VIRTUAL ONLY: Algebraic Geometry and Polyhedra
Institute for Computational and Experimental Research in Mathematics (ICERM)
April 12, 2021  April 16, 2021
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Monday, April 12, 2021

9:45  10:00 am EDTWelcomeVirtual
 Brendan Hassett, ICERM/Brown University

10:00  10:45 am EDTThe Foundation of a MatroidVirtual
 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 EDTBreakCoffee Break

11:15 am  12:00 pm EDTLagrangian geometry of matroidsVirtual
 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 ChernSchwartzMacPherson 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 HodgeRiemann relations. Together, these imply conjectures of Brylawski and Dawson about the logconcavity of the hvectors of the broken circuit complex and independence complex of a matroid.

12:00  1:30 pm EDTLunch/Free Time

1:30  2:15 pm EDTTautological classes of matroidsVirtual
 Speaker
 Christopher Eur, Stanford University
 Session Chair
 Sam Payne, University of Texas at Austin
Abstract
We introduce certain torusequivariant 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 Chowtheoretic description and a logconcavity property for a 4variable transformation of the Tutte polynomial, and by establishing an exceptional HirzebruchRiemannRochtype formula for permutohedral varieties that translates between Ktheory and Chow theory. This is joint work with Andrew Berget, Hunter Spink, and Dennis Tseng.

2:30  2:45 pm EDTBreakCoffee Break

2:45  3:30 pm EDTKazhdanLusztig theory and singular Hodge theory for matroidsVirtual
 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 EDTGathertown ReceptionReception  Virtual
Tuesday, April 13, 2021

9:00  9:45 am EDTGathertown Morning CoffeeCoffee Break  Virtual

10:00  10:45 am EDTReal phase structures on matroid fansVirtual
 Speaker
 Kristin 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 nonsingular tropical variety, the real part is a PLmanifold. Moreover, for a nonsingular 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 EDTBreakCoffee Break

11:15 am  12:00 pm EDTCombinatorics and real lifts of bitangents to tropical quartic curvesVirtual
 Speaker
 María Angélica Cueto, Ohio State University
 Session Chair
 Melody Chan, Brown University
Abstract
Smooth algebraic plane quartics over algebraically closed fields have 28 bitangent lines. By contrast, their tropical counterparts have infinitely many bitangents. They are grouped into seven equivalence classes, one for each linear system associated to an effective tropical theta characteristic on the tropical quartic curve. In this talk, I will discuss recent work joint with Hannah Markwig (arXiv:2004.10891) on the combinatorics of these bitangent classes and its connection to the number of real bitangents to real smooth quartic curves characterized by Pluecker. We will see that they are tropically convex sets and they come in 41 symmetry classes. The classical bitangents map to specific vertices of these polyhedral complexes, and each tropical bitangent class captures four of the 28 bitangents. We will discuss the situation over the reals and show that each tropical bitangent class has either zero or four lifts to classical bitangent defined over the reals, in agreement with Pluecker's classification.

12:00  1:30 pm EDTLunch/Free Time

1:30  2:15 pm EDTTropical psi classesVirtual
 Speaker
 Renzo Cavalieri, Colorado State University
 Session Chair
 Dhruv Ranganathan, University of Cambridge
Abstract
We introduce a tropical geometric framework that allows us to define $\psi$ classes for moduli spaces of tropical curves of arbitrary genus. We prove correspondence theorems between algebraic and tropical $\psi$ classes for some onedimensional families of genusone tropical curves.

2:30  2:45 pm EDTBreakCoffee Break

2:45  3:30 pm EDTWhen are multidegrees positive?Virtual
 Speaker
 Federico Castillo, University of Kansas
 Session Chair
 Dhruv Ranganathan, University of Cambridge
Abstract
The notion of multidegree for multiprojective varieties extends that of degree for projective varieties. They can be defined in geometric terms, using intersection theory, or alternatively in algebraic terms, via multigraded hilbert polynomial. We study the problem of their positivity and establish a combinatorial description using polyhedral geometry. We will show applications for Schubert polynomials and mixed volumes. This is joint work with Y.CidRuiz, B.Li, J.Montano, and N.Zhang.
Wednesday, April 14, 2021

9:00  9:45 am EDTGathertown Morning CoffeeCoffee Break  Virtual

10:00  10:45 am EDTWallcrossing phenomenon for NewtonOkounkov bodiesVirtual
 Speaker
 Laura Escobar, Washington University St. Louis
 Session Chair
 Lara Bossinger, Mathematics Institute UNAM, Oaxaca
Abstract
A NewtonOkounkov body is a convex set associated to a projective variety, equipped with a valuation. These bodies generalize the theory of Newton polytopes and the correspondence between polytopes and projective toric varieties. Work of KavehManon gives an explicit link between tropical geometry and NewtonOkounkov bodies. We use this link to describe a wallcrossing phenomenon for NewtonOkounkov bodies. As an example, we describe wallcrossing formula in the case of the Grassmannian Gr(2,m). This is joint work with Megumi Harada.

