Here is a (incomplete) list of talks that were held in our AAGS seminar since 2019.
Title: Instability of Tangent Bundles of Gorenstein-Fano Cones.
Abstract: Gorenstein-Fano varieties are an interesting class of varieties, having anticanonical divisors which are both ample and Cartier. An important invariant of these varieties is K-(poly)stability. For Gorenstein-Fano varieties, K-stability implies slope stability of the tangent sheaf with respect to the anticanonical polarization. Thus, slope stability is a necessary condition that one can sometimes check more easily. We will focus on a certain class of Gorenstein-Fano varieties, namely cones over other Gorenstein-Fano varieties, and show that their tangent sheaves are unstable with respect to the anticanonical polarization. As an application, we will show that for toric Gorenstein-Fano varieties, a sufficiently high Fano index guarantees the instability of the tangent sheaf with respect to the anticanonical divisor.
Abstract: We will define toric varieties, drawing materials from Cox-Little- Schenck Ch. 1 or Telen §2. Recommended preparation: read CLS §1.0 (pp. 3-7). The first seminar will be led by Mahrud Sayrafi.
Abstract: We continue our discussion on toric varieties following Cox-Little- Schenck Ch. 1 or Telen §2. The seminar will be led by Mahrud Sayrafi.
Abstract: I will discuss recent work done with Elena Guardo (Catania) and Adam Van Tuyl (McMaster) on ideals of points in P1 × P1. Let I_X be the bihomogeneous ideal of a set of finite points X ⊆ P1 × P1. We investigate various notions of “splittings” of the ideal I_X, namely finding ideals J and K such that I_X = J +K, where J and K have prescribed algebraic or geometric properties. We pay particular attention to arithmetically Cohen-Macaulay (ACM) sets of points and introduce the framework of ideals of unions of lines and ACM points in P1 × P1 which allows us to better ”split” these ideals. We also discuss some consequences for the graded Betti numbers of IX in terms of these splittings.
Feb 24, 2026 Paul Ayers (McMaster)
Abstract: I will discuss the combinatoric structure of the wavefunction in many-body quantum mechanics and some known links to algebraic geometry. I'll present several conjectures and open questions, explaining their conceptual and computational importance.
March 9, 2026, Dharm Veer (Dalhousie)
Title: Complements and complementary homologies
Abstract: n a finite lattice $L$, two elements are said to be complements if their meet is the minimal element $\hat{0}$ of $L$ and their join is the maximal element $\hat{1}$ of $L$. For $m\in L$, let $C(m)$ denote the set of all complements in $L$. It is known that if $L$ non-complemented, i.e., there exists an element in $L$ that does not have a complement, then the poset $\Bar{L} = (\hat{0}, \hat{1})$ is contractible. One of our main results shows that if $\widetilde{H}_k(\Bar{L}, K) \neq 0$ for some $k>0$ and some field $K$, then for every every $m\in \Bar{L}$, there exists an $m' \in C(m)$, an $\n \in (\hat{0},m] \cap C(m')$ such that $m'$ and $n$ have complementary homologies, i.e., $\widetilde{H}_{a-2}((\hat{0},n),K)\neq 0$ and $\widetilde{H}_{b-2}((\hat{0},n),K)\neq 0$ for some $a, b>0$ with $k+2=a+b$.
We apply above result to derive consequences for the Betti numbers of monomial ideals.
This is an ongoing joint project with Sara Faridi, and Volkmar Welker.
March 24, 2026 Javier Gonzalez Anaya (Santa Clara University)
Abstract: We will discuss two novel moduli spaces of labeled points in flags of affine spaces. The first space parametrizes distinct weighted points up to translation and scaling. This construction provides a simultaneous generalization of the Hassett moduli spaces of weighted stable rational curves, as well as the Chen-Gibney-Krashen moduli space of point configurations in affine space modulo translation and scaling. The second moduli space allows points to collide freely, without any notion of equivalence between configurations. We will see that the first admits a toric compactification coinciding with the polypermutohedral variety of Crowley-Huh-Larson-Simpson-Wang, while the second is already toric and coincides with the polystellahedral variety of Eur-Larson. This is joint work with P. Gallardo and J.L. Gonzalez.
