Combinatorics seminar (Fridays 3:35)
Student Combinatorics and Algebra Seminar (Thursdays 4:40)
Topology Seminar (Mondays at 2:30)
Syzygies and Mirror Symmetry Virtual Seminar (Thursdays at 1:00)
Landau-Ginzburg (LG) models consist of the data of a quotient stack X and a regular complex-valued function W on X. Here, geometry is encapsulated in the singularity theory of W. One can find that LG models are deformations of Calabi-Yau complete intersections in some sense. Exoflops essentially create new GIT problems of partial compactifications of X, expanding the tractable birational geometries related to a given Calabi-Yau complete intersection. We will explain this technique, provide some foundational results about this and then some new applications proven recently for Calabi-Yau orbifolds. This talk contains results from a series of joint works with D. Favero (UMinn), C. Doran (Bard/Alberta), A. Malter (BIMSA).
A landmark theorem of Orlov relates the singularity category of a graded Gorenstein algebra to the derived category of the associated noncommutative projective scheme. I will discuss a generalization of this theorem to the setting of differential graded algebras. This is joint work with Prashanth Sridhar.
Ehrhart theory is about convex polytopes P with vertices in the integer lattice, and the number of lattice points in their integer dilates mP. It explains why this number grows as a polynomial in m of degree d=dim(P). Its methods often use either combinatorics, or use an associated affine semigroup ring, which turns out to be Noetherian, Cohen-Macaulay, and has an easily identified canonical module. This talk will discuss a new q-deformation of this story (joint with B. Rhoades; see arxiv:2407.06511), replacing the number of lattice points in mP with a polynomial in q, and giving analogous conjectures on its behavior. These conjectures would follow from the properties of a new commutative ring, conjectured to be Noetherian, Cohen-Macaulay and with an analogous description for its canonical module.
The Coherent-Constructible Correspondence provides a categorical equivalence between the complexes of coherent sheaves on toric varieties and complexes of certain constructible sheaves on the mirror torus. In this talk, I will explain how to use this equivalence and the stratification of the mirror torus introduced by Bondal to relate the topology of the torus to sheaf resolutions in geometry and algebra. This talk is based on my work with David Favero (https://arxiv.org/pdf/2411.17873).
To any category we can assign an associative algebra known as a category algebra. Various types of algebras that naturally arise in (non)commutative algebra and algebraic geometry such as polynomial rings, incidence algebras, matrix algebras, and homotopy path algebras are examples of category algebras. In this talk, I will establish a connection between the Koszulity of a certain class of category algebras and the locally Cohen-Macaulay property. Next, I will introduce a class of functors preserving the locally Cohen-Macaulay property (and hence Koszullity). As applications,I will generalize a result of Reiner and Stamate and discuss Koszulity for full strong exceptional collections of line bundles on toric varieties. This is based on joint work with David Favero.
A toric vector bundle on a toric variety is a vector bundle endowed with a torus action compatible with the one on the underlying space. Almost all familiar examples of bundles on toric varieties are toric, from (co)tangent bundles to kernel bundles associated to complete linear series, which control syzygies of the corresponding embedding. In contrast to the case of line bundles, these are not purely combinatorial objects, as they also depend on linear algebraic information associated to cones known as the Klyachko data. We will discuss a new effective vanishing result for twists of toric vector bundles by line bundles, taking into account both the linear algebra and combinatorics via Batyrev’s primitive relations. This vanishing recovers many familiar results, and our techniques lead to new ways to study N_{p} properties for smooth projective toric varieties. This is joint work in progress with Gregory G. Smith.
We discuss, by example, the birational geometry of smooth projective toric varieties and describe how it interacts with the construction of the resolution of the diagonal due to Hanlon--Hicks--Lazarev. Ultimately, we construct a category above all birational models of such a toric variety, where a version of this resolution enjoys strong properties. This yields a birational interpretation of King's Conjecture, the subject of next week's talk. This is joint work with Matthew Ballard, Michael Brown, Lauren Cranton Heller, Daniel Erman, David Favero, Sheel Ganatra, Andrew Hanlon, and Jesse Huang.
King's conjecture proposed that sheaves on projective toric varieties can be understood using noncommutative algebra. Or more formally, that the derived category of a toric variety is equivalent to modules over a noncommutative ring (without proposing a specific ring). While this conjecture has been proven in many cases, it turns out that it is false in general with counterexamples by Hille-Perling, Efimov, and Michalek. Nevertheless, I will discuss a proof of King's conjecure (after some modification of the statment). This is a sequel to Christine's talk based on joint work with Matthew Ballard, Michael Brown, Lauren Cranton Heller, Daniel Erman, David Favero, Sheel Ganatra, Andrew Hanlon, and Jesse Huang.
Algebraic tori are widely studied due to their connection with Galois cohomology and other algebraic invariants. The connection with Galois cohomology gives us a bijection between Severi-Brauer varieties and central simple algebras. Blunk has given a classification of the forms of two-dimensional tori in del Pezzo surfaces of degree 6 in terms of separable algebras over an arbitrary field F. Duncan has proved that there exists an injective map from the forms of retract rational tori to isomorphism classes of separable algebras but there is no explicit description of the image of this map as for Blunk. In this talk, we discuss how to use the structure of Mackey functors to address this question.
Alternating sign matrices (ASMs) form the MacNeille completion of the strong Bruhat order on the symmetric group. There is a natural interpretation of this poset as the containment order on ASM varieties, which are generalized determinantal ideals. In 2018, Hamaker and Reiner defined weak Bruhat order on ASMs, which when restricted to the symmetric group, is the usual weak order. We initiate a geometric study of weak order on ASMs varieties, focusing on how combinatorial properties of this poset describe geometric properties of ASM varieties. This is joint work with Laura Escobar and Patricia Klein.
In applications of algebraic geometry including algebraic statistics, rigidity theory, and the study of chemical reaction networks, the algebraic matroid of a variety contains important data about the independence of sets of variables. In both algebraic statistics and rigidity theory, secant varieties play an important role. I'll discuss the relationship between the algebraic matroids of a variety and its secant varieties. I'll emphasize the motivating example, the Cayley-Menger varieties, and also apply our results to toric varieties. This is joint work with Fatemah Mohammadi and Louis Theran.
I will explain how perverse sheaves can be used to produce simpler proofs of stronger versions of known results and to prove some new results on local cohomology. This seems to be the first application of perverse sheaves to commutative algebra. Joint work with Bhargav Bhatt, Manuel Blickle, Anurag Singh and Wenliang Zhang.