11:00  11:15 am EDTBreakCoffee Break  Virtual

11:15 am  12:00 pm EDTOn combinatorics of Arthur's trace formula, convex polytopes, and toric varietiesVirtual
 Speaker
 Kiumars Kaveh, University of Pittsburgh
 Session Chair
 Lara Bossinger, Mathematics Institute UNAM, Oaxaca
Abstract
I start by discussing two beautiful wellknown theorems about decomposing a convex polytope into an signed sum of cones, namely the classical BrianchonGram theorem and LawrenceVarchenko theorem. I will then explain a generalization of the BrianchonGram which can be summerized as ""truncating a function on the Euclidean space with respect to a polytope"". This is an extraction of the combinatorial ingredients of Arthur's ''convergence'' and ''polynomiality'' results in his famous trace formula. Arthur's trace formula concerns the trace of left action of a reductive group $G$ on the space $L^2(G / \Gamma)$ where $\Gamma$ is a discrete (arithmetic) subgroup. The combinatorics involved is closely related to compactifications of ''locally summetric spaces'' (which btw are hyperbolic manifolds). Our ''combinatorial truncation'' can be thought of as an analogue of Arthur's truncation over a toric variety (in place of a compactification of a locally symmetric space). If there is time, I will briefly sketch geometric interpretations of our combinatorial truncation as a measure and a Lefschetz number on a toric variety respectively. This is a joint work in progress with Mahdi Asgari (Oklahoma State).

12:00  1:30 pm EDTLunch/Free Time

1:30  2:30 pm EDTPoster SessionVirtual
Abstract
Ideal Preserving Operations on Chemical Reaction Networks
Mark Curiel, University of Hawaii at Manoa
Under the assumption of mass action kinetics, the associated dynamical system of a reaction network is polynomial. We consider the ideals generated by these polynomials, which are called steadystate ideals. Steadystate ideals appear in multiple contexts within the chemical reaction network literature, however they have yet to be systematically studied. To begin such a study, we ask and partially answer the following question: when do two reaction networks give rise to the same steadystate ideal? In particular, our main results describe three operations on the reaction graph that preserve the steadystate ideal. Furthermore, since the motivation for this work is the classification of steadystate ideals, monomials play a primary role. To this end, combinatorial conditions are given to identify monomials in a steadystate ideal, and we give a sufficient condition for a steadystate ideal to be monomial.
Construction and properties of Kanev surfaces in toric 3folds
Julius Giesler, University of Tuebingen
In this poster Kanev surfaces, which are surfaces of general type, are considered, that arise as nondegenerate hypersurfaces in toric 3folds. First such an hypersurface might have singularities but we show how to construct a minimal and a canonical model with toric methods. After this construction we consider nondegenerate hypersurfaces with fixed Newton polytope, thus obtaining a family of Kanev surfaces, and we both compute their number of moduli and check whether the infinitesimal Torelli theorem holds for such a family. The results constitute part of the author's doctoral thesis.
On the shifted LittlewoodRichardson coefficients and the LittlewoodRichardson coefficients
Nguyen Khanh, Institut Camille Jordan
We give a new interpretation of the shifted LittlewoodRichardson coefficients $f_{\lambda\mu}^\nu$ ($\lambda,\mu,\nu$ are strict partitions). The coefficients $g_{\lambda\mu}$ which appear in the decomposition of Schur $Q$function $Q_\lambda$ into the sum of Schur functions $Q_\lambda = 2^{l(\lambda)}\sum\limits_{\mu}g_{\lambda\mu}s_\mu$ can be considered as a special case of $f_{\lambda\mu}^\nu$ (here $\lambda$ is a strict partition of length $l(\lambda)$). We also give another description for $g_{\lambda\mu}$ as the cardinal of a subset of a set that counts LittlewoodRichardson coefficients $c_{\mu^t\mu}^{\tilde{\lambda}}$. This new point of view allows us to establish connections between $g_{\lambda\mu}$ and $c_{\mu^t \mu}^{\tilde{\lambda}}$. More precisely, we prove that $g_{\lambda\mu}=g_{\lambda\mu^t}$, and $g_{\lambda\mu} \leq c_{\mu^t\mu}^{\tilde{\lambda}}$. We conjecture that $g_{\lambda\mu}^2 \leq c^{\tilde{\lambda}}_{\mu^t\mu}$ and formulate some conjectures on our combinatorial models which would imply this inequality if it is valid.
Computational complexity, Newton polytopes, and Schubert polynomials
Colleen Robichaux, University of Illinois at UrbanaChampaign
Schubert polynomials form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. The nonvanishing problem asks if a coefficient of a Schubert polynomial is nonzero. We give a tableau criterion for nonvanishing, from which we deduce the first polynomial time algorithm. These results are obtained from new characterizations of the Schubitope, a generalization of the permutahedron defined for any subset of the nxn grid. This is joint work with Anshul Adve and Alexander Yong.
Enumeration of algebraic and tropical singular hypersurfaces
Uriel Sinichkin, Tel Aviv University
We develop a version of Mikhalkin's lattice path algorithm for projective hypersurfaces of arbitrary degree and dimension, which enumerates singular tropical hypersurfaces passing through appropriate configuration of points. By proving a correspondence theorem combined with the lattice path algorithm, we construct a $ \delta $ dimensional linear space of degree $ d $ real hypersurfaces containing $ \frac{1}{\delta!}(\gamma_nd^n)^{\delta}+O(d^{n\delta1}) $ hypersurfaces with $ \delta $ real nodes, where $ \gamma_n $ are positive and given by a recursive formula.
This is asymptotically comparable to the number $ \frac{1}{\delta!} \left( (n+1)(d1)^n \right)^{\delta}+O\left(d^{n(\delta1)} \right) $ of complex hypersurfaces having $ \delta $ nodes in a $ \delta $ dimensional linear space. In the case $ \delta=1 $ we give a slightly better leading term. 
2:30  3:15 pm EDTOn NewtonOkounkov bodies associated to GrassmanniansVirtual
 Speaker
 Alfredo Nájera Chávez, Mathematics Institute UNAM, Oaxaca
 Session Chair
 Travis Mandel, University of Oklahoma
Abstract
In this talk I will elaborate on a certain class of NewtonOkounkov bodies that one can associate to "nice" compactifications of cluster varieties. In particular, I will explain how this approach recovers RietschWilliams' construction of NewtonOkounkov bodies for Grassmannians. In order to make the precise connection it will be necessary to explain how the MarshRietsch potential and the GrossHackingKeelKontsevich potential for Grassmannians are related. Finally, I will draw some consequences from this relation such as an isomorphism of the toric degenerations obtained by RietschWillimas and the toric degenerations obtained by the celebrated "principal coefficient" construction. Time permitting, I will briefly elaborate on the interpretation of these results from the viewpoint of the representation theory of the associated dimer algebra.

