Welcome to the webpage for the online seminar "Algebra, Geometry, and Combinatorics"!
This seminar occurs approximately biweekly on Mondays at 11am Eastern over Zoom. At each meeting of the seminar, there is a 50 minute colloquium-style talk followed by a question period. Speakers emphasize background and context, so that talks are accessible to mathematicians with a broad range of interests in algebra, geometry, and combinatorics.
Notes and slides from past talks can be found here.
Since many of us will be transitioning back to in-person commitments on campus this fall, we have decided to end this seminar in September 2021. All remaining AGC talks are listed below. We hope you can join us before the seminar wraps up!
In order to attend the talks you need to register in advance at this link. (If you have done so in the past, you shouldn't have to register again.)
Please use your academic email account for the registration process, for verification and security purposes. (For participants without an academic account, there is a box in the registration form to tell us about your situation.)
We will be approving the registrations manually, so please register in a timely manner if you wish to attend the talk on Monday.
Once we approve your registration, you will receive a confirmation email containing access information, i.e. a personalized link which is unique to you, and it will work for all future AGC colloquia (until further notice).
Title: Gröbner bases, symmetric matrices, and type C Kazhdan-Lusztig varieties
Abstract: Woo and Yong introduced Kazhdan--Lusztig ideals as ideals of affine neighbourhoods of type A Schubert varieties at torus-fixed points. This talk is concerned with the type C analogue. In certain cases, those where the opposite cell is indexed by a 123-avoiding permutation, we have a description of type C Kazhdan--Lusztig ideals in terms of minors of symmetric matrices, from which we recover Gr\"obner bases, prime decompositions, and K-polynomial pipe dream formulae for these ideals.
Title: Tautological classes of matroids
Abstract: A variety of geometric constructions for matroids can be naturally interpreted torus equivariantly via certain "tautological Chern classes", despite oftentimes lacking a natural torus action. Together with an exceptional Hirzebruch-Riemann-Roch formula for permutohedral toric varieties, this synthesizes most directions of research on matroids in algebraic geometry over the past 10 years.
Abstract: We present a combinatorial algorithm on trivalent trees that we call a lazy tournament, which gives rise to a new geometric interpretation of the multidegrees of a projective embedding of the moduli space M_{0,n}-bar of stable n-marked genus 0 curves. We will show that the multidegrees are enumerated by disjoint sets of boundary points of the moduli space that can be seen to total (2n-7)!!, giving a natural proof of the value of the total degree. These sets are compatible with the forgetting maps used to derive the previously known recursion for the multidegrees.
As time permits, we will discuss an alternative combinatorial construction of (non-disjoint) sets of boundary points that enumerate the multidegrees, via slide rules, that can in fact be achieved geometrically via a degeneration of intersections with hyperplanes in the projective embedding. These combinatorial rules further generalize to give a positive expansion of any product of psi or omega classes on M_{0,n}-bar in terms of boundary strata.
This is joint work with Sean Griffin and Jake Levinson.
Title: A survey of combinatorics for K-orbits
Abstract: A subgroup K of a reductive algebraic group G is spherical if its action on G/B has finitely many orbits. The geometry of these K-orbits shares many common features with the more frequently studied Borel orbits. They are indexed by combinatorial objects, determine a Bruhat order on these objects and have surprisingly nice cohomology representatives. In this talk, we survey combinatorial approaches to understanding K-orbits with a special focus on the case G = GL_n. This includes joint work with Eric Marberg and Brendan Pawlowski.
Title: A diagrammatic Carlsson-Mellit algebra
Abstract: The $A_{q,t}$ algebra was introduced by Carlsson and Mellit in their proof of the celebrated shuffle theorem, which gave a combinatorial formula for the Frobenius character of the space of diagonal harmonics in terms of certain symmetric functions indexed by Dyck paths. This algebra arises as an extension of two copies of the affine Hecke algebra by certain raising and lowering operators. Carlsson and Mellit constructed an action via plethystic operators on the space of symmetric functions which was then realized geometrically on parabolic flag Hilbert schemes by them and Gorsky. The original algebraic construction was then extended to an infinite family of actions by Mellit and shown to contain the generators of elliptic Hall algebra. However, despite the various formulations of $A_{q,t}$, performing computations within it is complicated and non-intuitive.
