Program

TBA

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.

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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.

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