Hodge theory applied to tropical curves.
problem Developing Hodge theory for tropical curves.
method Analytical approach using tropical differential forms and cohomologies.
result Construction of Hodge theory analog on tropical curves.
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48 results for “tropical Hodge theory”
Tropical Clemens-Schmid sequence and existence of tropical cycles with a given cohomology classmath.AG
Proves a tropical version of Clemens-Schmid sequence for tropical varieties.
problem Existence of tropical cycles with specific cohomology classes.
method Tropical Hodge theory and Clemens-Schmid sequence analogy.
result Proves tropical Hodge conjecture for rationally triangulable varieties.
Proves tropical Hodge theory for smooth projective varieties, conditional on Laplacian regularity.
problem Proving log-concavity of characteristic polynomials of matroids.
method Combinatorial approach, conditional proof of Kähler package.
result Conditional proof of Kähler package for tropical cohomology.
Proves cohomology theorems for tropical varieties.
problem Cohomology of smooth projective tropical varieties.
method Introduces and proves new results in tropical geometry.
result Establishes tropical analogs of three fundamental theorems.
Homology of tropical fansmath.AG
The paper studies homology of tropical fans and introduces smoothness.
problem Homological properties of tropical fans and smoothness.
method Proposes a notion of smoothness in tropical geometry, proving the Hodge isomorphism theorem.
result Chow rings of smooth unimodular tropical fans are isomorphic to tropical cohomology rings.
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…
Tropical geometry aids in computing topological quantum field theories.
problem Computing Gromov-Witten invariants using tropical geometry.
method Using mathematical techniques of tropical geometry to compute topological quantum field theories of pseudoholomorphic maps.
result Identifies the tropicalization of localization equations and studies the geometry and symmetries of the theory.
This survey consists of two parts. Part 1 is devoted to amoebas. These are images of algebraic subvarieties in the complex torus under the logarithmic moment map. The amoebas have essentially piecewise-linear shape if viewed at large. Furthermore, they degenerate to certain piecewise-linear objects called tropical vari…
To a tropical -cycle in , we naturally associate a normal closed and -dimensional current on denoted by . Such a "tropical current" will not be an integration current along any analytic set, si…
Perfect pairing for tropical cycles on integral affine manifolds.
problem Computing period integrals and versality of Calabi-Yau degenerations.
method Introducing a cap product pairing and using simplicial methods for constructible sheaves.
result The pairing is perfect in degree one for symplectic singularities.
Almost complex Hodge theorymath.DG
Study on Hodge theory for almost complex manifolds.
problem Determining Hodge numbers for almost complex manifolds.
method Review and analysis of recent developments in Hodge theory for almost complex manifolds.
result Hodge numbers are almost complex, almost Kähler, or birational invariants in dimension four.
Refines Hurwitz numbers with a two-parameter theory.
problem Understanding polynomial structure and tropicalization of -Hurwitz numbers.
method Introducing CJT-refinement of symmetric functions on Fock space.
result Derives tropicalization of -Hurwitz numbers and solves an open problem.
TML package uses tropical geometry for machine learning tasks.
problem Statistical learning problems.
method Tropical convexity computations, Hit and Run sampler, tropical metrics.
result First R package for tropical geometric machine learning.
Tropical SVM tackles phylogenomics by classifying multi-locus data.
problem Classifying multi-locus data sets for phylogenetic analysis.
method Proposes tropical support vector machines (SVMs) for phylogenomics, formulated as linear programming problems.
result Developed methods for hard and soft margin tropical SVMs, proving necessary and sufficient conditions for separation.
We introduce in this paper the concept of tropical mirror hypersurfaces and we prove a complex tropical localization Theorem which is a version of Kapranov's Theorem \cite{K-00} in tropical geometry. We give a geometric and a topological equivalence between coamoebas of complex algebraic hypersurfaces defined by a maxi…
Lecture notes from the Concentrated Graduate Course preceding the Workshop on Hodge Theory in String Theory at the Fields Institute in Toronto, November 11--15, 2013.