In this talk I will discuss joint work with Matt Hogancamp where we construct a new topological formulation of $A_{q,t}$ (at t=-1) and its representation as certain braid diagrams on an annulus. In this setting many of the complicated algebraic relations of $A_{q,t}$ and applications to symmetric functions are trivial consequences of the skein relation imposed on the pictures. In particular, many difficult computations become simple diagrammatic manipulations in this new framework. If time permits I will discuss how this purely diagrammatic formulation allows us to lift the operators as certain functors, thus providing a categorification of the $A_{q,t}$ action on the derived trace of the Soergel category.
Title: Chromatic symmetric functions for Dyck paths and q-rook theory
Abstract:
Given a graph and a set of colors, a coloring is a function that associates each vertex in the graph with a color. In 1995, Stanley generalized this definition to symmetric functions by looking at the number of times each color is used and extending the set of colors to $\mathbb{Z}^+$. In 2012, Shareshian and Wachs introduced a refinement of the chromatic functions for ordered graphs as q-analogues.
In the particular case of Dyck paths, Stanley and Stembridge described the connection between chromatic symmetric functions of abelian Dyck paths and square hit numbers, and Guay-Paquet described their relation to rectangular hit numbers. Recently, Abreu-Nigro generalized the former connection for the Shareshian-Wachs q-analogue, and in unpublished work, Guay-Paquet generalized the latter.
In this talk, I want to give an overview of the framework and present another proof of Guay-Paquet's identity using q-rook theory. Along the way, we will also discuss q-hit numbers, two variants of their statistic, and some deletion-contraction relations. This is recent work with Alejandro H. Morales and Greta Panova.
Title: The hypersimplex and the m=2 amplituhedron: Eulerian numbers, sign flips, triangulations
Abstract: Physicists Arkani-Hamed and Trnka introduced the amplituhedron to better understand scattering amplitudes in N=4 super Yang-Mills theory. The amplituhedron is the image of the totally nonnegative Grassmannian under the "amplituhedron map", which is induced by matrix multiplication. Examples of amplituhedra include cyclic polytopes, the totally nonnegative Grassmannian itself, and cyclic hyperplane arrangements. In general, the amplituhedron is not a polytope. However, Lukowski--Parisi--Williams noticed a mysterious connection between the m=2 amplituhedron and the hypersimplex, and conjectured a correspondence between their fine positroidal subdivisions. I'll discuss joint work with Matteo Parisi and Lauren Williams, in which we prove one direction of this correspondence. Along the way, we prove an intrinsic description of the m=2 amplituhedron conjectured by Arkani-Hamed--Thomas--Trnka; give a decomposition of the m=2 amplituhedron into chambers enumerated by the Eulerian numbers, in direct analogy to a triangulation of the hypersimplex; and find new cluster varieties in the Grassmannian.
Title: The power of moment graphs
Abstract: Moment graphs are combinatorial gadgets which arise as one-skeleta of torus actions on (particularly nice) varieties and which encode information about the cohomology of the underlying varieties. In this talk, I will review Goresky-Kottwitz-MacPherson's version of the localization theorem, and describe some of its consequences in the case of a flag variety acted upon by an appropriate torus. I will then explain how one can extend these techniques to a bigger class of varieties, obtained from quiver theory, leading to new research directions.
Title: An introduction to deformation theory and moduli
Abstract: The goal of this talk is to provide an introduction to deformation theory and to moduli spaces in algebraic geometry. This is a very vast subject and I will only mention very few topics. In particular, I will explain how deformation theory is the local study of moduli spaces and I will treat the case of curves (studied by Deligne and Mumford), I will mention some examples of deformations of singularities (together with some interplay with polytopes), and some recent applications to the local study of moduli of Fano varieties.
Title: Schubert polynomials from a polytopal point of view
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.
Abstract: The Peterson isomorphism directly relates the homology of the affine Grassmannian to the quantum cohomology of any finite flag variety. In the case of a partial flag, Peterson’s map is only a surjection, and one needs to quotient by a suitable ideal to map isomorphically onto the quantum cohomology. In this talk, we provide an exposition of this parabolic Peterson isomorphism in the case of the Grassmannian. Time permitting, we will also relate the Peterson isomorphism via Postnikov’s strange duality to several quantum-to-affine correspondences on the k-Schur functions representing the homology of the affine Grassmannian. This talk includes joint work with J. Cookmeyer, as well as L. Chen and J. Morse.
