The geometry of geometries: matroid theory, old and new

DOI10.4171/icm2022/144arXiv2111.08726OpenAlexW4389775055MaRDI QIDQ6118131

Federico Ardila

Publication date: 20 March 2024

Published in: International Congress of Mathematicians (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2111.08726

zbMATH Keywords

submodular functioncombinatorial geometryHopf algebramatroidTutte polynomiallog-concavitygeneralized permutahedrontropical geometrycombinatorial Hodge theoryLefschetz fanpolyhedral valuation

Mathematics Subject Classification ID

Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) (52B40) Combinatorial aspects of tropical varieties (14T15) Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry (14C17) Combinatorial aspects of matroids and geometric lattices (05B35) Dissections and valuations (Hilbert's third problem, etc.) (52B45) Connections of Hopf algebras with combinatorics (16T30)

Cites Work

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• Generalized permutohedra, \(h\)-vectors of cotransversal matroids and pure \(O\)-sequences
• Ehrhart polynomial and arithmetic Tutte polynomial
• Positively oriented matroids are realizable
• Vector partition functions and index of transversally elliptic operators
• Isotropical linear spaces and valuated Delta-matroids
• Convexity and Steinitz's exchange property
• \(K\)-classes for matroids and equivariant localization
• The \(f\)-vector of a representable-matroid complex is log-concave
• On the toric ideals of matroids of a fixed rank
• First steps in tropical intersection theory
• On the geometry of toric arrangements
• Antipodes and incidence coalgebras
• The algebra of continuous piecewise polynomials
• Computing the Tutte polynomial of a hyperplane arragement
• Matroid polytopes and their volumes
• Affine and toric hyperplane arrangements
• A matroid invariant via the \(K\)-theory of the Grassmannian
• The zonotope of a root system
• Combinatorial geometries, convex polyhedra, and Schubert cells
• Combinatorics and topology of complements of hyperplanes
• A unique exchange property for bases
• Valuated matroids
• Chern classes for singular algebraic varieties
• The polytope algebra
• Subspace arrangements over finite fields: Cohomological and enumerative aspects
• Coxeter matroids. With illustrations by Anna Borovik
• Puzzles and (equivariant) cohomology of Grassmannians
• Hodge theory for combinatorial geometries
• Chow rings of toric varieties defined by atomic lattices
• Log-concavity of characteristic polynomials and the Bergman fan of matroids
• Tropical cycles and Chow polytopes
• Valuative invariants for polymatroids
• Tropical flag varieties
• The harmonic polytope
• Coxeter submodular functions and deformations of Coxeter permutahedra
• The positive Bergman complex of an oriented matroid
• The tropical totally positive Grassmannians
• Murphy's law in algebraic geometry: Badly-behaved deformation spaces
• Free Hopf algebras generated by coalgebras
• Characteristic polynomials of subspace arrangements and finite fields
• Intersection theory on toric varieties
• The Bergman complex of a matroid and phylogenetic trees
• The Kazhdan-Lusztig polynomial of a matroid
• Bergman complexes, Coxeter arrangements, and graph associahedra
• Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs
• Algebraic and geometric methods in enumerative combinatorics
• Valuations for Matroid Polytope Subdivisions
• A Tutte polynomial for toric arrangements
• The multivariate arithmetic Tutte polynomial
• On the local linear independence of translates of a box spline
• Compactification of the moduli space of hyperplane arrangements
• Permutohedra, Associahedra, and Beyond
• Tropical Linear Spaces
• Coalgebras and bialgebras in combinatorics
• Facing up to arrangements: face-count formulas for partitions of space by hyperplanes
• Weight Enumeration and the Geometry of Linear Codes
• Why the characteristic polynomial factors
• Chow quotients of Grassmannian I
• Equations Defining Toric Varieties
• Flawlessness ofh-vectors of broken circuit complexes
• Almost all matroids are nonrepresentable
• Enumerative tropical algebraic geometry in ℝ²
• Oriented Matroids
• Chirurgie des grassmanniennes
• Logarithmic limit sets of real semi-algebraic sets
• The motivic zeta functions of a matroid
• The universal valuation of Coxeter matroids
• Chern–Schwartz–MacPherson cycles of matroids
• The Tutte polynomial of symmetric hyperplane arrangements
• COMBINATORIAL APPLICATIONS OF THE HODGE–RIEMANN RELATIONS
• Grothendieck Classes and Chern Classes of Hyperplane Arrangements
• The Arithmetic Tutte Polynomials of the Classical Root Systems
• On dichromatic polynomials
• The Logarithmic Limit-Set of an Algebraic Variety
• A Combinatorial Model for Series-Parallel Networks
• [https://portal.mardi4nfdi.de/wiki/Publication:5731810 On the foundations of combinatorial theory I. Theory of M�bius Functions]
• Numbers of solutions of equations in finite fields
• Divisors on matroids and their volumes