# Algebra, Geometry, and Combinatorics

## An Online Seminar

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.

## Registration

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).

# Upcoming talks

**All talks will take place on ****Mondays**** ****at 11am Eastern**

**All talks will take place on**

**Mondays**

**at 11am Eastern**

**June 21: Laura Colmenarejo (UMass Amherst)**

**June 21: Laura Colmenarejo (UMass Amherst)**

Topic: TBA

**July 12: Nicolle Gonzalez (UCLA)**

**July 12: Nicolle Gonzalez (UCLA)**

Topic: TBA

**July 26: Zachary Hamaker (Florida)**

**July 26: Zachary Hamaker (Florida)**

Topic: TBA

**Aug. 9: Maria Gillespie (Colorado State)**

**Aug. 9: Maria Gillespie (Colorado State)**

Topic: TBA

# Past talks

**June 7: Melissa Sherman-Bennett (Berkeley)**

**June 7: Melissa Sherman-Bennett (Berkeley)**

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.

**May 24, 2021: Martina Lanini (****Università degli Studi di Roma Tor Vergata)**

**May 24, 2021: Martina Lanini (**

**Università degli Studi di Roma Tor Vergata)**

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.

**May 10, 2021: Andrea Petracci (****Freie Universität Berlin)**

**May 10, 2021: Andrea Petracci (**

**Freie Universität Berlin)**

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.

**April 19, 2021: ****Karola Mészáros (Cornell University)**

**April 19, 2021:**

**Karola Mészáros (Cornell University)**

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.

**March 29, 2021: ****Elizabeth Milićević (Haverford College)**

**March 29, 2021:**

**Elizabeth Milićević (Haverford College)**

### Title: The Peterson Isomorphism and Quantum Cohomology of the Grassmannian

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.

**March 8, 2021: David Anderson (Ohio State)**

**March 8, 2021: David Anderson (Ohio State)**

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.

**February 22, 2021: Ryan Kinser (University of Iowa)**

**February 22, 2021: Ryan Kinser (University of Iowa)**

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.

**February 8, 2021: Juliette Bruce (UC Berkeley and MSRI)**

**February 8, 2021: Juliette Bruce (UC Berkeley and MSRI)**

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.

**January 25, 2021: Christian Steinert (****Universität zu Köln)**

**January 25, 2021: Christian Steinert (**

**Universität zu Köln)**

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.

**January 11, 2021: Aram Dermenjian (York University)**

**January 11, 2021: Aram Dermenjian (York University)**

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.

**December 15, 2020****: Federico Ardila (SFSU)**

**December 15, 2020**

**: Federico Ardila (SFSU)**

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).

**December 1, 2020: Alexandra Seceleanu (Nebraska)**

**December 1, 2020: Alexandra Seceleanu (Nebraska)**

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.

**November 17, 2020: Rohini Ramadas (Brown)**

**November 17, 2020: Rohini Ramadas (Brown)**

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}.

**November 3, 2020: Carolina Benedetti (****Universidad de los Andes)**

**November 3, 2020: Carolina Benedetti (**

**Universidad de los Andes)**

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.

**October 20, 2020: Federico Castillo (MPI Leipzig)**

**October 20, 2020: Federico Castillo (MPI Leipzig)**

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.

**October 6, 2020: Enrica Mazzon (MPI Bonn)**

**October 6, 2020: Enrica Mazzon (MPI Bonn)**

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.

**September 22, 2020: Melody Chan (Brown)**

**September 22, 2020: Melody Chan (Brown)**

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.

**September 8, 2020: Patricia Klein (Minnesota)**

**September 8, 2020: Patricia Klein (Minnesota)**

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.

**September 1, 2020: Allen Knutson (Cornell)**

**September 1, 2020: Allen Knutson (Cornell)**

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.

**August 24, 2020: Federico Castillo (MPI Leipzig) **

**August 24, 2020: Federico Castillo (MPI Leipzig)**

****THIS TALK IS BEING POSTPONED DUE TO TECHNICAL ISSUES WITH ZOOM****

****THIS TALK IS BEING POSTPONED DUE TO TECHNICAL ISSUES WITH ZOOM****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.

**August 10, 2020: Timothy Magee (Birmingham)**

**August 10, 2020: Timothy Magee (Birmingham)**

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.

**July 27, 2020: Anna Weigandt (Michigan)**

**July 27, 2020: Anna Weigandt (Michigan)**

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.

***Tuesday* July 14, 2020: Matthew Satriano (Waterloo)**

***Tuesday* July 14, 2020: Matthew Satriano (Waterloo)**

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.

**June 29, 2020: Gregory G. Smith (Queen's)**

**June 29, 2020: Gregory G. Smith (Queen's)**

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).

**June 1, 2020: Lara Bossinger (Insituto de Matematicás, UNAM) **

**June 1, 2020: Lara Bossinger (Insituto de Matematicás, UNAM)**

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.

**May 18, 2020**:** Peter Crooks (Northeastern)**

**May 18, 2020**:

**Peter Crooks (Northeastern)**

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.

# Zoom Information

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.

**Zoom House Rules:**

**Zoom House 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.Please enter the meeting 5-10 minutes before the talks you want to attend since we need to approve each participant and we will not allow new participants to join once the talk starts.

Please keep your microphones muted to avoid background noise. (Having your video on is encouraged but not required.)

The preferred way to ask a question is using the Chat.

Alternatively, to ask a question you can raise your hand (click on Participants and then Raise Hand at the bottom of the Participants window). Either the speaker or the hosts will see that you have raised your hand and will get to you at the earliest opportunity.

To quickly mute/unmute yourself, you can use any of the following keyboard shortcuts:

Push to Talk. If you are using the Zoom desktop client on a computer, you can temporarily unmute yourself by pressing and holding the SPACE key on your keyboard. You will be unmuted for as long as you hold the space key.

Mac: Command(⌘)+Shift(⇧)+A

Windows: Alt+M

Linux: Alt+A

iOS/iPadOS with keyboard: Command(⌘)+Shift(⇧)+A

**Organizers:** Laura Escobar, Megumi Harada, and Jenna Rajchgot

**Contact: **agccolloq@gmail.com