The ICRA 2012 workshop was held from Wednesday, 08 August, to Saturday, 11 August 2012, in the lecture halls H4 and H7 in the main university building (UHG) of Bielefeld University (maps are available on the Travel Information page). It consisted of series of lectures given by the invited speakers:
All lectures were held in lectures hall H4 and H7 in the main university building (UHG).
| Wednesday 08 August |
Thursday 09 August |
Friday 10 August |
Saturday 11 August |
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|---|---|---|---|---|---|---|
| 9:00 | – | 10:00 | Registration | Orlov Lecture 1 |
Mozgovoy Lecture 2 |
Iyama Lecture 3 |
| 10:15 | – | 11:15 | Angeleri Hügel Lecture 1 |
Angeleri Hügel Lecture 2 |
Benson Lecture 3 |
Mozgovoy Lecture 3 |
| Coffee break | ||||||
| 12:00 | – | 13:00 | Benson Lecture 1 |
Benson Lecture 2 |
Angeleri Hügel Lecture 3 |
Orlov Lecture 3 |
| Lunch break | ||||||
| 15:00 | – | 16:00 | Brundan Lecture 1 |
Brundan Lecture 2 |
Orlov Lecture 2 |
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| Coffee break | ||||||
| 16:30 | – | 17:30 | Iyama Lecture 1 |
Iyama Lecture 2 |
Brundan Lecture 3 |
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| 17:45 | – | 18:45 | Mozgovoy Lecture 1 |
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| 19:00 | – | Welcome reception |
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Infinite dimensional tilting modules are abundant in representation theory. They occur when studying torsion pairs in module categories, when looking for complements to partial tilting modules of projective dimension greater than one, or in connection with the Homological Conjectures. They share many properties with classical tilting modules, but they also give rise to interesting new phenomena. For example, they induce localizations of derived categories rather than derived equivalences. Moreover, they often correspond to localizations of module categories or of categories of quasi-coherent sheaves.
In my talks, I will review the main features of infinite dimensional tilting modules. I will discuss the relationship with approximation theory and with localization. Finally, I will focus on some classification results.
The concept of a module of constant Jordan type was introduced in a 2008 paper of Carlson, Friedlander and Pevtsova. In the case of a finite elementary abelian p-group, these modules are closely connected with algebraic vector bundles on projective space. My plan for these talks is to introduce this class of modules, explain some of their elementary and less elementary properties, and then explain how representation theoretic information comes from algebraic geometry. In particular, the theory of Chern classes and the Hirzebruch-Riemann-Roch theorem play a major role. An extensive set of notes on this subject can be found on my home page.
I will explain how to realize some algebraic structures—quantized enveloping algebras and their canonical bases—via a certain tensor category—representations of the quiver Hecke algebras of Khovanov, Lauda and Rouquier. I expect to focus mainly on finite type and discuss the categorification of (dual) PBW bases via (proper) standard modules over these algebras. This has some interesting homological consequences for quiver Hecke algebras in finite type.
Generalizing classical hereditary algebras, we introduce a class of finite dimensional algebras of global dimension n which we call n-hereditary. They consists of two disjoint classes: n-representation finite (n-RF) algebras and n-representation infinite (n-RI) algebras. n-RF algebras are defined by existence of n-cluster tilting modules, while n-RI algebras are closely related to n-Fano algebras of Minamoto and Mori in non-commutative algebraic geometry. We discuss their basic properties and examples: n-AR translations, preprojective algebras, n-APR tilting, connection with cluster categories, n-regular modules, representation dimension and so on. This series of lectures is based on joint work with Herschend and Oppermann motivated by higher dimensional Auslander-Reiten theory.
Refined Donaldson-Thomas invariants were introduced by Kontsevich and Soibelman for 3-Calabi-Yau A∞-categories. The goal of this lecture is to introduce these invariants in the special case of derived categories over the DG algebras associated with quivers with potentials (Ginzburg DG algebras). We will define refined Donaldson-Thomas invariants and compute them in some simple cases. Then we will discuss their basic properties including integrality, positivity, and wall-crossing phenomena.
I am going to give a few lectures on Homological Mirror Symmetry. I plan to talk on categories of D-branes of type B in sigma-models and Landau-Ginzburg models. These categories are directly related to derived categories of coherent sheaves and triangulated categories of singularities. We will describe some different properties of such categories and relations between them, many of which are coming from physics. Useful notions of exceptional collections, classical and strong generators will be discussed.
I am also going to describe a procedure of constructing mirror symmetric models (known as Batyrev-Givental-Hori-Vafa procedure) and will describe mirror symmetry for weighted projective spaces, Del Pezzo surfaces and their non-commutative deformations. Categorical generalization of Strange Arnold Duality will be introduced and some results and conjectures will be given in the last lecture.