Quantum BPS invariants linked to combinatorics of Lyndon words.
problem Relating quantum BPS invariants to combinatorics on words.
method Constructing combinatorial models and using difference equations.
result BPS invariants expressed in terms of Lyndon words.
The paper studies combinatorics of injective words in the context of Temperley-Lieb algebras.
problem Combinatorial properties of injective words in the context of Temperley-Lieb algebras.
method Investigation of a chain complex of modules over the Temperley-Lieb algebra, focusing on Euler characteristic, homology modules, and Jacobsthal numbers.
result The Euler characteristic of the complex is the n-th Fine number, and the top-dimensional homology module is decomposed in terms of standard Young tableaux.
Here we are fixing an output of a trivial calculation based on Konsevich's differential 2-form for the Chern class of polygon bundle. As a result an interesting combinatorics and arithmetics jumps right out of a jukebox. The calculation gives very simple rational combinatorial characteristics (we call it "curvature") o…
A simple text model shows word lengths follow Zipf's law.
problem Understanding word statistics in large language models.
method A non-linguistic model of text with independent symbol draws.
result Word lengths follow a geometric distribution and Zipf's law.
The study of Farey polynomials connects geometry, topology, and combinatorics.
problem Understanding the combinatorics of Farey polynomials and their applications.
method Recursive definition of Farey polynomials, combinatorial analysis, and geometric/topological connections.
result New properties and recursive definition of Farey polynomials, providing practical solutions to classification problems.
Study on representations of four-punctured sphere group in hyperbolic spaces.
problem Understanding representations of the four-punctured sphere group.
method Investigation into simple-stable and Bowditch representations in Gromov-hyperbolic spaces.
result Simple-stable representations and Bowditch representations are equivalent.
Surveying connections between graph combinatorics and algebraic right-angled Artin groups.
problem Understanding the relationship between graph structures and algebraic properties of right-angled Artin groups.
method Analyzing the defining and extension graphs of right-angled Artin groups.
result Discovers connections to geometric group theory and complexity theory.
Survey explores topology and combinatorics of Higgs bundle spaces.
problem Understanding the structure of Higgs bundle moduli spaces.
method Examples and combinatorial analysis of cohomology rings.
result Interesting combinatorial questions arise from the moduli space structure.
Neural network uses local credit assignment to learn distant cause-effect relationships.
problem Learning distant cause-effect relationships in sequential data.
method Sparse coding in a recurrent neural network memory with local and immediate credit assignment.
result Network can predict partially-observable higher-order sequences and navigate mazes.
Estimates higher order derivatives using Lie derivatives and combinatorics.
problem Estimating higher order derivatives of Lie derivatives.
method Combines Lie derivatives, combinatorics of forests, and Dyck polynomials.
result Provides an estimate for higher order covariant derivatives of multiple Lie derivatives.
Survey on GKM theory in low dimensions, highlighting combinatorics-geometry interplay.
problem Generalizing classical ideas from quasi-toric manifolds to torus actions.
method GKM theory applied to low-dimensional cases.
result Particularly fruitful interaction between geometry and combinatorics in low dimensions.
New model for links uses meander diagrams and combinatorics.
problem Modeling and analyzing random links.
method Random meander model based on meander diagrams and graphs, proving properties using combinatorics.
result Trivial links are unlikely, and there's a lower bound on non-isotopic knots.
The paper combinatorizes spaces of nondegenerate spherical curves.
problem Understanding the homotopy type of spaces of nondegenerate spherical curves.
method Orthogonalization of Frenet frames, decomposition into Schubert cells, and construction of cell complexes.
result Spaces of nondegenerate curves are contractible topological submanifolds.
New method determines arrangement combinatorics from Milnor fiber boundary.
problem Determining arrangement combinatorics from Milnor fiber boundary.
method Explicit method using plumbing graph in normal form.
result Milnor fiber boundary determines arrangement combinatorics.
New discrete cmc surfaces defined from sphere packings and combinatorics.
problem Creating constant mean curvature surfaces from discrete data.
method Discrete cmc surfaces defined via sphere packings and combinatorial patterns.
result Construction of discrete cmc surfaces from orthogonal ring patterns.
Totally nonnegative Grassmannian and related spaces are shown to be like closed balls.
problem Understanding the topological structure of certain spaces in combinatorics.
method Proving homeomorphic to closed balls using advanced combinatorial and geometric techniques.
result Three significant spaces in combinatorics are proven to be topologically equivalent to closed balls.
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.
problem Existence of special almost-complex structures on almost-Kähler manifolds.
method Combinatorics of Vogan diagrams for classical semisimple Lie groups.
result Explicit formulae for special almost-complex structures.
The maximum hyperbolic polyhedron volume is found to be the rectification of its skeleton.
problem Finding the maximum volume of hyperbolic polyhedra with given combinatorics.
method Applying a volume-increasing flow to any hyperbolic polyhedron, handling degeneracies carefully.
result The supremum volume is always the volume of the rectification of the 1-skeleton.
Study restricts line arrangements with odd points using topological arguments.
problem Restrictions on line arrangements with singular points of odd multiplicity.
method Topological arguments on locally-flat spheres in 4-manifolds.
result No line arrangement with 13 lines and only triple points exists.