March 31, 2026 Sara Asensio Ferrero (Universidad de Valladolid)
Abstract: In 1990, R. H. Villarreal was the first to define edge ideals associated to graphs. In the same year, R. Fröberg characterized edge ideals with regularity 2, and since then, many authors have tried to generalize his results. In this talk, I will present a family of edge ideals with arbitrarily large regularity and projective dimension simultaneously, whose Betti diagrams have a special shape and which have emerged while working on the problem of characterizing edge ideals with regularity 4 (the first case that is still unknown). In particular, I will pay special attention to the combination of combinatorial and homological tools that has played an important role in this story. This talk is based on joint work with Ignacio García-Marco and Philippe Gimenez.
Abstract: A famous result of Stanley shows that every Artinian monomial complete intersection (CI) over a field of characteristic zero has the Strong Lefschetz Property (SLP). In positive characteristic, however, this result no longer holds. While Lundqvist-Nicklasson gives a complete characterization of monomial CIs with the SLP, the Weak Lefschetz Property (WLP) is still open. In this talk, I will provide a complete classification of monomial CIs with the WLP in characteristic 2, and discuss some current work to extend this classification to arbitrary p. This talk is based on joint work with C. Raicu, A. Kyomuhangi, and E. Reed.
Abstract: Algebraic shifting, introduced by Kalai in the 80's, is an operator that canonically associates a shifted complex to a given simplicial complex. The advantage of this operator is that it preserves many combinatorial, topological and algebraic properties of the starting complex and in doing so it translates the initial problem to a simpler instance. We show that among such properties is that of area rigidity, a generalization of graph rigidity, and that every triangulation of a surface with small genus is area rigid. For arbitrary surfaces we initiate a statistical study of the behavior of algebraic shifting, and in turn of area rigidity. We show that asymptotically almost surely the algebraic shifting of a random Delaunay triangulation of any given closed Riemannian surface is concentrated in a simplicial complex that depends only on the genus and the number of vertices. This talk is based on joint works with Eran Nevo and Yuval Peled.
Abstract: We review Klyachko's classification of torus equivariant vector bundles on toric varieties (toric vector bundles) in terms of combinatorial and linear algebra data. We then give a tropical reformulation of this classification which allows one to give a definition of a combinatorial object called "tropical vecor bundle" on a toric variety. It involves the notion of a matroid which I will briefly introduce. This is a joint work with Christopher Manon. We note that, independently and around the same time, Khan and Maclagan also have arrived at (basically) the same notion (of a tropical vector bundle).
Somethings which we will not have time to discuss: one defines equivariant K-theory and characteristic classes of these bundles. As a particular case, we can see that matroids come with tautological tropical toric vector bundles over the permutahedral toric variety and the corresponding equivariant K-classes and Chern classes recover the tautological classes of matroids constructed in the work of Berget-Eur-Spink-Tseng. In analogy with toric vector bundles, one can define sheaf of sections and Euler characteristic as well as positivity notions such as global generation, ampleness and nefness for tropical toric vector bundles and prove a kind of vanishing of higher cohomologies result.
Let E be an elliptic curve over the rationals given by an integral Weierstrass model and let P be a rational point of infinite order. The multiple nP has the form (A_n/B_n^2,C_n/B_n^3) where A_n, B_n, C_n are integers with A_nC_n and B_n coprime, and B_n positive. The sequence (B_n) is called the elliptic divisibility sequence generated by P. In this talk we answer the question posed in 2007 by Everest, Reynolds and Stevens: does the sequence (B_n) contain only finitely many perfect powers?
We discuss (mostly) recent work from several authors on epsilon and Hilbert Kunz multiplicities.
The classification of algebraic vector bundles of rank 2 over smooth affine fourfolds over algebraically closed fields is a notoriously difficult problem. In contrast to the situation in lower dimensions, rank 2 bundles over smooth affine fourfolds are no longer uniquely determined up to isomorphism by their Chern classes. In this talk, we survey classification results in low dimensions and analyze the cohomological obstructions for the set of isomorphism classes of rank 2 bundles with prescribed Chern classes over a fixed smooth affine fourfold to be finite (resp. a singleton). This enables us in many cases to enumerate isomorphism classes of rank 2 bundles with prescribed Chern classes; we can even completely classify rank 2 bundles over some concrete smooth affine fourfolds. The talk is based on joint work with Thomas Brazelton and Morgan Opie.
Title: An introduction to jets of graphs
The notion of jets comes from geometry, where it is used to study singularities. The space of jets of an affine variety can be constructed directly from the equations of the variety, and this procedure naturally extends to ideals in a polynomial ring. After reviewing this construction, we will examine how it applies to (squarefree) monomial ideals which leads to a new notion of jets for (hyper)graphs. Finally, we will explain how vertex covers of jet graphs are related to covers of the original graph.
