We review the string/gauge theory duality relating Chern-Simons theory and topological strings on noncompact Calabi-Yau manifolds, as well as its mathematical implications for knot invariants and enumerative geometry.
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Survey of geometry developments, including complex structures on surfaces.
problem Enumerative geometry and complex structures on surfaces.
method Differential and algebraic geometry, nonlinear elliptic PDEs.
result Extensions to 4-manifolds and complex structures on surfaces of general type.
The study counts minimal surfaces in 3-manifolds with positive Ricci curvature.
problem Counting minimal surfaces in 3-manifolds with positive Ricci curvature.
method An enumerative min-max theorem linking surface counts to topological properties.
result Every 3-sphere of positive Ricci curvature contains at least 4 embedded minimal surfaces of genus 2.
Develops resolvent degree theory for algebraic geometry problems.
problem Hilbert's 13th Problem and related conjectures.
method Extends Brauer's resolvent degree theory to algebraic geometry.
result Hilbert's 13th Problem and related conjectures are equivalent to enumerative geometry problems.
Review of instantons in noncommutative gauge theories across 4, 6, and 8 dimensions.
problem Understanding instantons in noncommutative gauge theories.
method Analysis of instantons in various dimensions, focusing on string theory and toric varieties.
result Geometric interpretations and applications of instantons in non-compact toric varieties.
Revisits SYM theory to compute Donaldson invariants using mock modular forms.
problem Computing Donaldson invariants in topological SYM theory.
method Uses mock modular forms and indefinite theta functions to evaluate correlation functions.
result Explicit evaluation of correlation functions leading to modular data predictions.
The paper connects algebraic geometry to threefold theories.
problem Enumerative geometry of nested Hilbert schemes.
method Study of threefold theories including Donaldson-Thomas, Vafa-Witten, and Seiberg-Witten.
result Connections between algebraic geometry and threefold theories.
Counting curves on surfacesmath.GT
In this paper we consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology. Consider a real 2-dimensional compact surface , and fix a number of points on its boundary. We ask: how many configurations of disjoint arcs are there on whose boundary is ? We find that thi…
Conjectures on universal structures in algebraic geometry enumerative invariants.
problem Understanding virtual classes in moduli spaces of stable objects.
method Defining virtual classes in homology over Q and proving a universal wall-crossing formula.
result Proving conjectures for quiver representations using Behrend-Fantechi virtual classes.
Paper proves 1-point recursions for various enumerative problems.
problem Enumerating gluings of polygons to orientable surfaces.
method Proves existence of 1-point recursions for specific classes of problems.
result Recover Harer-Zagier recursion and prove existence for new problems.
The topological recursion of Eynard and Orantin governs a variety of problems in enumerative geometry and mathematical physics. The recursion uses the data of a spectral curve to define an infinite family of multidifferentials. It has been conjectured that, under certain conditions, the spectral curve possesses a non-c…
Monograph compares curve signs in intrinsic and Pin-dependent definitions.
problem Comparing curve signs in intrinsic and Pin-dependent definitions.
method Intrinsic perspective on Spin- and Pin-structures, detailed constructions of orientations, explicit comparison.
result Explicit comparison between curve signs in intrinsic and Pin-dependent definitions.
Study on hypermaps and KP hierarchy, proving tau function and enumerative meaning.
problem Understanding the partition function of meromorphic functions on the Riemann sphere.
method Analysis of Hurwitz Dubrovin--Frobenius manifold structure and rational reductions of the KP hierarchy.
result The all genera partition function is a tau function of a rational reduction of the Kadomtsev--Petviashvili hierarchy.
This is a survey of some recent developments in the study of complements of line arrangements in the complex plane. We investigate the fundamental groups and finite covers of those complements, focusing on homological and enumerative aspects. The unifying framework for this study is the stratification of the character …
Sequences of Nilsson type appear in abundance in Algebraic Geometry, Enumerative Combinatorics, Mathematical Physics and Quantum Topology. We give an elementary introduction on this subject, including the definition of sequences of Nilsson type and the uniqueness, existence, and effective computation of their asymptoti…
Study real algebraic curves on real del Pezzo surfaces using degeneration methods.
problem Classify topological types of real algebraic curves on real del Pezzo surfaces.
method Degeneration methods and real enumerative geometry.
result Obstructions and constructions of real algebraic curves with prescribed topology.