3:30  3:45 pm EDTBreakCoffee Break

3:45  4:30 pm EDTBroken line convexityVirtual
 Speaker
 Timothy Magee, University of Birmingham
 Session Chair
 Daping Weng, Michigan State University
Abstract
In this talk, I'll give an overview of how convex polytopes generalizes from the toric world to the cluster world, where the "polytopes" live in a tropical space rather than a vector space. In this setting, "broken line convex polytopes" define projective compactifications of cluster varieties. After this overview, I'll focus on two exciting applications of this more general notion of convexity: 1) an intrinsic version of NewtonOkounkov bodies and 2) a possible cluster version of a classic toric mirror symmetry construction due to Batyrev. The overview is based on joint work with Mandy Cheung and Alfredo Nájera Chávez, and the applications are based on ongoing joints works with Mandy, Alfredo, Lara Bossinger, and Bosco Frías Medina.
Thursday, April 15, 2021

9:00  9:45 am EDTGathertown Morning CoffeeCoffee Break  Virtual

10:00  10:45 am EDTToric vector bundles  an overviewVirtual
 Speaker
 Milena Hering, The University of Edinburgh
 Session Chair
 Linda Chen, Swarthmore College
Abstract
I will give a brief introduction to toric vector bundles, an overview of what we know about them so far, and explain some more recent developments on the defining equations of embeddings of their projectivisations

11:00  11:15 am EDTBreakCoffee Break

11:15 am  12:00 pm EDTThe FultonMacPherson compactification is not a Mori dream spaceVirtual
 Speaker
 José González, University of California, Riverside
 Session Chair
 Linda Chen, Swarthmore College
Abstract
We show that the FultonMacPherson compactification of the configuration space of n distinct labeled points in certain varieties of arbitrary dimension d, including projective space, is not a Mori dream space for n greater than or equal to d+9.

12:00  1:30 pm EDTLunch/Free Time

1:30  2:15 pm EDTSchubert polynomials from a polytopal point of viewVirtual
 Speaker
 Karola Meszaros, Cornell University
 Session Chair
 Laura Escobar, Washington University St. Louis
Abstract
Schubert polynomials are multivariate polynomials representing cohomology classes on the flag manifold. Despite the beautiful formulas developed for them over the past three decades, the coefficients of these polynomials remained mysterious. I will explain Schubert polynomials from a polytopal point of view, answering, at least partially, the questions: Which coefficients are nonzero? How do the coefficients compare to each other in size? Are the Newton polytopes of these polynomials saturated? Are their coefficients logconcave along lines? Is there a polytope whose integer point transform specializes to Schubert polynomials? As the questions themselves suggest, we will find that polytopes play an outsized role in our understanding. The talk is based on joint works with Alex Fink, June Huh, Ricky Liu, Jacob Matherne and Avery St. Dizier.

