New method calculates colored HOMFLY polynomials for knots drawn as double fat graphs.
problem Calculating HOMFLY polynomials for knots drawn as double fat graphs.
method Using fusion and braiding matrices of four-strand braids, incorporating four-point conformal blocks in WZNW models.
result Conjectured and verified colored HOMFLY polynomials for many knots, including distinguishing and non-distinguisable mutants.
Introduces fat Lie theory for Lie groupoids and algebroids.
problem Representation theory of Lie groupoids and algebroids.
method Introduces fat extensions and abstract 2-term representations up to homotopy (ruths). Establishes correspondences and equivalences.
result One-to-one correspondence between fat extensions and abstract 2-term representations up to homotopy.
New proofs connect fat graph models to mapping class groups.
problem Proving equivalence of fat graph models to mapping class groups.
method Using contractibility of arc complex and new integral proofs.
result Established direct connection between fat graph models and mapping class groups.
Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surface (we call these admissible). There is a natural necessary condition on such grap…
Given a smooth closed manifold M with a family {L_i} of closed submanifolds, we consider the free loop space LM and the spaces PM(L_i,L_j) of open strings (paths g:[0,1]->M with g(0) in L_i, and g(1) in L_j). We construct string topology operations resulting in an open-closed TQFT on the family (h_*(LM),h_*(PM(L_i,L_j)…
The paper studies colored knot polynomials focusing on representation [2,1].
problem Systematic description of colored knot polynomials.
method Parametrization of knot families, evaluation of mixing matrices, tabulation of results.
result Evaluation of knots from a 7-parametric family with up to 10 intersections.
Graphs with fat minors have a limited large-scale structure.
problem Understanding the large-scale structure of graphs excluding certain minors.
method Introduced the concept of Baker-treewidth and used it to prove asymptotic dimension bounds.
result Every hereditary class of bounded-degree graphs excluding some graph as a fat minor has asymptotic dimension at most 2.
Two combinatorial models for moduli space are compared and homotopy equivalence proven.
problem Comparing and relating combinatorial models of moduli space.
method Using a critical graph map to produce an explicit homotopy equivalence between Bödigheimer's radial slit configurations and Godin's admissible fat graphs, and discussing natural compactifications.
result Induces a cellular homeomorphism between unilevel harmonic compactification and Sullivan diagrams.
Extends knot polynomial construction for braids with m strands.
problem Understanding structure of mysterious knot polynomials.
method Generalizes knot polynomials for arbitrary m-strand braids using R-matrices and mixing matrices.
result Explicit expressions for knot polynomials in sectors R^⊗3 → Q.
The paper tabulates knot polynomials for a specific class of knots.
problem Computing knot polynomials for arborescent knots efficiently.
method Family approach and Feynman diagram technique with auxiliary matrix model field theory.
result New tables of colored knot polynomials for arborescent knots.
Researchers found all embeddings of Kuratowski graphs on a double torus.
problem Characterizing embeddings of Kuratowski graphs K3,3 and K5 on the double torus. method Constructive approach using Burnside's Lemma and automorphism groups.
result 14 orientable and 17 non-orientable 2-cell embeddings of K5 on the double torus. Simplified approach to categorify knot polynomials using matrix models.
problem Evaluate quantum dimensions of knot polynomials in a simplified Chern-Simons theory.
method Use matrix model to reformulate quantum dimensions in terms of fat graphs and derive recursions.
result Explicit expressions for quantum dimensions of virtual knots, including negative ones.
The study combines graph-minors and metric spaces, answering some questions and conjectures.
problem Whether geodesic metric spaces without a fat H minor are quasi-isometric to graphs without H minor. method Combining graph-minors and coarse geometry, answering affirmatively for small H. result Affirmative answer for small H in the problem statement. New method extracts Racah matrices from antiparallel double-braid knots.
problem Extract exclusive Racah matrices for arborescent knots.
method Factorization of differential expansion for antiparallel double-braids.
result Found a way to reduce problem to twist knots and provide answers for R=[33]. Paper classifies fibers of fat Riemannian submersions with non-negative curvature.
problem Understanding fat bundles in Riemannian submersions.
method Structural analysis of fat Riemannian submersions with non-negative sectional curvature.
result Classification of fibers as symmetric spaces, isometric correspondence of foliations, rigidity of dual foliations.
New graph invariant measures embeddability in 3D.
problem Measuring embeddability of graphs in 3D.
method Defining freeness index to measure embeddability of graph complements.
result Cubic graphs satisfying orientable cycle double cover conjecture have freeness index at least two.