Lin and Sjamaar have used symplectic Hodge theory to obtain canonical equivariant extensions for Hamiltonian actions on closed symplectic manifolds that have the strong Lefschetz property. Here we obtain canonical equivariant extensions much more generally by means of classical Hodge theory.
The paper uses non-abelian Hodge theory to generalize Kodaira vanishing theorems.
problem Generalizing Kodaira vanishing theorems for non-abelian settings.
method Non-abelian Hodge theory and Mixed Twistor D-modules.
result Generalized Kodaira vanishing theorems for various settings.
Introduces a new Hodge theory using vector fields on manifolds.
problem Developing a new Hodge theory for manifolds with vector fields.
method Defines a vector field induced Hodge -inner product, codifferential, and Laplacian.
result Established de Rham-Hodge theory for closed and boundary manifolds.
Novel Hodge theory developed for Carrollian bundles, addressing black hole event horizons.
problem Developing Hodge theory on Carrollian bundles with degenerate metrics.
method Constructing Hodge star and Laplacian operators on Carrollian -bundles.
result Hodge--de Rham Laplacian defined on event horizons of Schwarzschild black holes.
A bit of tropical geometrymath.AG
This friendly introduction to tropical geometry is meant to be accessible to first year students in mathematics. The topics discussed here are basic tropical algebra, tropical plane curves, some tropical intersections, and Viro's patchworking. Each definition is explained with concrete examples and illustrations. To a …
We establish, for the first time, connections between feedforward neural networks with ReLU activation and tropical geometry --- we show that the family of such neural networks is equivalent to the family of tropical rational maps. Among other things, we deduce that feedforward ReLU neural networks with one hidden laye…
New method for constructing real algebraic surfaces from complex tropical hypersurfaces.
problem Constructing real algebraic surfaces from complex tropical hypersurfaces.
method Non-Abelian patchworking theory.
result The method reproduces all existing isotopy types of surfaces up to degree three.
This note is devoted to the definition of moduli spaces of rational tropical curves with n marked points. We show that this space has a structure of a smooth tropical variety of dimension n-3. We define the Deligne-Mumford compactification of this space and tropical -class divisors.
Develops Hodge theory on ALG manifolds, proving existence and vanishing results.
problem Existence and vanishing of certain cohomology groups on ALG manifolds.
method Fredholm Theory for Hodge Laplacian in weighted spaces on ALG manifolds.
result Non-existence of ALG manifolds with non-negative Ricci curvature at infinity.
Study of algebraic dynamics on Markov cubics in tropical geometry.
problem Understanding the dynamics of Markov cubics over non-archimedean fields.
method Tropicalization and -triangle reflection group on hyperbolic plane.
result Existence of Fatou domain and finitude of orbits with rational points over prime power denominators.
Mathematical analysis of Riemann surfaces and their moduli spaces using hybrid Laplacians.
problem Analyzing the asymptotics of Arakelov Green functions on Riemann surfaces near boundary of moduli spaces.
method Introducing hybrid Laplacian, solving hybrid Poisson equation, and defining hybrid Green functions.
result Layered description of asymptotics of Arakelov Green functions on Riemann surfaces near boundary of their moduli spaces.
Discretizes Hodge-Dirac operators on a torus.
problem Capturing geometric aspects of continuum Hodge theory in discrete settings.
method Discrete exterior calculus framework, Hodge-Dirac and Laplace operators.
result Proves discrete Hodge decomposition theorem on combinatorial torus.
This work uses tropical geometry to understand neural network decision boundaries.
problem Characterizing neural network decision boundaries with piecewise linear activations.
method Tropical geometry applied to a simple neural network model.
result Decision boundaries are a subset of a tropical hypersurface related to a polytope formed by zonotopes.
We refine the Morgan's work on mixed Hodge structures on Sullivan's --minimal models by using non-abelian Hodge theory. As an application, we give explicit representatives of real unipotent variations of mixed Hodge structures over compact K"ahler manifolds.
The paper provides a combinatorial criterion for realizing tropical pluri-canonical divisors.
problem Determining when a tropical pair corresponds to a smooth algebraic curve with a pluri-canonical divisor.
method Introducing tropical normalized covers and reducing the problem to their realizability.
result Generalizes previous work on tropical canonical divisors and incorporates recent progress on -differentials.
