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    The Institute for Computational and Experimental Research in Mathematics

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    Welcome to ICERM

    The Institute for Computational and Experimental Research in Mathematics

  • ICERM's lecture hall

    Welcome to ICERM

    The Institute for Computational and Experimental Research in Mathematics

  • ICERM's conference room

    Welcome to ICERM

    The Institute for Computational and Experimental Research in Mathematics

This Week at ICERM

Combinatorial Algebraic Geometry
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Schubert Seminar Series
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VIRTUAL ONLY: Safety and Security of Deep Learning
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VIRTUAL ONLY: Algebraic Geometry and Polyhedra
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April 11, 2021
  • 9:10 - 9:50 am EDT
    Deep Learning and Neural Networks: The Mathematical View
    Virtual
    • Speaker
    • Ronald DeVore, Texas A&M University
    • Session Chair
    • Anders Hansen, University of Cambridge
    Abstract
    Deep Learning is much publicized and has had great empirical success on challenging problems in learning. Yet there is no quantifiable proof of performance and certified guarantees for these methods. This talk will give an overview of Deep Learning from the viewpoint of mathematics and numerical computation.
  • 10:00 - 10:30 am EDT
    Gathertown Morning Coffee Break
    Coffee Break - Virtual
  • 10:30 - 11:10 am EDT
    Can we design deep learning models that are inherently interpretable?
    Virtual
    • Speaker
    • Cynthia Rudin, Duke University
    • Session Chair
    • Anders Hansen, University of Cambridge
    Abstract
    Black box deep learning models are difficult to troubleshoot. In practice, it can be difficult to tell whether their reasoning process is correct, and ""explanations"" have repeatedly been shown to be ineffective. In this talk I will discuss two possible approaches to create deep learning methods that are inherently interpretable. The first is to use case-based reasoning, through a neural architecture called ProtoPNet, where an extra ""prototype"" layer in the network allows it to reason about an image based on how similar it looks to other images (the network says ""this looks like that""). Second, I will describe ""concept whitening,"" a method for disentangling the latent space of a neural network by decorrelating concepts in the latent space and aligning them along the axes. This Looks Like That: Deep Learning for Interpretable Image Recognition. NeurIPS spotlight, 2019. https://arxiv.org/abs/1806.10574 Concept Whitening for Interpretable Image Recognition. Nature Machine Intelligence, 2020. https://rdcu.be/cbOKj
  • 11:20 am - 12:00 pm EDT
    Differential privacy, deep learning, and synthetic data generation
    Virtual
    • Speaker
    • Rachel Cummings, Columbia University
    • Session Chair
    • Anders Hansen, University of Cambridge
    Abstract
    Differential privacy is a parameterized notion of database privacy that gives a mathematically rigorous worst-case bound on the maximum amount of information that can be learned about an individual's data from the output of a computation. Recent work has provided tools for differentially private stochastic gradient decent, which enables differentially private deep learning. These in turn enable differentially private synthetic data generation, to provide synthetic versions of sensitive datasets that share statistical properties with the original data while additionally providing formal privacy guarantees for the training dataset. This talk will first give an introduction to differential privacy, and then survey recent advances in differentially private deep learning and its application to synthetic data generation.
  • 12:10 - 1:30 pm EDT
    Lunch/Free Time
    Virtual
  • 1:30 - 2:10 pm EDT
    Reliability, Robustness and Minipatch Learning
    Virtual
    • Speaker
    • Genevera Allen, Rice University
    • Session Chair
    • Ben Adcock, Simon Fraser University
    Abstract
    Many have noted and lamented a reproducibility crisis in science with more recent discussion and interest on the reproducibility and reliability of data science and machine learning techniques.  In this talk, I will introduce the Four R's, a tiered framework for discussing and assessing the reproducibility, replicability, reliability, and robustness of a data science or machine learning pipeline.  Then, I will introduce a new minipatch learning framework that helps to improve the reliability and robustness of machine learning procedures.  Inspired by stability approaches from high-dimensional statistics, random forests, and dropout training in deep learning, minipatch learning is an ensemble approach where we train on very tiny randomly or adaptively chosen subsets of both observations and features or parameters.  Beyond the obvious computational and memory efficiency advantages, we show that minipatch learning also yields more reliable and robust solutions by providing implicit regularization.  
  • 2:20 - 3:00 pm EDT
    Reliable predictions? Counterfactual predictions? Equitable treatment? Some recent progress in predictive inference
    Virtual
    • Speaker
    • Emmanuel Candes, Stanford University
    • Session Chair
    • Ben Adcock, Simon Fraser University
    Abstract