Graphs with nonnegative Bakry-Émery curvature have volume doubling and Poincaré inequalities.
problem Proving properties of graphs with specific curvature conditions.
method Graph-theoretic modified nonlinear heat-flow method, including point-mass consequences and diffusive exit-time control.
result Volume doubling and Poincaré inequalities for graphs with nonnegative Bakry-Émery curvature.
GNNs generalize better on homophilic graphs than heterophilic ones.
problem Understanding the generalization error of GNNs on graph data.
method Analytical tools from statistical physics and random matrix theory.
result Risk is shaped by graph noise, feature noise, and training labels.
The paper proves properties for random graphs based on geometric submanifolds.
problem Establishing measure-metric properties of random geometric graphs.
method Analyzing ε-neighborhood graphs with specific conditions on submanifold and distribution. result Volume doubling and local Poincaré inequalities hold for random geometric graphs with high probability.
A generic immersion of a planar graph into the 2-space is said to be knotted if there does not exist a trivial embedding of the graph into the 3-space obtained by lifting the immersion with respect to the natural projection from the 3-space to the 2-space. In this paper we show that if a generic immersion of a planar g…
Paper introduces fat CW complexes including all closed manifolds.
problem No specific problem stated, focuses on introducing new CW complexes.
method Introduces a new smooth version of CW complexes called fat CW complexes.
result Fat CW complexes include all closed manifolds and have desirable properties.
Constructs fat, shellable 3-spheres with specific f-vectors.
problem Defines and constructs fat 3-spheres.
method Constructs strongly regular CW 3-spheres that are both shellable and dual shellable.
result Constructs arbitrarily fat, shellable and dual shellable 3-spheres with specific f-vectors. Incomplete pressure metric on bordered surface Teichmüller space proved.
problem Incompleteness of pressure metric on Teichmüller space of bordered surfaces.
method Using moduli space of metric graphs and fat graphs associated to bordered surfaces.
result Pressure metric is not a constant multiple of Weil-Petersson metric.
Liouville theorem for minimal graphs on manifolds with specific properties.
problem Characterizing positive minimal graphic functions on specific Riemannian manifolds.
method Using volume doubling property and uniform Neumann-Poincaré inequality.
result Positive minimal graphic functions on the manifold are constants.
The paper solves the Nielsen realization problem for cyclic actions on surfaces.
problem Realizing finite order mapping classes as isometries of hyperbolic structures.
method Inductive procedure to construct hyperbolic structures and combinatorial perspective.
result Explicit solutions to the Nielsen realization problem for cyclic subgroups.
Finite subgraphs of extension graph are found within a sequence of induced subgraphs.
problem Identifying all finite subgraphs within the extension graph.
method Inductively define a sequence of induced subgraphs through 'doubling along a star'.
result Every finite induced subgraph of the extension graph is isomorphic to an induced subgraph of some Γi. Let Fg denote a closed oriented surface of genus g. A set of simple closed curves is called a filling of Fg if its complement is a disjoint union of discs. The mapping class group Mod(Fg) of genus g acts on the set of fillings of Fg. The union of the curves in a filling forms a graph on the surfa…
New rigidity result for fat bundles with equal vertical curvatures.
problem Equalizing vertical curvatures in fat bundles with specific constraints.
method Rigidity result for fat Riemannian foliations with bounded holonomy and curvature constraint.
result Established a rigidity result for fat fiber bundles with compact structure groups.
The article proves the existence of horizontal immersions into fat distributions and contact structures.
problem Proving the existence of horizontal immersions in fat distributions and contact structures.
method Gromov's sheaf theoretic and analytic techniques of h-principle. result Existence of horizontal immersions of an arbitrary manifold into degree 2 fat distributions and quaternionic contact structures.
Classifies fat bundles over homogeneous spaces with conditions.
problem Classifying fat bundles over compact homogeneous spaces.
method Generalizing Berard-Bergery's classification for arbitrary G-structures.
result Necessary conditions for the existence of fat bundles.
Study of harmonic functions on infinite penny graphs.
problem Characterizing harmonic functions on infinite penny graphs.
method Proving volume doubling and Poincaré inequalities, analyzing polynomial growth harmonic functions.
result Finite dimensional property of ancient solutions of the heat equation.
Improved uniform convergence bound with fat-shattering dimension reduces sample complexity gap.
problem Gap between upper and lower bounds on sample complexity for fat-shattering dimension.
method Provided an improved uniform convergence bound.
result Closed the gap between existing upper and lower bounds on sample complexity.
