Organizing Committee
 Federico Ardila
San Francisco State University  ManWai Cheung
Harvard University  Yoav Len
University of St Andrews  Sam Payne
University of Texas at Austin  Lauren Williams
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 NewtonOkounkov bodies, Ehrhart theory, toric geometry, tropical geometry, matroids, and interactions with mirror symmetry.
Confirmed Speakers & Participants
Talks will be presented virtually or inperson as indicated in the schedule below.
 Speaker
 Poster Presenter
 Attendee
 Virtual Attendee

Dan Abramovich
Brown University

Ashleigh Adams
UC Davis

Adam Afandi
Colorado State University

Rizal Afgani
Institut Teknologi Bandung

Tair Akhmejanov
University of CaliforniaDavis

Edvard Aksnes
University of Oslo

Doriann Albertin
Université Gustave Eiffel

Elie Alhajjar
US Military Academy

Ayah Almousa
Cornell University

David Anderson
Ohio State University

Nicholas Anderson
Queen Mary University of London

Federico Ardila
San Francisco State University

Muhammad Ardiyansyah
Aalto University

Xerxes Arsiwalla
Pompeu Fabra University & Wolfram Research

Ahmed Umer Ashraf
University of Western Ontario

Matthew Baker
Georgia Institute of Technology

Gonzalo Barranco Mendoza
Universidad Nacional Autonoma de Mexico

Victor Batyrev
Universität Tübingnen

Marko Berghoff
University of Oxford

Shivam Bhatt
University of Toronto

Christin Bibby
Louisiana State University

Sara Billey
University of Washington

Simone Billi
University of Bologna

Aram Bingham
Tulane University

Alex Black
UC Davis

Narasimha Chary Bonala
Ruhr University of Bochum

Lara Bossinger
Mathematics Institute UNAM, Oaxaca

Tom Braden
University of Massachusetts Amherst

Madeline Brandt
Brown University

Paul Breiding
Technische Universität Berlin

Michel Brion
Université Grenoble Alpes

Juliette Bruce
University of California, Berkeley / MSRI

Anders Buch
Rutgers University

Amanda Burcroff
Durham University

Mahir Bilen Can
Tulane University

Federico Castillo
University of Kansas

Eduardo Cattani
University of Massachusetts Amherst

Renzo Cavalieri
Colorado State University

Ian Cavey
The Ohio State University

Seth Chaiken
University at Albany

Melody Chan
Brown University

Rajarshi Chatterjee
Kazi Nazrul University

Anastasia Chavez
University of California, Davis

Enrique Chávez Martínez
Universidad Nacional Autónoma de México

Linda Chen
Swarthmore College

Sunita Chepuri
University of Michigan

ManWai Cheung
Harvard University

Daniel Corey
University of Wisconsin, Madison

Alvaro Cornejo
San Francisco State University

Ethan Cotterill
Federal Fluminense University

María Angélica Cueto
Ohio State University

Mark Curiel
University of Hawaii at Manoa

Xinle Dai
Harvard University

Colin Defant
Princeton University

Graham Denham
University of Western Ontario

Papri Dey
University of Missouri

Galen DorpalenBarry
University of Minnesota

Theo Douvropoulos
University of Massachusetts, Amherst

Anne Dranowski
Institute for Advanced Study

Nicholas Early
Institute for Advanced Study

Richard Ehrenborg
University of Kentucky

Laura Escobar
Washington University St. Louis

Christopher Eur
Stanford University

Eleonore Faber
University of Leeds

Lorenzo Fantini
Goethe University Frankfurt

Matthew Faust
Texas A&M University

Elisenda Feliu
University of Copenhagen

Luis Ferroni
University of Bologna

Tara Fife
Max Planck Institute for mathematics in the Sciences
 Alex Fink

Sergey Fomin
University of Michigan

Chris Fraser
University of Minnesota

Netanel Friedenberg
Yale University

William Fulton
University of Michigan

Nir Gadish
MIT

Christian Gaetz
MIT

Pavel Galashin
University of California, Los Angeles

Julius Giesler
University of Tuebingen

Andrey Glubokov
Ave Maria University

Rebecca Goldin
George Mason University

Sergio Gómez
CINVESTAV

José González
University of California, Riverside

Vassily Gorbounov
Higher School of Economics, Russia

Brent Gorbutt
George Mason University

Elisa Gorla
University of Neuchatel

Eugene Gorsky
UC Davis

Sean Griffin
Brown University

Sarah Griffith
Brown University

Samuel Grushevsky
Stony Brook University

Aziz Burak Guelen
The Ohio State University

Trevor Gunn
Georgia Institute of Technology

Ajeeth Gunna
The University of Melbourne

Christian Haase
Freie Universität Berlin

Iva Halacheva
Northeastern University

Kangjin Han
DaeguGyeongbuk Institute of Sciences and Technology (DGIST)

