Finite-Dimensional DG Algebras and their Properties
Dr. Dmitri Orlov
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
9:00am - 10:00am
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Abstract: The talk will focus on finite-dimensional DG algebras and categories of perfect complexes over such DG algebras. These categories can be considered as proper derived noncommutative schemes. We are going to discuss basic properties of these noncommutative schemes and to establish a connection between such categories and DG categories with (semi)exceptional collections.
Wall Finiteness Obstruction for DG Categories I & II
Dr. Alexander Efimov
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
10:15am - 11:15am
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Abstract: The classical Wall finiteness obstruction theorem gives a criterion where by a finitely dominated CW complex is homotopy equivalent to a finite CW complex. I will explain that an analogue of this result holds for DG categories over a field.
Namely, a homotopically finitely presented DG category is Morita equivalent to a finite cell DG category iff the class of its diagonal bimodule in $K_0$ is generated by the classes of (box) tensor products of left and right perfect modules. Moreover, finite cell DG categories are (up to Morita equivalence) exactly the quotients of proper DG categories with full exceptional collections, by a subcategory generated by a single object.
As an application, we show that any smooth and proper phantom DG category can be embedded fully faithfully into a proper DG category with a full exceptional collection. The same holds for the derived category of the Barlow surface. This disproves two conjectures of Orlov (since the Barlow surface is not rational). We also obtain that for any smooth proper algebraic variety with a stratification by affine spaces, its derived category admits a fully faithful embedding as above.
We will also sketch the proof of a similar result (Wall finiteness obstruction) for algebras over (colored) DG operads. In this case the class of the diagonal should be replaced by the class of the cotangent complex.
Rigid, Not Infinitesimally Rigid Surfaces of General Type with Ample Canonical Bundle – Lecture II
Dr. Christian Böhning
University of Warwick
2:00pm - 3:00pm
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Abstract: In the talk I will report on work in progress, joint with Roberto Pignatelli and Hans-Christian von Bothmer, that concerns the construction of surfaces of general type with ample canonical bundle and Kuranishi space (and possibly also Gieseker moduli space) a non-reduced point. The main tools are configurations of lines and their incidence schemes as well as the theory of abelian covers due to Pardini and others.
I apologize for the thematic remoteness from the main topic of the conference; however, nearly rational varieties help in the construction, and the topic is fun!
Rationality and Exceptional Collections for Toric Varieties
Dr. Matthew Ballard
University of South Carolina
3:30pm - 4:30pm
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Abstract: A conjecture of Orlov asks whether all varieties with full exceptional collections are rational. We will discuss two interpretations of this question over non-closed fields. The strict one, where End(E)=k is the definition of exceptional for X over k, holds for toric varieties over a general field. The looser one, End(E)/k is a separable extension of fields, does not guarantee the existence of a k-point much less rationality. The work discussed is joint with Alexander Duncan, Alicia Lamarche, and Patrick McFaddin.
Holomorphic Bisectional Curvature and Applications to Deformations and Rigidity of Mixed Hodge Structure
Dr. Gregory Pearlstein
Texas A&M University
4:45pm - 5:45pm
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Abstract: The curvature properties of period domains play a fundamental role in Hodge theory. In this talk, I will discuss recent work with Chris Peters on Hodge metrics arising in mixed Hodge theory, and applications to rigidity results for geometric classes of examples.