    Recent progress in machine learning provides us with many potentially effective tools to learn from datasets of ever increasing sizes and make useful predictions. How do we know that these tools can be trusted in critical and high-sensitivity systems? If a learning algorithm predicts the GPA of a prospective college applicant, what guarantees do I have concerning the accuracy of this prediction? How do we know that it is not biased against certain groups of applicants? This talk introduces statistical ideas to ensure that the learned models satisfy some crucial properties, especially reliability and fairness (in the sense that the models need to apply to individuals in an equitable manner). To achieve these important objectives, we shall not “open up the black box” and try understanding its underpinnings. Rather we discuss broad methodologies that can be wrapped around any black box to produce results that can be trusted and are equitable. We also show how our ideas can inform causal inference predictive; for instance, we will answer counterfactual predictive problems: i.e. predict the outcome of a treatment would have been given that the patient was actually not treated.
  • 3:10 - 4:00 pm EDT
    Gathertown Afternoon Coffee Break
    Coffee Break - Virtual
April 12, 2021
  • 9:45 - 10:00 am EDT
    Welcome
    Virtual
    • Brendan Hassett, ICERM/Brown University
  • 10:00 - 10:45 am EDT
    The Foundation of a Matroid
    Virtual
    • Speaker
    • Matthew Baker, Georgia Institute of Technology
    • Session Chair
    • Lauren Williams, Harvard University
    Abstract
    Matroid theorists are typically interested in questions concerning representability of matroids over fields. More generally, one can ask about representability over partial fields in the sense of Semple and Whittle. Pendavingh and van Zwam introduced the universal partial field of a matroid, which governs the representations of over all partial fields. Unfortunately, almost all matroids are not representable over any partial field, and in this case, the universal partial field gives no information. Oliver Lorscheid and I have introduced a generalization of the universal partial field which we call the foundation of a matroid. The foundation of is a type of algebraic object which we call a pasture; pastures include both hyperfields and partial fields. As a particular application of this point of view, I will explain the classification which Lorscheid and I have recently obtained of all possible foundations for matroids having no minor isomorphic to U(2,5) or U(3,5). Among other things, our classification provides a short conceptual proof of the 1997 theorem of Lee and Scobee which says that a matroid is both ternary and orientable if and only if it is dyadic.
  • 11:00 - 11:15 am EDT
    Break
    Coffee Break
  • 11:15 am - 12:00 pm EDT
    Lagrangian geometry of matroids
    Virtual
    • Speaker
    • Graham Denham, University of Western Ontario
    • Session Chair
    • Lauren Williams, Harvard University
    Abstract
    In joint work with Federico Ardila and June Huh, we introduce the conormal fan of a matroid, which is the Lagrangian analogue of the Bergman fan. We use it to give a Lagrangian interpretation of the Chern-Schwartz-MacPherson cycle of a matroid. We develop tools for tropical Hodge theory to show that the conormal fan satisfies Poincaré duality, the Hard Lefschetz property, and the Hodge--Riemann relations. Together, these imply conjectures of Brylawski and Dawson about the log-concavity of the h-vectors of the broken circuit complex and independence complex of a matroid.
  • 12:00 - 1:30 pm EDT
    Lunch/Free Time
  • 1:30 - 2:15 pm EDT
    Tautological classes of matroids
    Virtual
    • Speaker
    • Christopher Eur, Stanford University
    • Session Chair
    • Sam Payne, University of Texas at Austin
    Abstract
    We introduce certain torus-equivariant classes on permutohedral varieties which we call ``tautological classes of matroids'' as a new geometric framework for studying matroids. Using this framework, we unify and extend many recent developments in matroid theory arising from its interaction with algebraic geometry. We achieve this by establishing a Chow-theoretic description and a log-concavity property for a 4-variable transformation of the Tutte polynomial, and by establishing an exceptional Hirzebruch-Riemann-Roch-type formula for permutohedral varieties that translates between K-theory and Chow theory. This is joint work with Andrew Berget, Hunter Spink, and Dennis Tseng.
  • 2:30 - 2:45 pm EDT
    Break
    Coffee Break
  • 2:45 - 3:30 pm EDT
    Kazhdan-Lusztig theory and singular Hodge theory for matroids
    Virtual
    • Speaker
    • June Huh, Stanford University
    • Session Chair
    • Sam Payne, University of Texas at Austin
    Abstract
    There is a remarkable parallel between the theory of Coxeter groups (think of the symmetric group or the dihedral group) and matroids (think of your favorite graph or point configuration) from the perspective of combinatorial cohomology theories. I will give an overview of the similarity and report on recent my joint work with Tom Braden, Jacob Matherne, Nick Proudfoot, and Botong Wang on singular Hodge theory for combinatorial geometries: https://arxiv.org/abs/2010.06088
  • 3:45 - 4:45 pm EDT
    Gathertown Reception
    Reception - Virtual
April 13, 2021
  • 9:00 - 9:45 am EDT
    Gathertown Morning Coffee
    Coffee Break - Virtual
  • 10:00 - 10:45 am EDT
    Real phase structures on matroid fans
    Virtual
    • Speaker