The paper bounds entries of γ-vectors for flag homology spheres.
problem Bounding entries of γ-vectors for flag homology spheres.
method Structural and enumerative results for flag homology spheres.
result Supports a conjecture and characterizes structures of flag homology spheres.
Pruned Hurwitz numbersmath.GT
We define a new Hurwitz problem which is essentially a small core of the simple Hurwitz problem. The corresponding Hurwitz numbers have simpler formulae, satisfy effective recursion relations and determine the simple Hurwitz numbers. We also apply this idea of finding a smaller simpler enumerative problem to orbifold H…
In the symplectic category there is a `connect sum' operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds. This paper establishes a formula for the Gromov-Witten invariants of a symplectic sum Z=X#Y in terms of the relative GW invariants of X and Y. Several appl…
We define relative Gromov-Witten invariants of a symplectic manifold relative to a codimension two symplectic submanifold. These invariants are the key ingredients in the symplectic sum formula of [IP4]. The main step is the construction of a compact space of `V-stable' maps. Simple special cases include the Hurwitz nu…
The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in http://arxiv.org/abs/math.AG/0209253. The result is established with the help of the …
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.
RIn-Close_CVC2: an even more efficient enumerative algorithm for biclustering of numerical datasetscs.LG
RIn-Close_CVC2 improves biclustering efficiency by reducing memory usage.
problem Mining maximal biclusters in numerical datasets efficiently and without redundancy.
method Proposes RIn-Close_CVC2, a new version of RIn-Close_CVC that eliminates redundant biclusters without a symbol table.
result RIn-Close_CVC2 reduces memory usage and improves runtime compared to RIn-Close_CVC.
This paper reformulates double Hurwitz numbers using topological recursion.
problem Enumerating branched covers with specific ramification over zero and infinity.
method Topological recursion framework applied to double Hurwitz numbers.
result Double Hurwitz numbers are governed by the topological recursion.
Gromov-Witten invariants of a symplectic manifold are a count of holomorphic curves. We describe a formula expressing the GW invariants of a symplectic sum $X# Y$ in terms of the relative GW invariants of and . This formula has several applications to enumerative geometry. As one application, we obtain new relat…
Paper proves BGW tau-function can be represented as Q-polynomials.
problem Enumerative geometric interpretations of BGW tau-function.
method Proves BGW tau-functions are hypergeometric tau functions of BKP hierarchy.
result Original BGW tau-function can be represented as a linear combination of Schur Q-polynomials.
In the paper we formulate and derive the family blowup formula of family Seiberg-Witten invariants. The formula has been used in the enumerative application of counting singular curves on algebraic surfaces. We first give a topological derivation of the formula by using family index theorem. Then we define the algebrai…
We prove that the mirror map is trivial for the canonical formal families of Calabi-Yau varieties constructed by Gross and the second author. In other words, the natural coordinate in a canonical Calabi-Yau family is a canonical coordinate in the sense of Hodge theory. This implies that the higher weight periods direct…
We give a complete characterization of all possible pairs (v,e), where v is the number of vertices and e is the number of edges, of any simplicial triangulation of an S^k-bundle over S^1. The main point is that Kuhnel's triangulations of S^{2k+1} x S^1 and the nonorientable S^{2k}-bundle over S^1 are unique among all t…
Classical Hurwitz numbers count branched covers of the Riemann sphere with prescribed ramification data, or equivalently, factorisations in the symmetric group with prescribed cycle structure data. Monotone Hurwitz numbers restrict the enumeration by imposing a further monotonicity condition on such factorisations. In …
Lectures on link homology and its algebraic/geometric models.
problem Defining and understanding link homology.
method Definition and properties of Khovanov-Rozansky triply graded homology, and three geometric models.
result Three geometric models for link homology: braid varieties, Hilbert schemes, and affine Springer fibers.
