Schedule for: 21w5509 - Derived, Birational, and Categorical Algebraic Geometry

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

A Kleinian surface singularity can be `resolved' by a smooth orbifold, and the McKay correspondence famously relates this resolution with the more geometric minimal resolution. In type A it is easy to write down a VGIT problem that interpolates between the orbifold and the minimal resolution, but this has never been done for types D and E. I'll describe some work-in-progress with Tarig Abdelgadir where we fill this gap using a construction inspired by Tannaka duality. I'll also discuss some potential applications.

I will discuss forthcoming work with Matthew Ballard on using the derived category of coherent sheaves to detect the existence of rational points on a particular family of arithmetic toric Fano varieties. More precisely, I will explain how we show that a member of this family of varieties is rational if and only if its bounded derived category of coherent sheaves admits a full étale exceptional collection.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

A delightful special property of toric varieties is that the Frobenius morphism from positive characteristic naturally makes sense for any power in any characteristic. Moreover, the pushforward of the trivial sheaf splits into a direct sum of line bundles. We discuss a combinatorial proof that these line bundles generate the bounded derived category of coherent sheaves for a smooth toric stack (the case for proper smooth varieties is sometimes called "Bondal's conjecture”).

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will discuss new applications of equivariant Burnside groups, introduced in joint work with Kontsevich-Pestun and developed with A. Kresch.

The Bernstein-Gel’fand-Gel'fand correspondence, Beilnson’s resolution of the diagonal, and the Eisenbud-Floystad-Schreyer theory of Tate resolutions form a collection of powerful tools for the homological study of projective space and the graded polynomial ring. I will give an overview of some extensions of these ideas to toric varieties and multigraded polynomial rings, including recent joint work with Michael Brown.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Dimitrov—Haiden—Katzarkov—Kontsevich introduced the notion of categorical dynamical system and that of its associated entropy, which is a function from the real line to the extended real line. The value at zero of this function takes the name of categorical entropy and it is the categorical analogue of the topological entropy. It is known that the categorical entropy is always a positive (possibly infinite) real number, and it has an explicit lower bound. The question of when this lower bound is attained has attracted some interest, and it is intriguing if one looks at it via the philosophy of homological mirror symmetry. In this talk I will report on joint work with Jongmyeong Kim in which we propose a categorical interpretation of (one of the properties of) holomorphicity, and we show that this property ensures that the lower bound is attained (at least for mass-growth of objects).

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will be reporting on ongoing work joint with Sarah Frei where we have characterized the Galois action on a portion of the middle l-adic \'etale cohomology of hyperk\"ahler 4-folds of Kummer type by finding symplectic involutions on these 4-folds. I will also discuss the geometry of the fixed loci in an example and relationships between the construction and derived equivalence. This is an extension of work by Hassett and Tschinkel on generalized Kummer 4-folds.

The bounded derived category has proved to be a useful invariant for investigating rationality of low-dimensional varieties defined over algebraically closed fields. Less is known for varieties defined over non-closed fields, as the derived category has trouble distinguishing between isomorphism classes of some twisted forms. In this talk we will discuss a Galois-theoretic invariant of a reductive algebraic group G, defined in terms of separable algebras with G-action. For (twisted forms of) algebraic tori, this invariant detects retract rationality and leads to examples of varieties which are derived-equivalent to smooth points but which are not rational. This is joint work with M. Ballard, A. Duncan, and A. Lamarche.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will introduce a skeleton obtained directly from monomial relations in a finite quiver without cycles, and relate the construction to some classical examples in mirror symmetry. This is work in progress with David Favero.

Golod rings are rings for which the associated Koszul homology admits a trivializing higher order homology operation, known as a trivial Massey operation. Such rings are extremal objects from the perspective of commutative algebra and have more recently warranted closer study for their connections with moment-angle complexes in combinatorial algebraic topology. In this talk, we will introduce characteristic-free techniques for proving rings are Golod by deforming arbitrary quotient rings to those defined by monomial relations, and study conditions for which the associated Grobner degenerations induce isomorphisms on homology algebras. In particular, we give a concise proof of the fact that (any power of) ideals of maximal minors define Golod rings in arbitrary characteristic, a fact that was previously only known in characteristic 0.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

for a pair (X, Z) consisting of an algebraic variety X and a closed subset Z with smooth complement, we consider a birational morphism Y --> X resolving Z to a divisor D with simple normal crossings. We show that, under reasonable assumptions, the pushforward of logarithmic differential forms to X is independent of the choice of Y. Using recent work of Arkhipov-Ørsted we indicate how this gives braid relations for the affine Hecke category (proved earlier by Bezrukavnikov and Riche). We plan to finish with conjectural discussion of a triangulated category giving rise to the constructed sheaf on X via the Hochschild homology formalism.

In this talk we define some new birational invariants

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will review some recent results on existence and obstructions for semiorthogonal decomposition of varieties with ordinary double points. The constructions originate from geometry of blow ups and Sarkisov links, while the obstructions come from the negative K-theory groups. The theory is primarily developed for nodal Fano threefolds, with the index two case completely understood. The talk is based on several papers joint with Kalck, Karmazyn, Kuznetsov, Pavic.

I will give an overview of the theory of Categorical Enumerative Invariants (CEIs), as developed in joint work with Junwu Tu and Kevin Costello. The emphasis will be on the role played by the Givental group, which controls the effect of changes of splitting on the CEIs. As an application I will discuss a conjectural B-model LG/CY correspondence for elliptic curves, inspired by work of Li-Shen-Zhou in the A-model. The results presented are joint work with Yunfan He.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.