Megumi Harada
MCMASTER UNIVERSITY

Pamela E. Harris
Williams College

Paul Helminck
Swansea University

Oskar Henriksson
University of Copenhagen

Milena Hering
The University of Edinburgh

María Herrero
University of Buenos Aires  CONICET

Neha Hooda
Fairfield University

Serkan Hosten
San Francisco State University

Daoji Huang
Brown University

Yifeng Huang
University of Michigan

June Huh
Stanford University

Brian Hwang
Cornell University

Anthony Iarrobino
Northeastern University

Muhammad Imran
United Arab Emirates University

Bogdan Ion
University of Pittsburgh

Shashank Jaiswal
Purdue University

Delio Jaramillo
CinvestavIPN (MEXICO)

Manoel Jarra
IMPA

David Jensen
University of Kentucky

Minyoung Jeon
The Ohio State University

Shuai Jiang
Virginia Tech

Tong Jin
Georgia Institute of Technology

Michael Joswig
TU Berlin & MPI Leipzig

Nidhi Kaihnsa
Brown University

Siddarth Kannan
Brown University

Lars Kastner
Institute of Mathematics of the Technical University

Syu Kato
Kyoto University

Eric Katz
The Ohio State University

Kiumars Kaveh
University of Pittsburgh

Elizabeth Kelley
University of MInnesota

Gary Kennedy
Ohio State University

Nguyen Khanh
Institut Camille Jordan

Young Rock Kim
Hankuk University of Foreign Studies

Shinyoung Kim
Institute for Basic Science Center for Geometry and Physics

Patricia Klein
University of Minnesota

Allen Knutson
Cornell University

Kathlén Kohn
KTH Royal Institute of Technology

Aleksandr Kolpakov
University of Neuchâtel

Jakub Koncki
Institute of Mathematics, Polish Academy of Sciences

Lukas Kühne
Max Planck Institute for Mathematics in the Sciences

Thomas Lam
University of Michigan

Matthew Larson
Stanford University

Cédric Le Texier
Universitetet i Oslo

Eunjeong Lee
IBSCGP

Yoav Len
University of St Andrews

Shiyue Li
Brown University

YenKheng Lim
Xiamen University Malaysia

David LowryDuda
ICERM & Brown University

Antonio Macchia
Freie Universität Berlin

Diane Maclagan
University of Warwick

Timothy Magee
University of Birmingham

Saikat Maity
University of Calcutta

Travis Mandel
University of Oklahoma

Madhusudan Manjunath
INDIAN INSTITUTE OF TECHNOLOGY BOMBAY

Hannah Markwig
Eberhard Karls University of Tübingen

Mikhail Mazin
Kansas State University

Jason McCullough
Iowa State University

Alex McDonough
Brown University

Jodi McWhirter
Washington University in St. Louis

Karola Meszaros
Cornell University

Leonardo Mihalcea
Virginia Polytechnic Institute and State University

Elizabeth Milićević
Haverford College

Kalina Mincheva
Tulane University

Ali Mohammad Nezhad
Purdue University

Fatemeh Mohammadi
Ghent University

Jonathan Montaño
New Mexico State University

Héctor Montiel
UNAM

Alejandro Morales
University of Massachusetts, Amherst

Sophie Morel
CNRS/ENS Lyon

Ralph Morrison
Williams College

Jennifer Morse
University of Virginia

Tommi Muller
University Of British Columbia

Reshma Munbodh
Alpert Medical School of Brown University

Philippe Nadeau
Institut Camille Jordan

Alfredo Nájera Chávez
Mathematics Institute UNAM, Oaxaca

Hiroshi Naruse
University of Yamanashi

Benjamin Nativi
The University of Texas at Austin

Gleb Nenashev
Brown University

Benjamin Nill
University of Magdeburg

Mounir Nisse
Xiamen University Malaysia

Damián Ochoa
Universidad Nacional Autónoma de México

Jianping Pan
University of California, Davis

Theodoros Stylianos Papazachariou
University of Essex

Matteo Parisi
University of Oxford

Beatriz Pascual Escudero
Universidad Carlos III de Madrid

Sam Payne
University of Texas at Austin

Nicolas Perrin
Versailles SaintQuentinenYvelines University

Nathan Pflueger
Amherst College

Irem Portakal
OttovonGuerickeUniversitaet Magdeburg

Zhijun (George) Qiao
University of Texas Rio Grande Valley

Jenna Rajchgot
McMaster University

Rohini Ramadas
Brown University

Alondra Ramírez Sandoval
Universidad Nacional Autónoma de México

Dhruv Ranganathan
University of Cambridge

Margaret Readdy
University of Kentucky

Yue Ren
Swansea University

William Reynolds
University of Edinburgh

Harry Richman
University of Washington

Konstanze Rietsch
