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Organizing Committee
San Francisco State University
Harvard University
University of St Andrews
University of Texas at Austin
Harvard University
Abstract
The workshop will revolve around the interplay between algebraic geometry and combinatorial structures such as graphs, polytopes, and polyhedral complexes. In particular, the workshop will foster dialogue among groups of researchers who use similar combinatorial geometric tools for different purposes within algebraic geometry and adjacent fields. The topics covered will include Newton-Okounkov bodies, Ehrhart theory, toric geometry, tropical geometry, matroids, and interactions with mirror symmetry.
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Workshop Schedule
Monday, April 12, 2021
9:45 - 10:00 AM EDT
Welcome
Virtual
Brendan Hassett, ICERM/Brown University
10:00 - 10:45 AM EDT
The Foundation of a Matroid
Virtual
Speaker
Matthew Baker, Georgia Institute of Technology
Session Chair
Lauren Williams, Harvard University
Abstract
Matroid theorists are typically interested in questions concerning representability of matroids over fields. More generally, one can ask about representability over partial fields in the sense of Semple and Whittle. Pendavingh and van Zwam introduced the universal partial field of a matroid, which governs the representations of over all partial fields. Unfortunately, almost all matroids are not representable over any partial field, and in this case, the universal partial field gives no information.
Oliver Lorscheid and I have introduced a generalization of the universal partial field which we call the foundation of a matroid. The foundation of is a type of algebraic object which we call a pasture; pastures include both hyperfields and partial fields. As a particular application of this point of view, I will explain the classification which Lorscheid and I have recently obtained of all possible foundations for matroids having no minor isomorphic to U(2,5) or U(3,5). Among other things, our classification provides a short conceptual proof of the 1997 theorem of Lee and Scobee which says that a matroid is both ternary and orientable if and only if it is dyadic.
11:00 - 11:15 AM EDT
Break
Coffee Break
11:15 AM - 12:00 PM EDT
Lagrangian geometry of matroids
Virtual
Speaker
Graham Denham, University of Western Ontario
Session Chair
Lauren Williams, Harvard University
Abstract
In joint work with Federico Ardila and June Huh, we introduce the conormal fan of a matroid, which is the Lagrangian analogue of the Bergman fan. We use it to give a Lagrangian interpretation of the Chern-Schwartz-MacPherson cycle of a matroid. We develop tools for tropical Hodge theory to show that the conormal fan satisfies Poincaré duality, the Hard Lefschetz property, and the Hodge--Riemann relations. Together, these imply conjectures of Brylawski and Dawson about the log-concavity of the h-vectors of the broken circuit complex and independence complex of a matroid.
12:00 - 1:30 PM EDT
Lunch/Free Time
1:30 - 2:15 PM EDT
Tautological classes of matroids
Virtual
Speaker
Christopher Eur, Stanford University
Session Chair
Sam Payne, University of Texas at Austin
Abstract
We introduce certain torus-equivariant classes on permutohedral varieties which we call ``tautological classes of matroids'' as a new geometric framework for studying matroids. Using this framework, we unify and extend many recent developments in matroid theory arising from its interaction with algebraic geometry. We achieve this by establishing a Chow-theoretic description and a log-concavity property for a 4-variable transformation of the Tutte polynomial, and by establishing an exceptional Hirzebruch-Riemann-Roch-type formula for permutohedral varieties that translates between K-theory and Chow theory. This is joint work with Andrew Berget, Hunter Spink, and Dennis Tseng.
2:30 - 2:45 PM EDT
Break
Coffee Break
2:45 - 3:30 PM EDT
Kazhdan-Lusztig theory and singular Hodge theory for matroids
Virtual
Speaker
June Huh, Stanford University
Session Chair
Sam Payne, University of Texas at Austin
Abstract
There is a remarkable parallel between the theory of Coxeter groups (think of the symmetric group or the dihedral group) and matroids (think of your favorite graph or point configuration) from the perspective of combinatorial cohomology theories. I will give an overview of the similarity and report on recent my joint work with Tom Braden, Jacob Matherne, Nick Proudfoot, and Botong Wang on singular Hodge theory for combinatorial geometries: https://arxiv.org/abs/2010.06088
3:45 - 4:45 PM EDT
Gathertown Reception
Reception - Virtual
Tuesday, April 13, 2021
9:00 - 9:45 AM EDT
Gathertown Morning Coffee
Coffee Break - Virtual
10:00 - 10:45 AM EDT
Real phase structures on matroid fans
Virtual
Speaker
Kris Shaw, University of Oslo
Session Chair
Melody Chan, Brown University
Abstract
In this talk, I will propose a definition of real phase structures on
polyhedral complexes. I’ll explain that in the case of matroid fans, specifying a real phase structure is cryptomorphic to providing an orientation of the underlying matroid.
Then I’ll define the real part of a polyhedral complex with a real phase
structure. This determines a closed chain in the real part of a toric variety. In the case when the polyhedral complex is a non-singular tropical variety, the real part is a PL-manifold. Moreover, for a non-singular tropical variety with a real phase structures we can apply the same spectral sequence for tropical hypersurfaces, obtained by Renaudineau and myself, to bound the Betti numbers of the real part by the dimensions of the tropical homology groups.
This is joint work in progress with Johannes Rau and Arthur Renaudineau.
11:00 - 11:15 AM EDT
Break
Coffee Break
