89 citations · 207 across the 18 of their papers we have counts for
27 papers
The parity module of a matroid
Tom Braden, June Huh, Jacob P. Matherne +2
We introduce the parity module of a matroid with coefficients in a field . When has characteristic zero, we show that the parity module of a matroid is iso…
Stellahedral geometry of matroids
Christopher Eur, June Huh, Matt Larson
We use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety, and…
The Bergman fan of a polymatroid
Colin Crowley, June Huh, Matt Larson +2
We introduce the Bergman fan of a polymatroid and prove that the Chow ring of the Bergman fan is isomorphic to the Chow ring of the polymatroid. Using the Bergman fan, we establish…
Lagrangian combinatorics of matroids
Federico Ardila, Graham Denham, June Huh
The Lagrangian geometry of matroids was introduced in [ADH20] through the construction of the conormal fan of a matroid M. We used the conormal fan to give a Lagrangian-geometric i…
Singular Hodge theory for combinatorial geometries
Tom Braden, June Huh, Jacob P. Matherne +2
We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. As applicati…
Lagrangian geometry of matroids
Federico Ardila, Graham Denham, June Huh
We introduce the conormal fan of a matroid M, which is a Lagrangian analog of the Bergman fan of M. We use the conormal fan to give a Lagrangian interpretation of the Chern-Schwart…
A semi-small decomposition of the Chow ring of a matroid
Tom Braden, June Huh, Jacob P. Matherne +2
We give a semi-small orthogonal decomposition of the Chow ring of a matroid M. The decomposition is used to give simple proofs of Poincaré duality, the hard Lefschetz theorem, and…
Logarithmic concavity of Schur and related polynomials
June Huh, Jacob P. Matherne, Karola Mészáros +1
We show that normalized Schur polynomials are strongly log-concave. As a consequence, we obtain Okounkov's log-concavity conjecture for Littlewood-Richardson coefficients in the sp…
Logarithmic concavity for morphisms of matroids
Christopher Eur, June Huh
Morphisms of matroids are combinatorial abstractions of linear maps and graph homomorphisms. We introduce the notion of basis for morphisms of matroids, and show that its generatin…
Lorentzian polynomials
Petter Brändén, June Huh
We study the class of Lorentzian polynomials. The class contains homogeneous stable polynomials as well as volume polynomials of convex bodies and projective varieties. We prove th…
Hodge-Riemann relations for Potts model partition functions
Petter Brändén, June Huh
We prove that the Hessians of nonzero partial derivatives of the (homogenous) multivariate Tutte polynomial of any matroid have exactly one positive eigenvalue on the positive orth…
Correlation bounds for fields and matroids
June Huh, Benjamin Schröter, Botong Wang
Let be a finite connected graph, and let be a spanning tree of chosen uniformly at random. The work of Kirchhoff on electrical networks can be used to show that the eve…
Combinatorial applications of the Hodge-Riemann relations
June Huh
Why do natural and interesting sequences often turn out to be log-concave? We give one of many possible explanations, from the viewpoint of "standard conjectures". We illustrate wi…
Lefschetz classes on projective varieties
June Huh, Botong Wang
The Lefschetz algebra of a smooth complex projective variety is the subalgebra of the cohomology algebra of generated by divisor classes. We construct smooth compl…
Enumeration of points, lines, planes, etc
June Huh, Botong Wang
One of the earliest results in enumerative combinatorial geometry is the following theorem of de Bruijn and Erdős: Every set of points in a projective plane determines at least…
Hodge Theory for Combinatorial Geometries
Karim Adiprasito, June Huh, Eric Katz
We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative ring associated to an arbitrary matroid M. We use the Hodge-Riemann relations to resolve a con…
A tropical approach to a generalized Hodge conjecture for positive currents
Farhad Babaee, June Huh
Demailly showed that the Hodge conjecture is equivalent to the statement that any (p,p)-dimensional closed current with rational cohomology class can be approximated by linear comb…
A counterexample to the geometric Chevalley-Warning conjecture
June Huh
We construct a quartic threefold with L-rational singularities which has torsion in its middle homology group. This answers a question of Brown and Schnetz for all fields of charac…
Likelihood Geometry
June Huh, Bernd Sturmfels
We study the critical points of monomial functions over an algebraic subset of the probability simplex. The number of critical points on the Zariski closure is a topological invari…
Correspondences between projective planes
June Huh
We characterize integral homology classes of the product of two projective planes which are representable by a subvariety.
Positivity of Chern classes of Schubert cells and varieties
June Huh
We show that the Chern-Schwartz-MacPherson class of a Schubert cell in a Grassmannian is represented by a reduced and irreducible subvariety in each degree. This gives an affirmati…
Varieties with maximum likelihood degree one
June Huh
We show that algebraic varieties with maximum likelihood degree one are exactly the images of reduced A-discriminantal varieties under monomial maps with finite fibers. The maximum…
Milnor numbers of projective hypersurfaces with isolated singularities
June Huh
Let V be a projective hypersurface of fixed degree and dimension which has only isolated singular points. We show that, if the sum of the Milnor numbers at the singular points of V…
The maximum likelihood degree of a very affine variety
June Huh
We show that the maximum likelihood degree of a smooth very affine variety is equal to the signed topological Euler characteristic. This generalizes Orlik and Terao's solution to V…