vertices have the same degree. A connected combinatorial 2-manifold is called weakly regular if it has a vertex-transitive automorphism group. Clearly, a weakly regular combinatorial 2-manifold is degree-regular and a degree-regular combinator…
Mapping class group subgroups yield quasi-isometric curve complex.
problem Understanding the curve complex through coset intersections.
method Proving quasi-isometry and combinatorial equivalence of curve complex and coset intersection complex.
result Automorphism group of coset intersection complex is the extended mapping class group.
Lagrangian embeddings of the Klein bottle and combinatorial properties of mapping class groupsmath.SG
A proof of non-existence of Lagrangian embeddings of the Klein bottle K in \CP^2 is given. We exploit the existence of a special embedding of K in a symplectic Lefschetz pencil on \CP^2 and study its monodromy. As the main technical tool, we develop the theory of mapping class groups, considered as quotients of special…
The paper studies deformations of smooth functions on a 2-torus.
problem Analyzing the group of diffeomorphisms preserving a Morse function on a 2-torus.
method Computing the groups π0S'(f), G(f), and π0Δ'(f) for Morse functions on 2-torus.
result Computed groups π0S'(f), G(f), and π0Δ'(f) for Morse functions on 2-torus.
New definition of regular points for PL functions on manifolds.
problem Defining regular points for PL functions on combinatorial manifolds.
method Definition based on link of the point, stratification of Jacobi set, Stein factorization of Reeb space.
result Our definition of regularity is distinct from existing definitions.
Study on embeddings and their topological properties in R^d.
problem Topology of embeddings and their bounds in R^d.
method Combinatorial formula for upper bounds of embeddings into R^d.
result Simple combinatorial formula for upper bounds of embeddings into R^d.
Method uses neural networks to solve combinatorial problems.
problem Solving combinatorial problems on raw input data.
method Integrates blackbox combinatorial solvers into neural networks.
result Efficient backward pass through blackbox solvers implemented.
In this paper we present the Ricci curvature on cell-complexes and show the Gauss-Bonnnet type theorem on graphs and 2-complex that decomposes closed surface. The defferential forms on a cell complex is defined as linear maps on chain complex, and Laplacian operates this defferential forms. Then we construct the Bochne…
The space of topological decompositions into triangulations of a surface has a natural graph structure where two triangulations share an edge if they are related by a so-called flip. This space is a sort of combinatorial Teichmüller space and is quasi-isometric to the underlying mapping class group. We study this space…
Extends knot invariant to filtered grid complexes.
problem Knot invariants and grid complexes.
method Combining Ozsváth-Szabó-Stipsicz crossing-change maps with Alishahi-Eftekhary l(K) invariant.
result Combinatorial formulation of knot invariant.
This research connects combinatorial Teichmüller space geometry to Weil-Petersson geometry.
problem Understanding the geometry of combinatorial Teichmüller space.
method Developed a parallel between combinatorial Teichmüller space and Weil-Petersson geometry, using measured foliations and Fenchel-Nielsen coordinates.
result Established a geometric recursion and topological recursion for mapping class group invariants.
The Ptolemy groupoid is a combinatorial groupoid generated by elementary moves on marked trivalent fatgraphs with three types of relations. Through the fatgraph decomposition of Teichmüller space, the Ptolemy groupoid is a mapping class group equivariant subgroupoid of the fundamental path groupoid of Teichmüller space…
Simplified computation of SFT invariants for Legendrian links.
problem Computing SFT invariants for Legendrian links is combinatorially intractable.
method Left-right-simplification of Legendrian links and analysis of holomorphic maps.
result SFT invariants of Legendrian links are combinatorially computable using disks with ≤ 2 positive punctures.
We define Discrete Quasi-Einstein metrics (DQE-metrics) as the critical points of discrete total curvature functional on triangulated 3-manifolds. We study DQE-metrics by introducing some combinatorial curvature flows. We prove that these flows produce solutions which converge to discrete quasi-Einstein metrics when th…
We compare two combinatorial models for the moduli space of two-dimensional cobordisms: Bödigheimer's radial slit configurations and Godin's admissible fat graphs, producing an explicit homotopy equivalence using a "critical graph" map. We also discuss natural compactifications of these two models, the unilevel harmoni…
The present paper produces examples of Gauss diagram formulae for virtual knot invariants which have no analogue in the classical knot case. These combinatorial formulae contain additional information about how a subdiagram is embedded in a virtual knot diagram. The additional information comes from the second author's…
New combinatorial structures for Teichmüller spaces with Thurston's metric are explored.
problem Understanding the combinatorial structures of Teichmüller spaces with Thurston's metric.
method Analyzing the unit tangent and cotangent spheres of Teichmüller space, proving formulas for dimensions and codimensions of faces.
result The combinatorial structure of unit spheres in Teichmüller spaces is independent of the underlying point and is isomorphic to the extended mapping class group.