    • 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 non-singular tropical variety, the real part is a PL-manifold. Moreover, for a non-singular tropical variety with a real phase structures we can apply the same spectral sequence for tropical hypersurfaces, obtained by Renaudineau and myself, to bound the Betti numbers of the real part by the dimensions of the tropical homology groups. This is joint work in progress with Johannes Rau and Arthur Renaudineau.
  • 11:00 - 11:15 am EDT
    Break
    Coffee Break
  • 11:15 am - 12:00 pm EDT
    Combinatorics and real lifts of bitangents to tropical quartic curves
    Virtual
    • 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 EDT
    Lunch/Free Time
  • 1:30 - 2:15 pm EDT
    Tropical psi classes
    Virtual
    • 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 one-dimensional families of genus-one tropical curves.
  • 2:30 - 2:45 pm EDT
    Break
    Coffee Break
  • 2:45 - 3:30 pm EDT
    When 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.Cid-Ruiz, B.Li, J.Montano, and N.Zhang.
April 14, 2021
  • 9:00 - 9:45 am EDT
    Gathertown Morning Coffee
    Coffee Break - Virtual
  • 10:00 - 10:45 am EDT
    Wall-crossing phenomenon for Newton-Okounkov bodies
    Virtual
    • Speaker
    • Laura Escobar, Washington University- St. Louis
    • Session Chair
    • Lara Bossinger, Mathematics Institute UNAM, Oaxaca
    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 EDT
    Break
    Coffee Break - Virtual
  • 11:15 am - 12:00 pm EDT
    On combinatorics of Arthur's trace formula, convex polytopes, and toric varieties
    Virtual
    • Speaker
    • Kiumars Kaveh, University of Pittsburgh
    • Session Chair
    • Lara Bossinger, Mathematics Institute UNAM, Oaxaca
    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 EDT
    Lunch/Free Time
  • 1:30 - 2:30 pm EDT
    Poster Session
    Virtual
    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 EDT
    On Newton-Okounkov bodies associated to Grassmannians
    Virtual
    • 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 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.
  • 3:30 - 3:45 pm EDT
    Break
    Coffee Break
  • 3:45 - 4:30 pm EDT
    Broken line convexity
    Virtual
    • 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 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.
April 15, 2021
  • 9:00 - 9:45 am EDT
    Gathertown Morning Coffee
    Coffee Break - Virtual
  • 10:00 - 10:45 am EDT
    Toric vector bundles -- an overview
    Virtual
    • 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 EDT
    Break
    Coffee Break
  • 11:15 am - 12:00 pm EDT
    The Fulton-MacPherson compactification is not a Mori dream space
    Virtual
    • Speaker
    • José González, University of California, Riverside
    • Session Chair
    • Linda Chen, Swarthmore College
    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.
  • 12:00 - 1:30 pm EDT
    Lunch/Free Time
  • 1:30 - 2:15 pm EDT
    Schubert polynomials from a polytopal point of view
    Virtual
    • 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 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.
  • 2:30 - 2:45 pm EDT
    Break
    Coffee Break
  • 2:45 - 3:30 pm EDT
    Families of Gröbner degenerations
    Virtual
    • 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 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.
April 16, 2021
  • 9:00 - 9:45 am EDT
    Gathertown Morning Coffee
    Coffee Break - Virtual
  • 10:00 - 10:45 am EDT
    The logarithmic Hilbert scheme of curves
    Virtual
    • 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 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.
  • 11:00 - 11:15 am EDT
    Break
    Coffee Break
  • 11:15 am - 12:00 pm EDT
    Initial degenerations of Grassmannians
    Virtual
    • 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 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.
  • 12:00 - 1:30 pm EDT
    Lunch/Free Time
  • 1:30 - 2:15 pm EDT
    On the top-weight rational cohomology of A_g
    Virtual
    • 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 EDT
    Break
    Coffee Break
  • 2:45 - 3:30 pm EDT
    Tropical Flag Varieties
    Virtual
    • 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.
April 17, 2021

There are no events currently scheduled for April 17th.

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New ICERM Deputy Director, Misha Kilmer

March 31, 2021 - ICERM is pleased to announce that Misha Kilmer is joining ICERM as a Deputy Director, effective July 1, 2021. She is the William Walker Professor of Mathematics at Tufts University with expertise in numerical linear and multilinear algebra. Misha brings significant leadership experience to ICERM. Misha has received a number of awards and recognitions. She is a current member of ICERM's Scientific Advisory Board. https://mkilme01.pages.tufts.edu/

ICERM Completes Major Website Schedules Upgrade

March 1, 2021 -

Last week ICERM completed a major upgrade to our website schedule system. This upgrade brings the redesigned schedules on the homepage and all 2021 program pages, along with new print views and iCalendar feeds. Read more

NSF Grant Renews ICERM through 2025

July 13, 2020 - ICERM received the largest National Science Foundation grant in Brown University's history. The $23.7M grant will support the institute's continuing mission to broaden the relationship between mathematics and computation for the next five years and enable the expansion of cutting-edge programming. Read more