C.A.T.S.:

Ayah Almousa: Standard monomials and Gröbner bases for positroid varieties

Abstract: One way to probe the structure of an algebraic variety is to understand the anatomy of its coordinate ring at each degree, which can be achieved by constructing a Gröbner basis. Influential work of Hodge in the 1940s paved the way for using Gröbner bases to combinatorially study the Grassmannian. In this talk, we will follow Hodge’s approach in order to investigate certain subvarieties of the Grassmannian called positroid varieties. Introduced by Knutson–Lam–Speyer in 2013, positroid varieties provide a stratified decomposition of the Grassmannian into subvarieties which enjoys many advantages over other previously studied decompositions. We will see that a Gröbner basis approach to investigating positroid varieties has powerful applications in algebra, geometry, and combinatorics. This is joint work with Shiliang Gao and Daoji Huang.

Yairon Cid-Ruiz: Log-concavity of polynomials arising from equivariant cohomology

Abstract: A remarkable result of Brändén and Huh tells us that volume polynomials of projective varieties are Lorentzian polynomials. The dual notion of covolume polynomials was introduced by Aluffi by considering the cohomology classes of subvarieties of a product of projective spaces. In this talk, we shall address the equivariant cohomology classes of torus-equivariant subvarieties of the space of matrices. For a large class of torus actions, we shall show that the polynomials representing these classes (up to suitably changing signs) are covolume polynomials in the sense of Aluffi. If time permits, we shall present a description of the cohomology rings of smooth complex varieties in terms of a generic Macaulay inverse system over the integers. This is based on joint work with Yupeng Li and Jacob Matherne.

Kyungyong Lee: Positivity of generalized cluster algebras

Abstract: This is joint work with Amanda Burcroff and Lang Mou. The generalized cluster algebras form a large subclass of commutative algebras. Each generalized cluster algebra is equipped with distinguished generators called cluster variables. It has been conjectured that these cluster variables are Laurent polynomials with positive coefficients. We prove this conjecture. No background on cluster algebras is needed.

Justin Lyle: Vanishing of Tor and Depth of Tensor Products Over Cohen-Macaulay Rings

Abstract: Let R be a commutative Noetherian local ring and let M,N be finitely generated-modules. We say R satisfies the condition (dep) if the depth formula depth_R(M ⊗ N)+depth(R)=depth_R(M)+depth_R(N) holds whenever Tor^R_i(M,N)=0 for all i greater than 0. Recent work of Kimura-Lyle-Otake-Takahashi shows this condition need not hold in general, and in fact characterizes the so-called AB property for Gorenstein rings of positive dimension. Replacing the equality in the above formula by either the inequality > or the inequality <, the (dep) condition bifurcates into two conditions which we call (ldep) and (rdep), respectively. We show when R is Cohen-Macaulay that (ldep) is tied to the so-called uniform Auslander condition, and that (ldep) implies (rdep) for Gorenstein rings of positive dimension. We also describe how these conditions behave under standard operations in commutative algebra, such as localization, completion, and modding out by a regular sequence.

Prashanth Sridhar: Differential Graded Noncommutative Geometry

Abstract: Pioneering work of Artin-Tate-Van den Bergh-Zhang extends important aspects of projective geometry to the noncommutative (nc) setting. In particular, the derived category of a nc scheme shares many features with the derived category of a classical one. In this talk, I'll discuss an ongoing program to extend classical and modern results in the theory of nc projective geometry to nc spaces associated to differential graded algebras. The focus will be on its applications to projective varieties: for instance, this approach results in a generalization of a celebrated theorem of Orlov concerning the derived category of a projective complete intersection.

William D. Taylor: New Variations on Frobenius and Integral Closure

Abstract: Inspired by theories in algebraic geometry, N. Hara and K.-I. Yoshida introduced "a^t-tight closure", a variation on the classical tight closure of an ideal I involving another ideal a and a positive real number t. The Hara-Yoshida constructions fail to be closures in general because they are not idempotent. Later, A.\ Vraciu introduced a similar construction which gives true closure operations on ideals. Inspired by these variations on tight closure, we define four similar constructions which are adjustments of the Frobenius closure and integral closure of ideals, using the Hara-Yoshida and Vraciu patterns for each. We show that the arising operations have many good properties and even in simple cases give closures which are distinct from the previously studied ones. We show further that these closures can be effectively computed for monomial ideals in semigroup rings. This is based on joint work with Kriti Goel and Kyle Maddox.

Kevin Tucker: Plus-pure thresholds of some cusp-like singularities

Abstract: The log canonical threshold (lct) is an important numerical invariant of singularities in complex algebraic geometry, with analytic origins. Via standard reduction to characteristic p techniques, it is closely related to the F-pure threshold in positive characteristic defined in terms of the Frobenius endomorphism. These equal characteristic thresholds admit an analogue in the developing theory of singularities in mixed characteristic, which is known as the plus-pure threshold. In this talk, I will review these notions and discuss a computation of the plus-pure thresholds of some mixed characteristic cusp-like singularities (such as p^2+x^3 in Z_p[[x]]). This talk is based on joint work with Hanlin Cai, Suchitra Pande, Eamon Quinlan-Gallego, and Karl Schwede.