We study topological recursion on the irregular spectral curve , which produces a weighted count of dessins d'enfant. This analysis is then applied to topological recursion on the spectral curve , which takes the place of the Airy curve to describe asymptotic behaviour of enumerative proble…
In this note we derive enumerative formulas for several types of labelled acyclic directed graphs by slight modifications of the familiar recursive formula for simple acyclic digraphs. These considerations are motivated by, and based upon, recent combinatorial results in geometric topology obtained by S.Choi, who estab…
This note presents a formula for the enumerative invariants of arbitrary genus in toric surfaces. The formula computes the number of curves of a given genus through a collection of generic points in the surface. The answer is given in terms of certain lattice paths in the relevant Newton polygon. If the toric surface i…
For each sphere with three orbifold points, we construct an algorithm to compute the open Gromov-Witten potential, which serves as the quantum-corrected Landau-Ginzburg mirror and is an infinite series in general. This gives the first class of general-type geometries whose full potentials can be computed. As a conseque…
Algorithm to compute cohomology groups of real flag manifolds, proving torsion and Schubert varieties.
problem Computing cohomology groups of real flag manifolds.
method Algorithm based on Schubert cells and incidence coefficients.
result Results on torsion classes and Schubert varieties for real flag manifolds.
Researchers found all embeddings of Kuratowski graphs on a double torus.
problem Characterizing embeddings of Kuratowski graphs and 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 on the double torus.
Geometrically, Legendrian surfaces related by surgery have related skein-valued cluster spaces.
problem Understanding the skein-valued cluster transformation in Legendrian surfaces.
method Geometric considerations of moduli of holomorphic curves.
result Skein-valued cluster transformation of Legendrian surfaces related by surgery.
Fast, fully-automated histograms for large data sets.
problem Efficiently constructing histograms for large-scale data.
method G-Enum histograms using Minimum Description Length (MDL) principle for model selection and a greedy search heuristic.
result Construct histograms in linearithmic time, significantly faster than previous methods.
Computes the Euler characteristic and rational homology of moduli spaces of multi-monopoles.
problem Enumerative invariant of 3-manifolds and geometry of algebraic curves.
method Spectral curves and Fulton-Lazarsfeld techniques.
result Completes a computation of Doan's initiated Euler characteristic.
We make a precision test of a recently proposed conjecture relating Chern-Simons gauge theory to topological string theory on the resolution of the conifold. First, we develop a systematic procedure to extract string amplitudes from vacuum expectation values (vevs) of Wilson loops in Chern-Simons gauge theory, and then…
We study the following question: given a set P of 3d-2 points and an immersed curve G in the real plane R^2, all in general position, how many real rational plane curves of degree d pass through these points and are tangent to this curve. We count each such curve with a certain sign, and present an explicit formula for…
Constructs Morse homology for complex algebraic varieties.
problem Homology of vanishing cycles in complex algebraic varieties.
method Morse homology groups associated with a perturbed function on a compactified variety.
result Morse homology groups are isomorphic to the homology of vanishing cycles.
Jointly learning to localize and repair variable-misuse bugs improves program repair.
problem Variable-misuse bugs in programs.
method Multi-headed pointer networks for joint localization and repair.
result Joint model significantly outperforms an enumerative solution.
Strong positivity for the skein algebras of the -punctured sphere and of the -punctured torusmath.GT
Researchers prove positivity of skein algebra structure constants for specific surfaces.
problem Positivity of structure constants in skein algebras of specific surfaces.
method Mirror symmetry construction based on higher genus Gromov-Witten theory applied to a complex cubic surface.
result Proved positivity of structure constants for skein algebras of the 4-punctured sphere and 1-punctured torus.
Let be a compact toric Kähler manifold with nef. Let be a regular fiber of the moment map of the Hamiltonian torus action on . Fukaya-Oh-Ohta-Ono defined open Gromov-Witten (GW) invariants of as virtual counts of holomorphic discs with Lagrangian boundary condition . We prove a formula…
Improved biclustering algorithm reduces memory usage and runtime.
problem Efficiently enumerating maximal biclusters in numerical datasets.
method Online partitioning to guide biclustering results.
result RIn-Close_CVC3 reduces memory usage and runtime, handles missing values.
New operations defined on moduli spaces for bundles with orientations.
problem Pushforward operations for principal bundles with orientations.
method Developed a general theory of pushforward operations for principal -bundles, constructing specific operations for .
result Classified all stable pushforward operations and showed they are generated by the projective Euler and rank operations.