Title: Mutations of Exceptional Collections on Projective Varieties
Utilizing recent joint work with Michael Brown, Souvik Dey, and Guanyu Li on computing extensions of bounded complexes of coherent sheaves on projective varieties, we present several applications involving computations in D^b(X), including manipulations of exceptional collections consisting of complexes and computing explicit monads based on them. We will also discuss related problems and conjectures from Helix theory in triangulated categories, particularly D^b(P^n).
Title: Gapfree graphs and powers of their edge ideals
Abstract: Powers of edge ideals often reflect the combinatorics of the underlying graph. Let $G$ be a graph and let $I(G)$ be its edge ideal. For gapfree graphs, a conjecture of Nevo–Peeva states that high enough powers of the edge ideal have linear resolutions. In this talk, I will describe our approach to this conjecture via a new conjecture involving linear quotients. Our conjecture is monotone: once one power $I(G)^q$ has linear quotients, then every higher power $I(G)^s$ (for $s\geq q$) should as well. I’ll present partial progress, including hypotheses under which the problem reduces to checking just the second power $I(G)^2$. It is known that forbidding a cricket, a diamond, and a 4-cycle forces $I(G)^q$ to have a linear resolution for all $q\geq 2$. So, one of the results we will discuss is about a construction of gapfree graphs that do contain those subgraphs (and a 5-cycle), yet still have linear quotients for every $q\geq 2$. The talk will be example-driven, with quick primers on linear resolutions and linear quotients, and open questions at the end. Joint work with Erey, Faridi, Hà, Hibi, and Morey.
Title: Square-free monomial ideals via open neighborhoods of graphs.
A key branch of commutative algebra, combinatorial commutative algebra, focuses on the study of square-free monomial ideals using combinatorial structures. Many techniques have been developed to analyze these ideals, particularly through the use of simplicial complexes and (hyper)graphs. This talk explores square-free monomial ideals through the lens of open neighborhood ideals of finite simple graphs. We will explore which types of tree graphs have a Cohen-Macaulay open neighborhood ideal, and how these ideals and their corresponding graphs arise in different algebraic or combinatorial settings. Furthermore, we will answer what types of square-free monomial ideals can be realized as the open neighborhood ideal of a finite simple graph.
Title: Long Live the King's Conjecture!
Abstract: King’s conjecture proposed that every smooth projective toric variety admits a full strong exceptional collection of line bundles - equivalently, a tilting bundle composed of line bundles. Although the original conjecture is known to fail in general, recent advances on resolution of diagonal toric varieties inspired by homological mirror symmetry suggest a new perspective: it can be proved that a birational reformulation of the King’s conjecture is indeed true!
In this talk, we will discuss the new “birational King”, and its new implications on the coherent-constructible correspondence and homological properties of Bondal-Thomson monads. This talk is based on joint works with Favero, and Ballard-Berkesch-Brown-Cranton Heller-Erman-Favero-Ganatra-Hanlon.
Title: An introduction to multivariate cryptography
Abstract: Multivariate cryptography is a promising candidate for post-quantum digital signatures. In practice, a multivariate cryptographic instance corresponds to a system of multivariate polynomial equations, so the security of such protocols depends on the complexity of solving these systems. In this talk, I will discuss two key invariants that play a central role in this context. Finally, I will examine the notion of generic systems as it appears in the cryptographic literature.
Title: Dynamics of Projectivized Toric Vector Bundles
Abstract: Understanding the dynamics (self-maps) of a variety can often reveal information about its geometry. For example, the only curves with non-trivial surjective endomorphisms are of genus zero or one. In this talk, I will discuss the dynamics of projectivized vector bundles, where a similar classifying phenomenon seems to arise: the existence of non-trivial surjective endomorphisms suggests that the bundle splits into a direct sum of line bundles. This is joint work with Javier González Anaya and Brett Nasserden.
Abstract: Symbolic powers of ideals in a Noetherian commutative ring have long been a topic of interest in commutative algebra. They are directly related to the associated primes, and always contain the ordinary powers. The symbolic defect function is a numerical function designed to measure the ``closeness” of symbolic powers and the ordinary powers. In particular, symbolic defect measures the minimal number of generators of the quotient of symbolic powers with ordinary powers. In this talk, we will introduce tools to help us study minimal generation of ordinary powers, symbolic powers, and symbolic defect, particularly for monomial ideals. We compare the latter function to a related invariant, the integral symbolic defect, and then discuss some recent results.