Hardware-accelerated RBM solves large combinatorial problems and integer factorization.
problem Solving large combinatorial optimization and integer factorization problems.
method Logically synthesized RBM architecture, hardware acceleration, and efficient training methods.
result Hardware-accelerated RBM factorizes 16-bit numbers with 10000x speed and 32x power improvements.
The study examines Morse diagrams and their behavior under Murasugi sums, leading to contact structure classifications.
problem Understanding Morse diagrams and their behavior under Murasugi sums.
method Examination of combinatorial Morse structures, open book decompositions, and contact structures.
result Diagrammatic criterion for detecting overtwisted contact structures and classification of Morse diagrams for one-holed torus pages.
Classifies theories with eight supercharges using pseudo-periodic maps and Riemann surfaces.
problem Classifying theories with eight supercharges using mathematical tools.
method Assumes theories are given by genus g fibrations of Riemann surfaces, uses pseudo-periodic maps of negative type in mapping class group.
result Identifies dual graphs and 3d mirror quivers, unifies various SCFTs in combinatorial framework.
According to an analogy to quasi-Fuchsian groups, we investigate topological and combinatorial structures of Lyubich and Minsky's affine and hyperbolic 3-laminations associated with the hyperbolic and parabolic quadratic maps. We begin by showing that hyperbolic rational maps in the same hyperbolic component have quasi…
Infinitesimal calculations link fundamental groups to Lie algebras.
problem Calculating logarithm maps in fundamental groups.
method Hopf invariants defined by Harrison cohomology of commutative cochains.
result Zeroth Harrison cohomology is a universal dual to Malcev Lie algebra.
In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to …
In all possible cases, we prove that local embeddings between two curve complexes whose complexities do not increase from domain to codomain are induced by surface homeomorphism. This is our first main result. From this we can deduce our second, a strong local co-Hopfian result for mapping class groups.
Tucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT-manifolds, i.e. for manifolds that satisfy BUT. Proofs rel…
We study the periods mapping from the moduli space of real hyperelliptic curves with marked point on an oriented oval to the euclidean space. The mapping arises in the analysis of Chebyshev construction used in the constrained optimization of the uniform norm of polynomials and rational functions. The decomposition of …
Paper proves circle packings converge to Riemann mapping for Jordan domains.
problem Proving discrete conformal maps converge to Riemann mapping.
method Establishing solvability theorem for inversive distance circle packings.
result Bowers-Stephenson's conjecture for Jordan domains is proven.
Extends Thurston's combinatorial characterization to all branched coverings of the 2-sphere.
problem Characterizing branched coverings of the 2-sphere.
method Generalizing Thurston's local balancing to all branched coverings.
result Provides a new proof for a theorem concerning real rational functions.
We give completely combinatorial proofs of the main results of [3] using polygons. Namely, we prove that the mapping class group of a surface with boundary acts faithfully on a finitely-generated linear category. Along the way we prove some foundational results regarding the relevant objects from bordered Heegaard Floe…
We obtain a combinatorial formula for the Miller-Morita-Mumford classes for the mapping class group of punctured surfaces and prove Witten's conjecture that they are proportional to the dual to the Witten cycles. The proportionality constant is shown to be exactly as conjectured by Arbarello and Cornalba [J. Alg. Geom.…
A vertex-transitive map is a map on a surface on which the automorphism group of acts transitively on the set of vertices of . If the face-cycles at all the vertices in a map are of same type then the map is called a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is …
Combinatorial aspects of the Torelli-Johnson-Morita theory of surface automorphisms are extended to certain subgroups of the mapping class groups. These subgroups are defined relative to a specified homomorphism from the fundamental group of the surface onto an arbitrary group . For abelian, there is a combinato…