A classic problem in physics is the origin of fat tailed distributions generated by complex systems. We study the distributions of stock returns measured over different time lags τ. We find that destroying all correlations without changing the τ=1 d distribution, by shuffling the order of the daily returns, causes…
This work is devoted to new constructions of symplectically fat fiber bundles. The latter are constructed in two ways: using the Kirwan map and expressing the fatness condition in terms of the isotropy representation related to the G-structure over some homogeneous spaces.
Proposes NDIG model to capture bitcoin volatility and option pricing.
problem Capturing the volatility and option pricing of cryptocurrency Bitcoin.
method Doubly subordinated Levy process (NDIG) to model Bitcoin time series properties.
result NDIG model perfectly captures observed in-sample volatility.
Expands small recommendation datasets to industrial scale.
problem Disconnection between academic and industrial data scales.
method Randomized fractal expansions using Kronecker Graph Theory.
result Generated synthetic data sets with 1.2B ratings, 2.2M users, and 855K items.
Study horizontal discs in fat distributions, proving their existence.
problem Existence of embedded horizontal discs in fat distributions.
method Analyzing nonlinear PDEs and proving local invertibility.
result Existence of germs of embedded horizontal discs.
Study on manifolds with flat sections and their geodesics.
problem Characterizing geodesics in manifolds with flat sections.
method Analyzing closed non-positively curved Riemannian manifolds with fat k-flats.
result Existence of uncountably many closed geodesics in manifolds with fat k-flats.
New example disproves complex contact theory for fat distributions with Reeb directions.
problem Whether fat (4,6)-distributions with Reeb directions always come from complex contact structures. method Constructed a counterexample of a fat distribution with two Reeb directions that does not support a complex contact structure.
result The space of complex-contact germs has infinite codimension within the space of fat (4,6)-distribution germs with Reeb directions. Estimates fat-shattering dimension of aggregated function classes.
problem Understanding the complexity of aggregated function classes.
method Analyzes fat-shattering dimension of k-fold aggregations of real-valued function classes. result Provides upper and lower bounds on fat-shattering dimension for linear and affine function classes.
Graphs satisfy Li-Yau inequality under CD(0,n) curvature condition.
problem Proving Li-Yau inequality for graphs under CD(0,n) condition. method Introduced modified heat equation and used Bakry Emery curvature condition.
result Proved Li-Yau inequality −Δut≤2tn under CD(0,n) condition. Geodesic nets on flat spheres are studied using Gauss-Bonnet theorem.
problem Existence and non-existence of specific geodesic nets on flat spheres.
method The theorem of Gauss-Bonnet is applied to demonstrate results.
result Existence and non-existence of geodesic nets on regular doubled polygons.
New bounds on inscribed triangles in arbitrary planar domains.
problem Finding inscribed triangles in arbitrary planar domains with specific angle constraints.
method Proving the existence of uniformly fat triangles and not-too-fat triangles in bounded open sets.
result Existence of a maximal number Θ (between 0 and 60) for inscribed triangles with angles ≥ Θ degrees.
Paper introduces a new fat-tail test using conditional second moments.
problem Measuring and assessing the impact of fat-tails on data dispersion.
method Constructs a goodness-of-fit statistic based on conditional second moments.
result Shows superior performance of the new test compared to existing methods.
Study finds recent Korean stock returns have fatter tails than before, especially for smaller companies.
problem Understanding the fat tails in financial return distributions.
method Empirical analysis of Korean stock market data, controlling for the 1997 foreign currency crisis and volatility clustering.
result Fat tails in stock return distributions persist even after controlling for market crashes and volatility clustering.
The h-principle fails for prelegendrians in fat distributions of corank 2.
problem Investigating the h-principle for fat distributions of corank 2.
method Developed the theory of prelegendrians, including front projection and pseudoholomorphic curve invariants.
result Found an infinite family of non-prelegendrian isotopic tori in the standard fat distribution.
New method extracts Racah matrices revealing hidden integrability in knot evolution.
problem Understanding hidden integrability in knot evolution.
method Evolution method to extract Racah matrices from 3-strand mixing matrices.
result Reveals unexpected integrability in the evolution of knots.
In these lecture notes we discuss a body of work in which Morse theory is used to construct various homology and cohomology operations. In the classical setting of algebraic topology this is done by constructing a moduli space of graph flows, using homotopy theoretic methods to construct a virtual fundamental class, an…