King's College London

Colleen Robichaux
University of Illinois at UrbanaChampaign

Alejandro Rodriguez Matta
Universidad Nacional Autónoma de México

Felix Röhrle
Goethe University Frankfurt

Felix Rydell
KTH

María Sabando Alvarez
Washington University in St. Louis

Francisco Santos
University of Cantabria

Mahrud Sayrafi
University of Minnesota

Hal Schenck
Auburn University

Bernd Schober
Carl von Ossietzky Universität Oldenburg

Nolan Schock
University of Georgia

Alexandra Seceleanu
University of NebraskaLincoln

Oswaldo Sevilla
Centro de Investigación en Matemáticas, Guanajuato Mexico

Aniket Shah
The Ohio State University

Kris Shaw
University of Oslo

Melissa ShermanBennett
UC Berkeley/Harvard

Farbod Shokrieh
University of Washington

Kevin Shu
Georgia Institute of Technology

Eugenii Shustin
Tel Aviv University

Sergio Silva
Universidad Distrital Francisco Jose de Caldas

Rob Silversmith
Northeastern University

Connor Simpson
University of Wisconsin  Madison

Uriel Sinichkin
Tel Aviv University

Gregory Smith
Queen's University

Ivan So
Michigan State University

MirunaStefana Sorea
SISSA, Trieste

Frank Sottile
Texas A&M University

Pedro Souza
Goethe Universität Frankfurt am Main

David Speyer
University of Michigan

Avery St. Dizier
Univeristy of Illinois at UrbanaChampaign

Bernd Sturmfels
MPI Leipzig

Changjian Su
University of Toronto

Yuri Sulyma
Brown University

Mariel Supina
University of California, Berkeley

Anna Tao
Brown University

Nicola Tarasca
Virginia Commonwealth University

Máté Telek
University of Copenhagen

Ayush Tewari
TU Berlin

Vasu Tewari
University of Hawaii

Joaquin Torres Henestroza
Universidad Nacional Autónoma de México, UNAM

Trang Tran
Ucsc

Ling Hei Tsang
The Ohio State University

Martin Ulirsch
Goethe University Frankfurt

Jeremy Usatine
Brown University

Ravi Vakil
Stanford University

Bart Van Steirteghem
CUNY and FAU ErlangenNuernberg

Lorenzo Vecchi
Alma Mater Studiorum  Università di Bologna

Emanuele Ventura
University of Bern

Duc Vo
Harvard University

Weikun Wang
Southern University of Science and Technology

Andrzej Weber
Uniwersity of Warsaw

Anna Weigandt
University of Michigan

Daping Weng
Michigan State University

Lauren Williams
Harvard University

Kaelyn Willingham
The Ohio State University

Corey Wolfe
Tulane University

Cameron Wright
University of Washington

Weihong Xu
Rutgers

Damir Yeliussizov
KazakhBritish Technical University

Alexander Yong
University of Illinois at UrbanaChampaign

Semin Yoo
University of Rochester

Chi Ho Yuen
Brown University

Claudia Yun
Brown University

Dmitry Zakharov
Central Michigan University

Mohammad Zaman Fashami
Amirkabir University of Technology

Yuanning Zhang
UC Berkeley

Changlong Zhong
State University of New York Albany

Paul ZinnJustin
The University of Melbourne
Workshop Schedule
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 (Virtual)
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 (Virtual)
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 (Virtual)
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 (Virtual)
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 (Virtual)
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 (Virtual)
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 (Virtual)
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 (Virtual)
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 (Virtual)
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 (Virtual)
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 (Virtual)
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 (Virtual)
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 (Virtual)
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 (Virtual)
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 (Virtual)
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 (Virtual)
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 (Virtual)
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 (Virtual)
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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