Seminar on higher AuslanderReiten theory (winter term 2015/2016)
The following article is an introduction to higher AuslanderReiten theory:

[1] Osamu Iyama:
AuslanderReiten theory revisited
(Proceeding of the ICRA XII conference, Torun, August 2007, 47 pages, arXiv version)
The goal of the seminar is to cover the main results and topics mentioned of Iyama's survey article.
The proofs of the results mentioned in [1] are given in the following two papers:

[2] Osamu Iyama:
Higher dimensional AuslanderReiten theory on maximal orthogonal
subcategories
(Adv. Math. 210 (2007), no. 1, 2250, arXiv version) 
[3] Osamu Iyama:
Auslander correspondence
(Adv. Math. 210 (2007), no. 1, 5182, arXiv version)
 [4] Osamu Iyama: Cluster tilting for higher Auslander algebras (Adv. Math. 226 (2011), no. 1, 161, arXiv version)
Seminar schedule
#1  20.10.2015  Wassilij Gnedin  Classical AuslanderReiten theory 
recall of classical AuslanderReiten theory, classical AuslanderReiten formula (Theorems 1.6 and 1.7 in [1]), examples for knitting AuslanderReiten quivers 

#2  20.10.2015  Julian Külshammer  Classical Auslander correspondence 
statement and examples for classical Auslander correspondence
(Theorem 1.2 in [1]), Yoneda's lemma in the context of Auslander algebras (first equivalence in the proof of Theorem 2.6 (a) in [2]), main steps in the proof of classical Auslander correspondence 

#3  3.11.2015  Ruari Walker  Clustertilting categories and their properties 
Definition of clustertilting category C (Subsection 2.1 in [1], or Subsections 2.1 and 2.2 in [2]), equivalent characterizations of clustertilting subcategories (Proposition 2.2 in [1] or Proposition 2.2.2 in [2]), basic examples (Example 2.4(a) in [1]). 

#4  10.11.2015  Armin Shalile  Higher AuslanderReiten formula 
Definition of higher AuslanderReiten translation, statement of higher AuslanderReiten formula for clustertilting categories (Theorems 2.8 and 2.9 in [1], or Theorem 1.4.1, 1.5 in [2]), basic examples, main steps in the proof of higher AuslanderReiten formula. 

#5  24.11.2015  Gustavo Jasso  A short proof of the defect formula for dexact sequences 
#6  1.12.2015  Pin Liu  Tautilting and clustertilting in higher cluster categories 
Relationship between higher clustertilting objects in higher cluster categories and tautilting modules over a clustertilted algebra.  
#7  8.12.2015  Steffen König  Higher AuslanderReiten sequences 
Existence and properties of higher AuslanderReiten sequences (Theorem 2.11 in [1] or Theorem 3.3.1 in [2]) 

#8  15.12.2015  Christian Lomp  Higher Auslander correspondence 
main steps in the proof of higher Auslander correspondence (Theorem 2.6 in [1] or Theorem 4.2.2 in [3])  
#9  12.1.2016  René Marczinzik  Clustertilting for selfinjective algebras 
Cluster tilting objects for representationfinite selfinjective algebras
(Section 4 in [2]), general results on the existence of clustertilting objects for selfinjective algebras (Theorem 2.22 in [1] by Erdmann and Holm) 

#10  19.1.2016  Yiping Chen  ncomplete algebras 
Inductive construction of algebras admitting cluster subcategories (Section 2.4 in [1], or [4])  
#11  26.1.2016  Bernhard Böhmler  Introduction to representation theory of orders 
Classical AuslanderReiten theory for CohenMacaulay modules over orders (Section 3.1 in [1])  
#12  26.1.2016  Wassilij Gnedin  Clustertilting modules over quotient singularities 
higher AuslanderReiten theory for CohenMacaulay modules over orders, main steps in the proof of Theorem 3.10 a) in [1]  
#13  2.2.2016  Wassilij Gnedin  Higher algebraic McKay correspondence 
main steps in the proof that McKay quiver is the higher AR quiver (Theorem 3.10 b), c) in [1] )  
#14  9.2.2016  Wassilij Gnedin  Higher Auslander orders 
the depth of cluster tilted algebras, higher Auslander correspondence for orders, derived equivalence of 2cluster tilted algebras (Theorem 3.15 in [1]), ;  
#15  9.2.2016  Steffen König  Noncommutative crepant resolutions 
geometric and algebraic notions of resolutions, BondalOrlov and van den Bergh's conjectures, main steps of the proof of Theorem 3.18 in [1] 
Talks related to higher AuslanderReiten theory at Stuttgart:
DarstellungstheorieTage:  13.11.2015, 16:00, (V57.04)  Julian Külshammer  Higher Nakayama algebras 
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