Title: Infinite flags and Schubert polynomials
Abstract: Using the geometry of certain infinite-dimensional flag varieties, I will describe a variation on the back-stable Schubert polynomials introduced by Lam, Lee, and Shimozono. New morphisms among these varieties become apparent in the infinite limit, and they imply certain properties of the corresponding Schubert polynomials. The direct sum morphism plays an especially interesting role.
This is based on joint work with William Fulton.
Title: Representation varieties of algebras with nodes
Abstract: This is a report on joint work arXiv:1810.10997 with András C. Lőrincz. I will start with a brief recollection of varieties of complexes, introduced in the 1970s. One generalization which received a lot of attention in the 1980s through recent years is representation varieties of type A quivers, which are defined by more general ideals of minors coming from sequences of matrices with variable entries. I will recall some highlights of this story for historical motivation.
In my work with Lőrincz, we generalize some results about varieties of complexes in a different direction, replacing the sequences of matrices with any finite directed graph (quiver) having matrices assigned to each arrow. The setup can be neatly formalized in the language of representation varieties of algebras (or quivers with relations), though I will emphasize the concrete matrix-based approach in this talk.
Our joint results are in characteristic 0, including that these varieties have rational singularities, and a concrete description of their prime defining ideals (which end up requiring more than just minors as generators!). Independent work of Lőrincz arXiv:2008.08270 extends our main results to positive characteristic, as one application of his extension of Kempf's famous "vector bundle collapsing" technique to positive characteristic.
Title: Syzygies in higher dimensions
Abstract: I will discuss recent large-scale computations, which utilize numerical linear algebra and highly distributed, high-performance computing to generate data about the syzygies of various algebraic surfaces. Further, I will discuss how this data has led to several new conjectures.
Title: The Art of Counting Lattice Points
Abstract: It is a core principle of mathematics to translate problems in one area of research to another area and solve them there. An especially fruitful realization of this principle is the connection between algebraic and polyhedral geometry — essentially via toric varieties. In this talk we want to present this connection in one particular example.
From our everyday quest to pay for our morning coffee at our favorite coffee shop by cash to the peak of recent research, lattice point counting problems appear in many different branches of mathematics. The formalization of this problem leads us to a beautiful theory by Eugène Ehrhart, whose applications to algebraic geometry and representation theory are far from exhausted. As our main application we will study the problem whether a given polarized Gorenstein Fano variety can be degenerated to an anticanonically polarized Gorenstein Fano toric variety. This should allow for further research in mirror symmetry.
It should be noted that the connection between algebraic and polyhedral geometry via Ehrhart Theory is not a one-way street. As an example we will show that for a decent class of varieties, Serre Duality and Ehrhart–Macdonald Reciprocity are equivalent statements, hence one of the most famous theorems from algebraic geometry can actually be proved by counting lattice points.
Title: Sign Variations and Descents
Abstract: In this talk we consider a poset structure on projective sign vectors. We show that the order complex of this poset is partitionable and give an interpretation of the h-vector using type B descents of the type D Coxeter group.
Title: Measuring polytopes through their algebraic structure.
Abstract: Generalized permutahedra are a beautiful family of polytopes with a rich combinatorial structure, and strong connections to optimization and algebraic geometry. We prove they are the universal family of polyhedra with a certain Hopf-algebraic structure. We also prove that this Hopf-algebraic structure is compatible with the theory of valuations of polytopes; these are the measures of polytopes that satisfy the natural inclusion-exclusion relations.
Our construction provides a unifying framework to organize and study many combinatorial families and derive different kinds of results; for example:
1. It uniformly answers open questions and recovers known Hopf-algebraic results about graphs, posets, matroids, hypergraphs, simplicial complexes, and others.
2. It shows that permutahedra and associahedra “know" how to compute the multiplicative and compositional inverses of power series.
3. It explains the mysterious fact that many combinatorial and algebro-geometric invariants of matroids can also be thought of as measures on polytopes, satisfying the inclusion-exclusion relations. These include the Tutte polynomial, the Chern-Schwartz-MacPherson cycles, and the class of a matroid in the Chow ring of the permutahedral variety.
I will make the talk as elementary and self-contained as I am able to. This is joint work with Marcelo Aguiar (2017) and Mario Sanchez (2020).