This work uncovers the tropical analogue for measured laminations of the convex hull construction of decorated Teichmueller theory, namely, it is a study in coordinates of geometric degeneration to a point of Thurston's boundary for Teichmueller space. This may offer a paradigm for the extension of the basic cell decom…
Explains Hodge theory and Kodaira embedding theorem for complex manifolds.
problem Understanding complex manifold properties and their geometric implications.
method Expository review of harmonic forms, Hodge theory, and Kodaira embedding theorem.
result Establishes connections between de Rham cohomology, Dolbeault cohomology, and projective varieties.
Anabelian geometry reformulated using Hodge theory for hyperbolic curves.
problem Determining varieties over number fields using their étale fundamental groups.
method Formulating a Hodge-theoretic version of anabelian conjecture, replacing Galois action with -action.
result Proved a Hodge-theoretic analog of Mochizuki's theorem for smooth projective hyperbolic curves over .
Knots and tropical curvesmath.GT
Using elementary ideas from Tropical Geometry, we assign a a tropical curve to every -holonomic sequence of rational functions. In particular, we assign a tropical curve to every knot which is determined by the Jones polynomial of the knot and its parallels. The topical curve explains the relation between the AJ Con…
New coordinates for SL3-web graphs on surfaces defined by Fock-Goncharov.
problem Characterizing functions on SL3-character varieties of surfaces.
method Introduced tropical Fock-Goncharov coordinates based on Knutson-Tao rhombus inequalities and congruence conditions.
result Tropical coordinates naturally index the commutative algebra of functions on SL3-character varieties.
We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex algebraic varieties, we show th…
Tropical curves match to special Lagrangian shapes.
problem Connecting tropical geometry to special Lagrangian shapes.
method Gluing construction that matches tropical local models to Lagrangian shapes.
result Locally planar tropical curves can be realized as special Lagrangian limits.
We study cohomologies and Hodge theory for complex manifolds with twisted differentials. In particular, we get another cohomological obstruction for manifolds in class of Fujiki. We give a Hodge-theoretical proof of the characterization of solvmanifolds in class of Fujiki, first proven by D.…
Paper constructs braid invariants using tropical Ptolemy equation.
problem Constructing invariants of braids.
method Tropical version of the Ptolemy equation.
result Invariants of braids constructed.
From string theory, the notion of deformed Hermitian Yang-Mills connections has been introduced by Mariño, Minasian, Moore and Strominger. After that, Leung, Yau and Zaslow proved that it naturally appears as mirror objects of special Lagrangian submanifolds via Fourier-Mukai transform between dual torus fibrations. In…
Develops tropical geometry for weighted Hurwitz numbers, generalizing previous results.
problem Understanding weighted Hurwitz numbers across various cases.
method Tropical geometry framework for weighted Hurwitz numbers.
result Generalized structural results for weighted Hurwitz numbers.
We use Hodge theory and a construction of Merkulov to construct structures on de Rham cohomology and Dolbeault cohomology.
We present a method to develop a Hodge theory for tangential cohomology of foliations by mimicing Witten's approach to ordinary Morse theory by perturbations of the Laplacian
We use recent results by Bainbridge-Chen-Gendron-Grushevsky-Moeller on compactifications of strata of abelian differentials to give a comprehensive solution to the realizability problem for effective tropical canonical divisors in equicharacteristic zero. Given a pair consisting of a stable tropical curve …
This paper illustrates a computational approach to Culler-Morgan-Shalen theory using ideal triangulations, spun-normal surfaces and tropical geometry. Certain affine algebraic sets associated to the Whitehead link complement as well as their logarithmic limit sets are computed. The projective solution space of spun-nor…
Proves Hodge structures on modular functors, providing formulas for Hodge numbers.
problem Existence and uniqueness of Hodge structures on modular functors.
method Non-Abelian Hodge correspondence and Ocneanu rigidity.
result Explicit formulas for Hodge numbers in modular functors of level 2 times an odd number.
We use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. We apply this construction to the tropical curves obtained from the 2875 lines on the quintic Calabi-Yau threefold. Each admissible tropical curve gives a Lagrang…