Workshop
August 1013, 2016 (WednesdaySaturday)
Schedule
All workshop talks were held in Auditorium 001 of the Life Sciences Building (LSB).
August 10  August 11  August 12  August 13  

9:00  10:00  Registration  Amiot (1)  Takahashi (2)  Amiot (3) 
10:15  11:15  Marsh (1)  Marsh (2)  Linckelmann (3)  Takahashi (3) 
11:15  11:45  Coffee Break  Coffee Break  Coffee Break  Coffee Break 
11:45  12:45  Linckelmann (1)  Linckelmann (2)  Amiot (2)  Pevtsova (3) 
12:45  14:30  Lunch  Lunch  Lunch  
14:30  15:30  Pevtsova (1)  Pevtsova (2)  Brion (3)  
15:30  16:00  Coffee Break  Coffee Break  Coffee Break  
16:00  17:00  Brion (1)  Brion (2)  Marsh (3)  
17:15  18:15  Takahashi (1)  
18:30  
Welcome Reception at Genesee Grande 
Abstracts
Claire Amiot (Université Grenoble Alpes): Cluster categorification and applications to tilting theory
This series of talks is based on joint works with Oppermann, Grimeland, Labardini and Plamondon. Cluster categories are triangulated categories where quiver mutation appears as a natural operation. A first class of example is given by cluster categories associated with surfaces with marked points. A second class is constructed using the derived category of finite dimensional algebras of global dimension 2. Mixing both constructions, one may consider surface cut algebras, that are algebras of global dimension 2 constructed from a surface and show how cluster combinatorics permits to deduce information on their derived category.
Michel Brion (Institut Fourier): The isogeny category of commutative algebraic groups
The commutative algebraic groups over a prescribed field k form an abelian category C_{k}; the finite commutative algebraic groups form a full subcategory F_{k}, stable under taking subobjects, quotients and extensions. This minicourse will study the categories C_{k} and C_{k}/F_{k} (the isogeny category) from a homological viewpoint, emphasizing the analogies and differences with categories of representations. In particular, we will show that C_{k}/F_{k} has homological dimension 1, and we will describe the projective and the injective objects in C_{k} and C_{k}/F_{k}.
Markus Linckelmann (City University London): Hochschild cohomology and modular representation theory
Modular representation theory of finite groups seeks to understand, and possibly classify, the algebras  called block algebras of finite groups  which arise as indecomposable direct factors of finite group algebras over a complete local principal ideal domain with residue field of prime characteristic p. The expectation is that `few' algebras should arise in this weay, and that this should in turn lead to significant structural connections between finite groups and their block algebras.
The key feature of block algebras of finite groups is the dichotomy of invariants attached to these algebras.
On the one hand, they have all the typical algebra theoretic invariants  module categories, their derived categories and stable categories, as well as numerical invariants such as the numbers of isomorphism classes of simple modules, and cohomologivcal invariants such as their Hochschild cohomology.
On the other hand, they have plocal invariants, due to their provenance from group algebras  reminiscent of the local structure of a finite group which includes its Sylow psubgroups and its associated fusion systems.
Essentially all prominent conjectures which drive modular representation theory revolve around the interplay between these two types of invariants. We describe this interplay with a focus on Hochschild cohomology and analogous cohomology rings which are defined plocally. This involves a variety of angles  Hochschild cohomology is graded commutative, hence methods and notions from commutative algebra will play a role. Hochschild cohomology in positive degree is also a Lie algebra. We will investigate connections between the algebra structure of block algebras and the Lie algebra structure of its first Hochschild cohomology space.
Robert Marsh (University of Leeds): Dimer models and cluster categories of Grassmannians
The homogeneous coordinate ring of the Grassmannian Gr(k,n) has a beautiful structure as a cluster algebra, by a result of J. Scott. Central to this description is a collection of clusters containing only Pluecker coordinates, which are described by certain diagrams in a disc, known as Postnikov diagrams or alternating strand diagrams. Recent work of B. Jensen, A. King and X. Su has shown that the Frobenius category of CohenMacaulay modules over a certain algebra, B, can be used to categorify this structure.
In joint work with Karin Baur and Alastair King, we associate a dimer algebra A(D) to a Postnikov diagram D, by interpreting D as a dimer model with boundary. We show that A(D) is isomorphic to the endomorphism algebra of a corresponding CohenMacaulay clustertilting Bmodule, i.e. that it is a clustertilted algebra in this context. The proof uses the consistency of the dimer model in an essential way.
It follows that B can be realised as the boundary algebra of A, that is, the subalgebra eAe for an idempotent e corresponding to the boundary of the disk. The general surface case can also be considered, and we compute boundary algebras associated to the annulus.
Julia Pevtsova (University of Washington): Support theories for the stable module category of a finite group scheme
We'll study the global structure of the stable module category StMod G or, equivalently, the category of singularities of representations of a finite group scheme G over a field of positive characteristic p. The goal of the lectures will be to classify the tensor ideal localizing subcategories in StMod G. The techniques involved in the classification include the theories of support and cosupport in modular representation theory, detection of projectivty for modules, BensonIyengarKrause theory of local cohomology functors, and new methods inspired by commutative algebra which allow to relate local cohomology at closed and arbitrary points. This is based on joint work with Eric Friedlander and Dave Benson, Srikanth Iyengar and Henning Krause.
Ryo Takahashi (Nagoya University): Thick tensor ideals of right bounded derived categories of commutative rings
Let R be a commutative Noetherian ring. Denote by D^{}(R) the derived category of cochain complexes X of finitely generated Rmodules with H^{i}(X)=0 for i>>0. Then D^{}(R) has a structure of a tensor triangulated category with tensor product ⊗_{R}^{L} and unit R. In this series of lectures, we study thick tensor ideals of D^{}(R), i.e., thick subcategories closed under the tensor action by each object in D^{}(R), and investigate the Balmer spectrum Spc D^{}(R) of D^{}(R), i.e., the set of prime thick tensor ideals of D^{}(R). Here is a plan.

 We give a complete classification of the (co)compactly generated thick tensor ideals of D^{}(R), establishing a generalized version of the HopkinsNeeman smash nilpotence theorem.
 We construct a pair of maps between the Balmer spectrum Spc D^{}(R) and the prime spectrum Spec R, and explore their topological properties.
 We compare several classes of thick tensor ideals of D^{}(R), relating them to specializationclosed subsets of Spec R and Thomason subsets of Spc D^{}(R).
If time permits, I would like to talk about the case where R is a discrete valuation ring. My lectures are based on joint work with Hiroki Matsui.