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…
New combinatorial method connects knot invariants to reflection groups.
problem Computing knot invariants using combinatorial techniques.
method Relating dual braid group generators, Hecke images of pure braids, and reflection groups.
result The (a,z=0)-HOMFLYPT polynomial can be computed as a solution to factorization problems. Graph dynamics link combinatorics to geometry, revealing manifold intersections and stability.
problem Understanding the geometry of graph dynamical systems with odd interactions.
method Proved geometry and stability of manifolds governed by graph homology and coverings.
result Derived upper and lower bounds on the dimension of the equilibrium set.
In this paper, we show that the volumes for a family of A-adequate closed braids can be bounded above and below in terms of the twist number, the number of braid strings, and a quantity that can be read from the combinatorics of a given closed braid diagram. We also show that the volumes for many of these closed braids…
Paper studies geometric and combinatorial properties of circular snakes.
problem Exploring geometric and combinatorial properties of circular snakes.
method Definition and investigation of outer Lipschitz geometry, decomposition of Valette link, construction of combinatorial objects, weakly outer Lipschitz classification.
result Existence of canonical decomposition and necessary/sufficient criteria for removing segments or Hölder triangles.
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 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 (0,2π], and the union of n copies of Rd, when the polyhedron has the combin…
Gordon-Litherland pairing connects combinatorics and topology.
problem Unifying quadratic forms in link theory.
method Picture proof using Kirby diagrams.
result Their theorem has numerous applications in low-dimensional topology.
Simplified proof of topological Tverberg conjecture for prime powers, with counterexamples for others.
problem Proving the topological Tverberg conjecture for all integers r, d > 1.
method Combining combinatorics, algebra, and topology to provide a simplified proof.
result The conjecture holds for prime powers r but not for others.
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 d=2m depends only on the parity of m. Gromov defined several hyperbolization functors which produce an aspherical manifold …
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.
problem Investigating embeddings of edge-colored dual graphs of balanced 3- and 4-manifolds.
method Introducing the concept of balanced genus and proving lower bounds for the genus of 3- and 4-manifolds.
result Established lower bounds for the balanced genus of 3- and 4-manifolds, and conditions for homeomorphism to spheres.
Study conic line arrangements of degree 7, finding their topology and connected components.
problem Understanding the topology of conic line arrangements of degree 7.
method Identifying a π1-equivalent Zariski pair to prove the existence of a conic line arrangement with specific combinatorics. result Determine the number of connected components of conic line arrangements of degree 7.
Square-tiled surfaces with fixed combinatorics become equidistributed in moduli spaces.
problem Equidistribution of square-tiled surfaces with specific combinatorics.
method Analyzing the asymptotic behavior of square-tiled surfaces and their contributions to moduli spaces.
result Square-tiled surfaces with fixed combinatorics become asymptotically equidistributed in moduli spaces.
Minimal surfaces in 3-sphere created by reflections from polygons, with new examples based on pentagons.
problem Constructing minimal surfaces in 3-sphere using reflections.
method Minimal n-gon solves free boundary problem; curvature lines combinatorics investigated. result New examples of minimal reflection surfaces based on pentagons.
The paper studies the topology and geometry of simple orbifolds, generalizing concepts from simple polytopes.
problem Understanding the topology and geometry of simple orbifolds.
method Generalizing concepts from simple polytopes to simple orbifolds, focusing on simple handlebodies.
result Characterization of orbifold-aspherical properties and the existence of rank-two free abelian subgroups in terms of combinatorics.
Study of hyperideal polyhedra in anti-de Sitter space.
problem Characterizing hyperideal polyhedra in anti-de Sitter space.
method Defined hyperideal polyhedra as intersections with convex polyhedra in projective model of anti-de Sitter space.
result Hyperideal polyhedra uniquely determined by combinatorics, dihedral angles, and induced metrics on boundary.
Simplified proof for approximations of set systems.
problem Approximations of set systems in various fields.
method Modular, self-contained proof using Chernoff's bound.
result Accessible proof for a wider audience.
The family Blow Up formula is recalled. Certain combinatoric graphs are introduced for the discussion of the counting of nodal curves on an Kahler surface.
The study shows that certain curve graphs are hierarchically hyperbolic but not Gromov hyperbolic.
problem Characterizing the hyperbolicity of curve graphs and their boundaries.
method Using hierarchical hyperbolicity and framed curves, the study examines the properties of curve graphs and their boundaries.
result The curve graphs and their boundaries are hierarchically hyperbolic but not Gromov hyperbolic.
Formulae for Seiberg-Witten invariants derived from graph combinatorics.
problem Calculating Seiberg-Witten invariants for 3-manifolds.
method Combining Poincaré series and counting functions of a graph to derive surgery formulae.
result The periodic constant difference of invariants for surgeries on 3-manifolds.
The study of Chern numbers on vector bundles uses combinatorial methods to establish bounds and ordering.
problem Establishing bounds and ordering for Chern numbers on vector bundles.
method Combinatorial ideas to study Chern numbers on ample and numerically effective vector bundles.
result An effective lower bound for Chern numbers of ample vector bundles and reverse dominance ordering for nef vector bundles.
Study on inflection points of plane curve shadows with fixed embedded shapes.
problem Minimum number of inflection points in plane curves with fixed embedded shadows.
method Finite coorientation problem on building polygons, dynamic programming, universal lower bound, tree-necklace shadows.
result Exact formula for minimum number of normalized inflections for tree-like shadows.
Paper learns identity-sensitive word embeddings from text corpora.
problem Lack of context-aware word embeddings.
method Constructs a heterogeneous network of words and identities, then embeds into a low-dimensional space.
result Identity-sensitive word embeddings capture different meanings of words.