2:30  2:45 pm EDTBreakCoffee Break

2:45  3:30 pm EDTFamilies of Gröbner degenerationsVirtual
 Speaker
 Lara Bossinger, Mathematics Institute UNAM, Oaxaca
 Session Chair
 Laura Escobar, Washington University St. Louis
Abstract
Let V be the weighted projective variety defined by a weighted homogeneous ideal J and C a maximal cone in the Gröbner fan of J with m rays. We construct a flat family over affine mspace that assembles the Gröbner degenerations of V associated with all faces of C. This is a multiparameter generalization of the classical oneparameter Gröbner degeneration associated to a weight. We explain how our family can be constructed from KavehManon's recent work on the classification of toric flat families over toric varieties: it is the pullback of a toric family defined by a Rees algebra with base the toric variety associated to cone C along its universal torsor. We apply this construction to the Grassmannians of planes with their Plücker embeddings and the Grassmannian Gr(3,6) with its cluster embedding. In each case there exists a unique maximal Gröbner cone whose associated initial ideal is the StanleyReisner ideal of the cluster complex. We show that the corresponding cluster algebra with universal coefficients arises as the algebra defining the flat family associated to this cone. This talk is based on joint work with F. Mohammadi and A. Nájera Chávez, arxiv:2007.14972.
Friday, April 16, 2021

9:00  9:45 am EDTGathertown Morning CoffeeCoffee Break  Virtual

10:00  10:45 am EDTThe logarithmic Hilbert scheme of curvesVirtual
 Speaker
 Dhruv Ranganathan, University of Cambridge
 Session Chair
 Federico Ardila, San Francisco State University
Abstract
Within the Hilbert scheme of curves in projective space is a subscheme of curves that are "tropical" in the sense of Tevelev: they interact well with the coordinate subspaces. I will explain why, from the point of view of tropical and logarithmic geometry, this locus ought to be the principal open cell in another moduli space, of which the Hilbert scheme is only an approximation. This "logarithmic Hilbert scheme" was recently constructed in work with Davesh Maulik (MIT) and is the core of a new theory of logarithmic DonaldsonThomas invariants. The story touches another major character in the story of polyhedral and algebraic geometry: the secondary polytope of Gel'fandKapranovZelevinsky. I'll try to give some sense for why.

11:00  11:15 am EDTBreakCoffee Break

11:15 am  12:00 pm EDTInitial degenerations of GrassmanniansVirtual
 Speaker
 Daniel Corey, University of Wisconsin, Madison
 Session Chair
 Federico Ardila, San Francisco State University
Abstract
We construct closed immersions from initial degenerations of Gr_0(d,n)the open cell in the Grassmannian Gr(d,n) given by the nonvanishing of all Plücker coordinatesto limits of thin Schubert cells associated to diagrams induced by the face poset of the corresponding tropical linear space. These are isomorphisms in many cases, including (d,n) equal to (2,n), (3,6) and (3,7). As an application, Gr_0(3,7) is schön, and the Chow quotient of Gr(3,7) by the maximal torus in PGL(7) is the log canonical compactification of the moduli space of 7 points in P^2 in linear general position, making progress on a conjecture of Hacking, Keel, and Tevelev. Time permitting, I will discuss recent work on extending these results to the Lietype D setting.

12:00  1:30 pm EDTLunch/Free Time

1:30  2:15 pm EDTOn the topweight rational cohomology of A_gVirtual
 Speaker
 Melody Chan, Brown University
 Session Chair
 Yoav Len, University of St Andrews
Abstract
I'll report on recent work using tropical techniques to find new rational cohomology classes in moduli spaces A_g of abelian varieties, building on previous joint work with Galatius and Payne on M_g. Joint work with Madeline Brandt, Juliette Bruce, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.

2:30  2:45 pm EDTBreakCoffee Break

2:45  3:30 pm EDTTropical Flag VarietiesVirtual
 Speaker
 Madeline Brandt, Brown University
 Session Chair
 Yoav Len, University of St Andrews
Abstract
Flag matroids are combinatorial abstractions of flags of linear subspaces, just as matroids are of linear subspaces. We introduce the flag Dressian as a tropical analogue of the partial flag variety, and give a correspondence between: (a) points on the flag Dressian, (b) valuated flag matroids, (c) flags of projective tropical linear spaces, and (d) coherent flag matroidal subdivisions of the flag matroid polytope. The ideas presented in this talk will be brought to life through examples.
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