Commutative Algebra Seminar

Log Gromov-Witten invariants, introduced by Abramovich-Chen-Gross-Siebert, are counts of curves in pairs (X,D) consisting of a smooth projective variety X together with a normal crossing divisor D, with prescribed tangency conditions along D. These invariants play a key role in mirror symmetry for log Calabi-Yau pairs (X,D), in which case D is an anticanonical divisor. After briefly reviewing log Gromov-Witten theory, I will explain a combinatorial recipe based on tropical geometry and wall-crossing algorithms to calculate such curve counts when (X,D) is obtained as a blow-up of a toric variety along hypersurfaces in the toric boundary divisor. This is based on joint work with Mark Gross.

A fundamental problem at the confluence of algebraic geometry, commutative algebra, and representation theory is to understand the structure and vanishing behavior of the cohomology of line bundles on (partial) flag varieties. I will describe an answer in the case of the incidence correspondence (the partial flag variety consisting of pairs of a point in projective space and a hyperplane containing it), and highlight surprising connections to other questions of interest: the splitting of jet bundles on the projective line, the Han-Monsky representation ring, or Lefschetz properties for Artinian monomial complete intersections. This is based on joint work with Annet Kyomuhangi, Emanuela Marangone, and Ethan Reed.

In this talk, I will discuss some examples of the relative Langlands duality (introduced by Ben-Zvi-Sakellaridis-Venkatesh) for strongly tempered spherical varieties. In some cases, I will introduce a relative trace formula comparison and prove the fundamental lemma/smooth transfer. This is a joint work with Zhengyu Mao and Lei Zhang.

Supercuspidal representations are the mysterious "elementary particles" from which all other representations of a reductive p-adic group are built. Residue characteristic two presents additional difficulties in the construction of these representations, and even for classical groups, our knowledge is incomplete. In this talk, based on joint work with Jessica Fintzen, I'll explain how to overcome one of these difficulties: the exceptional behavior of the Heisenberg group and Weil representation in characteristic two. Time permitting, I'll also explain how to overcome a second difficulty: disconnected Lie-algebra centralizers.

Consider the conjugation action of GL_2(K) on the polynomial ring K[X]. When K is an infinite field, the ring of invariants is a polynomial ring generated by the trace and the determinant. We describe the ring of invariants when K is a finite field, and show that it is a hypersurface.

The derived category of modules over a commutative ring captures many properties of the ring. One approach is to study its triangulated structure through finite building. An object $X$ finitely builds an object $Y$, if $Y$ can be obtained from $X$ by taking cones, suspensions and retracts. The $X$-level measures the number of cones required in this process. This recovers various classical invariants as projective dimension and Loewy length. I will explain the behavior of level with respect to tensor products in an enhanced triangulated category. I will further present applications to Koszul objects, which generalize Koszul complexes. This is joint work with Marc Stephan.

Matlis duality for modules over commutative rings gives rise to the notion of Matlis reflexivity. In my talk I will discuss basic properties, adding some new perspectives on a classical subject. For instance, I'll explain that Matlis reflexive modules form a Krull-Schmidt category. For noetherian rings the absence of infinite direct sums is a characteristic feature of Matlis reflexivity. This leads to a discussion of objects that are extensions of artinian by noetherian objects. Also, classifications of Matlis reflexive modules for some small examples are discussed.

TT geometry associates a geometric invariant - the Spectrum - to a tensor triangulated category. This invariant carries a lot of global structure information about the category; but the calculations are usually difficult. I'll describe “the fiber functor” technique which proved to be successful for calculating the spectrum in various settings. Motivations, as often happens, comes from commutative algebra, and examples include finite group schemes, Lie superalgebras, quantum groups, and even old fashioned modular representation theory of finite groups. If time allows, I'll also mention one recently constructed family of tensor triangulated categories where the fiber functor technique is destined to fail basically by definition.

Suppose R -> S is a surjective map of local Noetherian rings. In this talk I will discuss a notion of lifting S-modules to R-modules along this map. Classically, we say an S-module M lifts to an R-module M' if M' is isomorphic to M upon extending scalars via R -> S, and Tor_i(M', R) = 0 for i > 0. It turns out, it is interesting to consider a much weaker and more geometric notion of lifting modules. Instead of asking the higher Tors to vanish, we require that the lift M' of M is of the "correct codimension," that is, dim R - dim M' = dim S - dim M. This is mostly joint work with Andrew Soto Levins and partly ongoing joint work with Kesavan Mohana Sundaram, Ben Katz, and Ryan Watson.

I will describe a new construction of the spectral sequence of a filtration which makes transparent certain properties like multiplicativity. The construction uses an old t-structure on filtered complexes, the Beilinson t-structure, which I will explain in detail.

Lifting theory was studied by Auslander, Ding, and Solberg for modules and by Yoshino for complexes. Further progress on this theory has been made recently (in the works of Nasseh, Ono, Sather-Wagstaff, and Yoshino) in the context of differential graded (DG) modules in order to obtain a clearer insight on some major problems in commutative algebra. In this talk, I will survey recent developments on the lifting theory of DG modules and describe a relationship between this notion and a DG version of the notion of connections. This is an in-progress joint work with Maiko Ono and Yuji Yoshino.