Abstract: Matrix Schubert varieties, introduced by Fulton in the '90s, are affine varieties that "live above" Schubert varieties in the complete flag variety. They have many desirable algebro-geometric properties, such as irreducibility, Cohen--Macaulayness, and easily-computed dimension. They also enjoy a close connection with the symmetric groups.
Alternating sign matrix (ASM) varieties, introduced by Weigandt just several years ago, are generalizations of matrix Schubert varieties in two senses: (1) ASM varieties are unions of matrix Schubert varieties and (2) the defining equations of ASM varieties are determined by ASMs, which are generalizations of permutation matrices. ASMs have been important objects of study in enumerative combinatorics since at least the '80s and appear in statistical mechanics as the 6-vertex lattice model. Although ASMs have a robust combinatorial underpinning and although their irreducible components are matrix Schubert varieties, they are nevertheless much more difficult to get a handle on than matrix Schubert varieties themselves. In this talk, we will define ASMs, compare and contrast with matrix Schubert varieties, and state some open problems.
Abstract: Many fundamental properties of projective varieties are encoded by their Hilbert functions, which record dimensions of the graded pieces of their coordinate rings. A variety is called Hilbertian if its Hilbert function is a polynomial. We will introduce these notions from the ground up before explaining the ubiquity of Hilbertian varieties in combinatorial commutative algebra via Stanley-Reisner theory. If time permits we will also discuss possible generalizations of these ideas to the multigraded setting.
Abstract: In this talk we shall present a duality for sequences of numbers which interchanges superadditive and subadditive sequences, and inverts their asymptotic growths. We shall discuss at least two algebro-geometric contexts where this duality shows up: how it interchanges the sequence of initial degrees of symbolic powers of an ideal of points with the sequence of regularities of a family of ideals generated by powers of linear forms; and how it underpins the reciprocity between the Seshadri constant and the asymptotic regularity of a finite set of points. This is joint work with Michael DiPasquale and Alexandra Seceleanu.
Abstract: In this talk I will introduce Barile-Macchia resolutions, which is a special type of resolutions for monomial ideals constructed via discrete Morse theory. These resolutions are minimal for many classes of monomial ideals, including edge ideals of weighted oriented forests and (most) cycles. I will also discuss recent follow-up work on this topic.
Abstract: We can associate to a simplicial complex of dimension d a tuple f =(f_0,f_1,...,f_d),where f_i counts the number of faces of the simplicial complex of dimension i. The tuple f is called the f-vector of the simplicial complex. In this expository talk, I will describe the Kruskal-Katona theorem, which characterizes what vectors can be the f-vector of a simplicial complex. If there is time, I will explain some connections to commutative algebra.
Abstract: The regularity and projective dimension of combinatorially-defined ideals are frequently studied invariants in combinatorial commutative algebra. In particular, much work has been done towards understanding the values these invariants can achieve for toric ideals I_G associated to a graph G. In this talk, we fully describe the possible values of these invariants for I_G as G ranges over all bipartite graphs on a fixed number of vertices. As a corollary, we show that any pair of positive integers can be realized as the regularity and projective dimension of a toric ideal of a bipartite graph. Finally, we demonstrate how our main result allows us to determine the values all five major invariants studied in the literature for this family of graphs.
Abstract: In this talk we will introduce an ideal associated to a graph, called a binomial edge ideals. These ideals were first introduced in 2010, with an application to conditional independence statements. Since then a lot of work has been done studying various homological invariants of these ideals. Here, we will define these ideals and give some results on their minimal primes and grobner basis. If time permits, we will talk about some results on the Betti numbers and regularity of these ideals.
Abstract: In Yitang Zhang’s proof of bounded gaps between primes (2014), a certain 3-variable Kloosterman sum played a crucial role and was one of the deepest parts of his proof. Further improvements on bounded gaps will likely require deeper understanding of this sum. This particular exponential sum was originally studied by J. Friedlander and H. Iwaniec (1985) with upper bound first obtained by B. Birch and E. Bombieri (1985) by special arguments. Further improvements were made by N. Katz (1986) using ℓ-adic method, and very recently by C. Chen and X. Lin (2022) by p-adic method. In an attempt to understand a remark made by Katz at the end of his 1986 paper, M. Roth and I were able to find a way that leads to a slight improvement on the upper bound of this sum, as well as on a family of similar exponential sums. Our method is based on ℓ-adic cohomology, consisting of finer studying of the local monodromies at zero via representations of the inertia groups there. In this talk, I will survey the above-mentioned results and give a high-level overview of our procedure that gives rise to this new improvement. This is joint work in progress with M. Roth.
