Program

Deformation theory formalizes the idea of "varying parameters" in an algebraic or geometric object such as an associative algebra, a variety, or a triangulated category. In his ICM 2002 address, P. Seidel showed how partial compactifications of symplectic manifolds could be understood in terms of A∞ deformations of Fukaya categories, sketching a programme for studying the Fukaya category of a projective variety through deformations of the (wrapped) Fukaya category of an affine subvariety.

In such a context, deformation theory generally presents several obstacles, of both theoretical and practical nature. I will explain how for partially wrapped Fukaya categories of surfaces, all of these obstacles can be overcome. Moreover, all abstract A∞ deformations can be explained in terms of partial orbifold compactifications of the underlying surface, giving rise to a theory of Fukaya categories of orbifold surfaces as well as clarifying the role of stop data in Seidel's programme.

This talk is based on joint work with Sibylle Schroll and Zhengfang Wang.

LECTURE 1. NODAL ORDERS
Nodal orders provide a natural noncommutative generalization of the ring A=k[[x,y]]/(xy), where k is a field. In 1968, Gelfand and Ponomarev — and, shortly thereafter and independently, Nazarova and Roiter — classified all indecomposable finite-dimensional A-modules, thereby proving in particular that A is of tame representation type. In 1990, Drozd showed that nodal orders are precisely the pure Noetherian k-algebras of tame representation type. Later, in 2002, Drozd and I proved that nodal orders are also derived-tame. In this lecture, I will explain the main ring-theoretic properties of nodal orders, as well as their classification over both algebraically closed fields and the field of real numbers.

LECTURE 2. TAME NONCOMMUTATIVE NODAL CURVES AND RELATED FINITE-DIMENSIONAL ALGEBRAS
Tame noncommutative nodal projective curves can be viewed as the “global” counterparts of nodal orders. In this talk, I shall discuss the definition and construction of (tame) noncommutative nodal curves and explain their classification over an algebraically closed field. I shall then describe the construction of the Auslander curve associated with a noncommutative nodal curve, which itself turns out to be nodal again. The derived category of an Auslander nodal curve admits a distinguished tilting object whose endomorphism algebra can be computed explicitly. In the tame case, this approach yields special classes of gentle and skew-gentle algebras providing an interpretation of noncommutative nodal curves as the B-side of homological mirror symmetry for graded oriented (orbifold) surfaces with marked boundaries.

Cohen-Macaulay modules over a Gorenstein ring provide a canonical enhancement of the singularity category. Tilting theory offers a modern framework for studying Cohen-Macaulay representation theory. In this talk, based on joint works with Buchweitz, Yamaura, Kimura, and Ueyama, we present a detailed study of (Artin-Schelter) Gorenstein algebras A of dimension one. We show that the generically projective Z-graded singularity category of A admits a silting object if and only if the degree zero part A_0 has finite global dimension. Moreover, it admits a tilting object if and only if either A is regular or the average Gorenstein parameter g of A is non-positive. We also give explicit constructions of silting objects, and illustrate the results with examples including noncommutative quadrics and tiled orders.

In 2003, Eisenbud and Schreyer introduced Ulrich bundles as geometric analogues of maximal Cohen-Macaulay modules having the maximal number of generators. In the same paper they proposed two questions: (1) whether a given projective variety X embedded in a projective space supports an Ulrich sheaf; (2) if yes, what is the smallest possible rank of it. The first question is nowadays often called the Eisenbud-Schreyer conjecture, in a connection with the study of Boij-Soederberg cones. The second question is called the Ulrich complexity problem, which is widely open even for smooth cubic hypersurfaces. In this talk, we first discuss how to construct an Ulrich bundle of rank 6 on a smooth cubic fourfold X, by using deformation theory on the Kuznetsov component of X. And then we also discuss a computational approach to achieve Ulrich bundles of various ranks on some special cubic fourfolds. A part of the talk is based on a joint work with Daniele Faenzi.

I will explain how to compute autoequivalence groups of derived categories of so-called gentle algebras, a class of algebras which have been studied since the 1980s and which eventually were given a geometric interpretation in the context of Fukaya categories of surfaces in work by Bocklandt and Haiden-Katzarkov-Kontsevich some years ago. In the first lecture, I will give an overview of how topological models and results from surface topology can be combined to study the derived autoequivalence groups of Fukaya categories, derived categories of gentle algebras and stacky nodal curves. My second talk will then explain how this all naturally leads to a very general analogy between Lie groups, Lie algebras, autoequivalence groups and Hochschild cohomology.

TBA

Starting from the Weierstrass elliptic function, we study the associated Frobenius structure, incorporating the perspective of derived categories, particularly that of homological mirror symmetry. In particular, we examine the relationship between the degree of the Lyashko–Looijenga map modulo the modular group and the number of full exceptional collections up to the braid group action and translations, as well as the associated Gamma-integral structure.

It was expected for some time that the infinity-category of A-infinity categories should be equivalent to the infinity-category of dg categories, over any fixed base ring R. I will discuss a proof of this result, along with applications such as the computation of mapping spaces invariant under quasi-equivalences, tensor products, and formal properties. A central role is played by a cofibrancy condition (of having homotopically projective hom spaces) for A-infinity categories.