Organizing Committee
- Federico Ardila
San Francisco State University - Man-Wai 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 Newton-Okounkov bodies, Ehrhart theory, toric geometry, tropical geometry, matroids, and interactions with mirror symmetry.

Confirmed Speakers & Participants
Talks will be presented virtually or in-person 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 California-Davis
-
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
-
Man-Wai 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 Dorpalen-Barry
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
Daegu-Gyeongbuk 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
Cinvestav-IPN (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
IBS-CGP
-
Yoav Len
University of St Andrews
-
Shiyue Li
Brown University
-
Yen-Kheng Lim
Xiamen University Malaysia
-
David Lowry-Duda
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 Saint-Quentin-en-Yvelines University
-
Nathan Pflueger
Amherst College
-
Irem Portakal
Otto-von-Guericke-Universitaet 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 Urbana-Champaign
-
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 Nebraska-Lincoln
-
Oswaldo Sevilla
Centro de Investigación en Matemáticas, Guanajuato Mexico
-
Aniket Shah
The Ohio State University
-
Kris Shaw
University of Oslo
-
Melissa Sherman-Bennett
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
-
Miruna-Stefana 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 Urbana-Champaign
-
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 Erlangen-Nuernberg
-
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
Kazakh-British Technical University
-
Alexander Yong
University of Illinois at Urbana-Champaign
-
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 Zinn-Justin
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 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 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 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 EDTBreakCoffee Break
-
2:45 - 3:30 pm EDTKazhdan-Lusztig 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 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 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 one-dimensional families of genus-one 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.Cid-Ruiz, 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 EDTWall-crossing phenomenon for Newton-Okounkov bodiesVirtual
- Speaker
- Laura Escobar, Washington University- St. Louis
- Session Chair
- Lara Bossinger, Mathematics Institute UNAM, Oaxaca (Virtual)
Abstract
A Newton-Okounkov 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 Kaveh-Manon gives an explicit link between tropical geometry and Newton-Okounkov bodies. We use this link to describe a wall-crossing phenomenon for Newton-Okounkov bodies. As an example, we describe wall-crossing 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 well-known theorems about decomposing a convex polytope into an signed sum of cones, namely the classical Brianchon-Gram theorem and Lawrence-Varchenko theorem. I will then explain a generalization of the Brianchon-Gram 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 steady-state ideals. Steady-state 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 steady-state ideal? In particular, our main results describe three operations on the reaction graph that preserve the steady-state ideal. Furthermore, since the motivation for this work is the classification of steady-state ideals, monomials play a primary role. To this end, combinatorial conditions are given to identify monomials in a steady-state ideal, and we give a sufficient condition for a steady-state ideal to be monomial.
Construction and properties of Kanev surfaces in toric 3-folds
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 3-folds. 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 Littlewood-Richardson coefficients and the Littlewood-Richardson coefficients
Nguyen Khanh, Institut Camille Jordan
We give a new interpretation of the shifted Littlewood-Richardson 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 Littlewood-Richardson 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 Urbana-Champaign
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\delta-1}) $ 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)(d-1)^n \right)^{\delta}+O\left(d^{n(\delta-1)} \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 Newton-Okounkov 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 Newton-Okounkov bodies that one can associate to "nice" compactifications of cluster varieties. In particular, I will explain how this approach recovers Rietsch--Williams' construction of Newton--Okounkov bodies for Grassmannians. In order to make the precise connection it will be necessary to explain how the Marsh--Rietsch potential and the Gross--Hacking--Keel--Kontsevich potential for Grassmannians are related. Finally, I will draw some consequences from this relation such as an isomorphism of the toric degenerations obtained by Rietsch-Willimas 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.
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3:30 - 3:45 pm EDTBreakCoffee Break
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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 Newton-Okounkov 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
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9:00 - 9:45 am EDTGathertown Morning CoffeeCoffee Break - Virtual
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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
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11:00 - 11:15 am EDTBreakCoffee Break
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11:15 am - 12:00 pm EDTThe Fulton-MacPherson 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 Fulton-MacPherson 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.
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12:00 - 1:30 pm EDTLunch/Free Time
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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 log-concave 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.
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2:30 - 2:45 pm EDTBreakCoffee Break
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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 m-space that assembles the Gröbner degenerations of V associated with all faces of C. This is a multi-parameter generalization of the classical one-parameter Gröbner degeneration associated to a weight. We explain how our family can be constructed from Kaveh--Manon's recent work on the classification of toric flat families over toric varieties: it is the pull-back 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 Stanley--Reisner 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
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9:00 - 9:45 am EDTGathertown Morning CoffeeCoffee Break - Virtual
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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 Donaldson-Thomas invariants. The story touches another major character in the story of polyhedral and algebraic geometry: the secondary polytope of Gel'fand-Kapranov-Zelevinsky. I'll try to give some sense for why.
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11:00 - 11:15 am EDTBreakCoffee Break
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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 coordinates---to 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 Lie-type D setting.
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12:00 - 1:30 pm EDTLunch/Free Time
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1:30 - 2:15 pm EDTOn the top-weight 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.
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2:30 - 2:45 pm EDTBreakCoffee Break
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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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