Abstract: The Hilbert scheme of points in affine n-dimensional space, which parametrizes zero-dimensional subschemes with a fixed degree, is a fundamental parameter space in algebraic geometry. Quot schemes are a generalization of Hilbert schemes, parametrizing finite length quotients of a locally free sheaf. We will explore some interesting phenomena and problems about these spaces that are specific to the three-dimensional case, focusing on the tangent space and recent progress on the parity conjecture of Okounkov and Pandharipande. This is joint work with Alessio Sammartano.
Abstract: In this talk, after a review of the definition of and facts about Kaehler differentials, I will give some history behind the classic Lipman-Zariski Conjecture and the generalized Lipman-Zariski questions of Graf. Then I’ll give some results on the torsion and cotorsion of exterior powers of the module of Kaehler differentials over complete intersection rings, and say how these are used to prove a generalized Lipman-Zariski result under certain conditions. This is joint work with Sophia Vassiliadou.
Abstract: Schubert determinantal ideals are a class of generalized determinantal ideals which include the classical determinantal
ideals. In this talk, we use the approach of "Grobner basis via linkage" to give a new proof of a well known result of Knutson and Miller: the essential minors of every Schubert determinantal ideal form a Grobner basis with respect to a certain term order. We also adapt the Grobner basis via linkage technique to the multigraded setting and use this to show that the essential minors of every Kazhdan-Lusztig ideal form a Grobner basis with respect to a certain term order, thereby giving a new proof of a result of Woo and Yong.
Abstract: In a recent pre-print (arXiv:2008.13656), Benjamin Hoffman and I extended some earlier results of Harada and Kaveh about toric degenerations and gradient-Hamiltonian vector fields. Our main application is to symplectic manifolds equipped with Hamiltonian group actions.
Abstract: I will discuss recent work of Favacchio, Van Tuyl and myself which demonstrates that we may construct toric ideals associated with graphs which have regularity r and h-polynomials of degree d for any 4 \leq r\leq d. The ideas involved rely on Gröbner bases and combinatorics. I will also discuss how we can use this to recover a result of Hibi, Higashitani, Kimura, and O’Keefe.
Abstract: Many geometric properties of Schubert varieties in the full flag manifold GL_n/B can be described using pattern avoidance or interval pattern avoidance conditions. These convenient characterizations reduce difficult computations to simple combinatorics involving permutations. For example, a Schubert variety associated to a permutation w is smooth if and only if w pattern avoids 3412 and 4231. Similar descriptions exist for checking factoriality or Gorensteinness, among other properties.
I will start by providing any necessary background on Schubert varieties followed by an overview of known results regarding pattern avoidance. I will also discuss newer research on a combinatorial subword description for the Gorenstein property.
Abstract: Harmonic metrics on vector bundles mediate the nonabelian Hodge correspondence between categories of irreducible representations of fundamental groups and semi-stable Higgs bundles. By uniformizing the underlying base one obtains a description of harmonic metrics as smooth matrix-valued automorphic forms satisfying a nonlinear PDE. The main step in establishing the nonabelian Hodge correspondence lies in proving the existence of solutions to these PDEs.
In this talk we will discuss a construction of (families of) metrics for bundles on hyperbolic curves that is inspired by the theory of Eisenstein series. We will discuss cases where harmonic metrics can be obtained explicitly as residues of these series. This talk will be partly expository and will assume no prior knowledge of the subject. In particular, we will begin with a summary of basic facts on the Riemann zeta function and classical Eisenstein series.
Abstract: Geometric vertex decomposition (a degeneration technique) and liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties. In this talk, I will review these two techniques and explain why they are useful for studying certain types of questions in algebraic geometry and commutative algebra. I will then discuss recent joint work with Patricia Klein on an explicit connection between geometric vertex decomposition and liaison, as well as applications of this connection.
Abstract: For an ideal I in a commutative Noetherian ring we can define its symbolic power denoted by I^(n). We will discuss how the symbolic power compares to the ordinary power (i.e. I^n) in the context of polynomial ideals and their associated varieties. The Waldschmidt constant is an invariant of I which measures the growth of the I^(n) relative to I^n as n increases. We will demonstrate how the Waldschmidt constant can be computed as the value of a linear program when I is a monomial ideal.