Title: Monomials, convex bodies, and optimization
Abstract: A classical construction associates to each monomial ideal a convex body termed the Newton polyhedron. This is the convex hull of the exponent vectors for all monomials in the ideal. We present new ways of obtaining convex bodies from monomial ideals based on primary decompositions for these ideals. This leads to a construction known as a symbolic polyhedron. Two problems, one originating from algebraic geometry and the other from combinatorial optimization can be expressed in terms of linear optimization on this polyhedron. We survey progress regarding these problems obtained in collaboration with the participants of the 2020 Polymath REU.
Title: Dynamics on the moduli space M_{0,n}
Abstract: A rational function f(z) in one variable determines a self-map of P^1. A rational function is called post-critically finite (PCF) if every critical point is either pre-periodic or periodic. PCF rational functions have been studied for their special dynamics, and their special distribution within the moduli space of all rational maps. By works of W. Thurston and S. Koch, every PCF map (with a well-understood class of exceptions) arises as an isolated fixed point of an algebraic dynamical system on the moduli space M_{0,n} of point-configurations on P^1; these dynamical systems are called Hurwitz correspondences.
I will introduce Hurwitz correspondences and their connection to PCF rational maps, and discuss how the dynamical complexity of Hurwitz correspondence can be studied via combinatorial compactifications of M_{0,n}.
Title: Quotients of lattice path matroids
Abstract: Matroids are a combinatorial object that generalize the notion of linear independence. One way to characterize matroids is via polytopes, as shown in the work of Gelfand, Goresky, MacPherson, Serganova.
In this talk we will focus on a particular class of matroids called Lattice Path Matroids (LPMs). We will show when a collection M_1,...,M_k of LPMs are a flag matroid, using their combinatorics. Part of our work will show that the polytope associated to such a flag can be thought as an interval in the Bruhat order, and thus provides a partial understanding of flags of LPMs from a polytopal point of view.
We will not assume previous knowledge on matroids nor quotients. This is joint work with Kolja Knauer.
Title: When are multidegrees positive?
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 wonderful combinatorial description that relates to 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.
Title: Introduction to Berkovich geometry.
Abstract: The goal of this talk is to provide an introduction and some motivations to the study of Berkovich geometry. This theory was developed by Berkovich as an approach to the study of varieties over non-archimedean valued fields. It has quickly found many applications, from algebraic and arithmetic geometry, to tropical geometry and complex dynamics.
I will focus on how to visualize Berkovich spaces, via simple normal crossing degenerations and their associated dual complex. Then I will present recent progress in the interplay between this theory and open questions from mirror symmetry.
Title: Tropical abelian varieties
Abstract: I'll give an introduction to the moduli space of tropical abelian varieties, assuming no background in tropical geometry. Lots of different combinatorics arises, including the beautiful century-old combinatorics of Voronoi reduction theory, perfect quadratic forms, regular matroids, and metric graphs. On the geometric side, it relates to toroidal compactifications of the classical moduli space A_g of abelian varieties. I'll explain how, and, time permitting, I'll report on work-in-progress in which we use tropical techniques to find new rational cohomology classes in A_g in a previously inaccessible range. Joint work with Madeline Brandt, Juliette Bruce, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.
Title: Geometric vertex decomposition and liaison
Abstract: Geometric vertex decomposition and liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties. In this talk, we will describe an explicit connection between these approaches. In particular, we describe how each geometrically vertex decomposable ideal is linked by a sequence of elementary G-biliaisons of height 1 to an ideal of indeterminates and, conversely, how every G-biliaison of a certain type gives rise to a geometric vertex decomposition. As a consequence, we can immediately conclude that several well-known families of ideals are glicci, including Schubert determinantal ideals, defining ideals of varieties of complexes, and defining ideals of graded lower bound cluster algebras.
This connection also gives us a framework for implementing with relative ease Gorla, Migliore, and Nagel’s strategy of using liaison to establish Gr\"obner bases. We describe briefly, as an application of this work, a proof of a recent conjecture of Hamaker, Pechenik, and Weigandt on diagonal Gr\"obner bases of Schubert determinantal ideals, which Weigandt discussed in her talk in this colloquium series several weeks ago.
This talk is based on joint work with Jenna Rajchgot.
Title: Stable map resolutions of Richardson varieties
Abstract: To a simple normal crossings divisor (sncd), one associates its "dual simplicial complex", with a vertex for each component and face for each nonempty intersection. For example, Escobar's brick manifolds (which among other things, provide resolutions of Richardson varieties) come with an sncd whose dual complex is a spherical subword complex.
To a simple normal crossings divisor (sncd), one associates its "dual simplicial complex", which in good cases (such as the sncd in Escobar's brick manifolds) is a sphere. Björner-Wachs showed that the order complex of a Bruhat interval (u,v) is a sphere. I'll define a space of equivariant stable maps from P^1 to the Richardson variety X_u^v, and prove that this space is a smooth orbifold, which comes with a natural sncd whose dual is the Björner-Wachs complex. There are no choices (e.g. of reduced words). In the Grassmannian case this space is GKM, and I describe its GKM graph.
Title: When are multidegrees positive?
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 wonderful combinatorial description that relates to 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.
Title: Convexity in tropical spaces and compactifications of cluster varieties
Abstract: Cluster varieties are a relatively new, broadly interesting class of geometric objects that generalize toric varieties. Convexity is a key notion in toric geometry. For instance, projective toric varieties are defined by convex lattice polytopes. In this talk, I'll explain how convexity generalizes to the cluster world, where "polytopes" live in a tropical space rather than a vector space and "convex polytopes" define projective compactifications of cluster varieties. Time permitting, I'll conclude with 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. Based on joint work with Man-Wai Cheung and Alfredo Nájera Chávez.
Topic: Gröbner geometry of Schubert polynomials through ice
Abstract: The geometric naturality of Schubert polynomials and the related combinatorics of pipe dreams was established by Knutson and Miller (2005) via antidiagonal Gröbner degeneration of matrix Schubert varieties. We consider instead diagonal Gröbner degenerations. In this dual setting, Knutson, Miller, and Yong (2009) obtained alternative combinatorics for the class of vexillary matrix Schubert varieties. We will discuss general diagonal degenerations, relating them to an older formula of Lascoux (2002) in terms of the 6-vertex ice model. Lascoux's formula was recently rediscovered by Lam, Lee, and Shimozono (2018), as "bumpless pipe dreams." We will explain this connection and discuss conjectures and progress towards understanding diagonal Gröbner degenerations of matrix Schubert varieties. This is joint work with Zachary Hamaker and Oliver Pechenik.
Title: New types of heights with connections to the Batyrev-Manin and Malle Conjectures
Abstract: The Batyrev-Manin conjecture gives a prediction for the asymptotic growth rate of rational points on varieties over number fields when we order the points by height. The Malle conjecture predicts the asymptotic growth rate for number fields of degree d when they are ordered by discriminant. The two conjectures have the same form and it is natural to ask if they are, in fact, one and the same. We develop a theory of point counts on stacks and give a conjecture for their growth rate which specializes to the two aforementioned conjectures. This is joint work with Jordan Ellenberg and David Zureick-Brown. No prior knowledge of stacks will be assumed for this talk.
Title: Geometry of smooth Hilbert schemes
Abstract: How can we understand the subvarieties of a fixed projective space? Hilbert schemes provide the geometric answer to this question. After surveying some features of these natural parameter spaces, we will classify the smooth Hilbert schemes. Time permitting, we will also describe the geometry of nonsingular Hilbert schemes by interpreting them as suitable generalizations of partial flag varieties. This talk is based on joint work with Roy Skjelnes (KTH).
Title: Cluster structures and tropicalization
Abstract: I will explain the two concepts in the title: cluster structures and tropicalization. Both concepts offer a range of tools that can be used for example, to construct toric degenerations of projective varieties. In this talk I will focus on examples such as Grassmannians and flag varieties to explain how the two notions are related.
Title: Some geometric aspects of Hessenberg varieties
Abstract: Hessenberg varieties form a broad and well-studied class of closed subvarieties in the flag variety. Their study is central to ongoing research at the interface of algebraic geometry, combinatorics, geometric representation theory, and symplectic geometry. I will review the more manifestly geometric aspects of this study, emphasizing joint works with each of Hiraku Abe, Ana Balibanu, Steven Rayan, and Markus Röser. This presentation will not presuppose any familiarity with Hessenberg varieties.
The goal of the seminar is to reach as wide as possible a group of students and faculty, but some protection is necessary to avoid annoying interference on Zoom. Thus we have the following rules.
In order to attend the seminar you must be on our mailing list. Students and faculty are invited to send an email requesting membership from a university account to agccolloq@gmail.com.
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Organizers: Laura Escobar, Megumi Harada, and Jenna Rajchgot
Contact: agccolloq@gmail.com