Prof. Dr. Stefan Schwede

Director of Graduate Studies / Vice-Coordinator HCM

E-mail: schwede(at)math.uni-bonn.de
Phone: +49 228 73 3158
Homepage: http://www.math.uni-bonn.de/~schwede/
Room: 4.008
Location: Mathematics Center
Institute: Mathematical Institute
Research Areas: Research Area C
Research Area F*
Former Research Area E
Date of birth: 23.Jun 1969
Mathscinet-Number: 623322

Publication List

Academic Career

1996

Dr. math., University of Bielefeld

1996

Postdoc, University of Chicago, IL, USA

1997 - 1998

Postdoc, Massachusetts Institute of Technology, Cambridge, MA, USA

1998 - 2001

Assistant Professor (C1), University of Bielefeld

2001

Habilitation, University of Bielefeld

2002 - 2003

Head of Junior Research Group, DFG Collaborative Research Center SFB 478 “Geometric structures in mathematics”, University of Münster

Since 2003

Professor (C4), University of Bonn

Research Profile

My main area of expertise is algebraic topology, specifically stable homotopy theory. I contributed to the foundations of stable homotopy theory (comparison of models for the stable homotopy category, rigidity theorem for the stable homotopy category, foundations of the theory of symmetric spectra). Further research concerned basic questions about triangulated categories, in particular the existence and uniqueness of models for triangulated categories, the distinction of algebraic and topological triangulated categories and examples of exotic triangulated categories.

Much of my current and future research is in equivariant stable homotopy theory, in particular ''global'' phenomena, i.e., spaces or spectra with simultaneous and compatible actions of all compact Lie groups, up to deformations that preserve all symmetries. I introduced a framework for global equivariant homotopy theory based on orthogonal spectra which opens the door for a rigorous study of global stable homotopy types. The global perspective reveals systematic patterns and facilitates equivariant calculations, for example in the rank filtrations of equivariant infinite symmetric products or global equivariant K-theory.
Future goals include a better understanding of the universal properties of global K-theory (both algebraic and topological) and global equivariant bordism. Alongside, we want to further exploit naturally occuring global structures for new computations.

Selected Publications

[1] Fernando Muro, Stefan Schwede, Neil Strickland
Triangulated categories without models
Invent. Math. , 170: (2): 231--241
2007
DOI: 10.1007/s00222-007-0061-2
[2] Stefan Schwede
Algebraic versus topological triangulated categories
Triangulated categories
of London Math. Soc. Lecture Note Ser. : 389--407
Publisher: Cambridge Univ. Press, Cambridge
2010
DOI: 10.1017/CBO9781139107075.010
[3] Stefan Schwede
On the homotopy groups of symmetric spectra
Geom. Topol. , 12: (3): 1313--1344
2008
DOI: 10.2140/gt.2008.12.1313
[4] Stefan Schwede
The stable homotopy category is rigid
Ann. of Math. (2) , 166: (3): 837--863
2007
DOI: 10.4007/annals.2007.166.837
[5] Stefan Schwede
Equivariant properties of symmetric products
DOI:10.1090/jams/879
to appear in J. Amer. Math. Soc.
2017
[6] Stefan Schwede
The n-order of algebraic triangulated categories
J. Topol.
, 6: (4): 857--867
2013
DOI: 10.1112/jtopol/jtt014
[7] Stefan Schwede
The p-order of topological triangulated categories
J. Topol. , 6: (4): 868--914
2013
DOI: 10.1112/jtopol/jtt018
[8] M. A. Mandell, J. P. May, S. Schwede, B. Shipley
Model categories of diagram spectra
Proc. London Math. Soc. (3) , 82: (2): 441--512
2001
DOI: 10.1112/S0024611501012692
[9] Stefan Schwede, Brooke E. Shipley
Algebras and modules in monoidal model categories
Proc. London Math. Soc. (3) , 80: (2): 491--511
2000
DOI: 10.1112/S002461150001220X
[10] Stefan Schwede, Brooke Shipley
Stable model categories are categories of modules
Topology , 42: (1): 103--153
2003
DOI: 10.1016/S0040-9383(02)00006-X
[11] Stefan Schwede
An exact sequence interpretation of the Lie bracket in Hochschild cohomology
J. Reine Angew. Math. , 498: : 153--172
1998
DOI: 10.1515/crll.1998.048

Selected Invited Lectures

2007

Plenary talk, Joint International Meeting UMI - DMV, Perugia, Italy

2008

Plenary talk, European Mathematical Society - Joint Mathematical Weekend Copenhagen, Denmark

Editorships

• Documenta Mathematica (2003 - 2016)
• Mathematische Zeitschrift (2006 - 2012)
• Geometry & Topology (since 2016)

Research Projects and Activities

DFG Research Training Group GRK 1150 “Homotopy and Cohomology”
Scientific Member

DFG Priority Program SPP 1786 “Homotopy Theory and Algebraic Geometry”
Initiator

Series of Oberwolfach Workshops on “Homotopy theory” (2007, 2011, 2015)
Organizer

Abel Symposion, 2007
Organizer

HIM-Trimester, 2015
Organizer

Semester program “Homotopy Harnessing Higher Structures” at Isaac Newton Institute, 2018
Organizer

“Bonn International Graduate School of Mathematics”
Director

DFG Cluster of Excellence “Hausdorff Center for Mathematics”
Vice-Coordinator (since 2017) and Principal Investigator

Contribution to Research Areas

Research Area C
My contribution consists of foundational results about triangulated categories and their models. In joint work with Muro and Strickland [1], we exhibited the first examples of triangulated categories without models. In another direction, I introduced the notion of ‘n-order’, a numerical invariant of triangulated categories that measures ‘how strongly’ objects of the form X/n (a mapping cone of the n times the identity map) are annihilated by n, see [2]. This invariant allows to prove that the stable homotopy category, localized at an odd prime, is not algebraic.
Former Research Area E
My contributions are to foundational aspects of stable homotopy theory. In [3], this refers to the theory of symmetric spectra, where extra algebraic structure on the homotopy groups, in the form of a tame action of the monoid of injective selfmaps of the natural numbers, was discovered and exploited.
The paper [4] proves that the stable homotopy category is rigid, i.e., that every other model (in the sense of Quillen's closed model categories) with an equivalent homotopy category is already Quillen equivalent.
Research Area F*
My contributions to Research Area F* are in equivariant stable homotopy theory, in particular a rigorous new framework for ''global'' phenomena, i.e., spaces or spectra with simultaneous and compatible actions of all compact Lie groups, up to deformations that preserve all symmetries. I introduced a framework for global equivariant homotopy theory based on orthogonal spectra which opens the door for a rigorous study of global stable homotopy types. The global perspective reveals systematic patterns that lead to concrete equivariant calculations, for example of the equivariant homotopy groups of the symmetric product filtration of spheres [5].

Habilitations

Christian Ausoni (2008), now Professor, University of Paris 13, France

Gerald Gaudens (2010)

Steffen Sagave (2013), now Assistant Professor, Radboud University Nijmegen, Netherlands

Selected PhD students

Steffen Sagave (2006): “Universal Toda Brackets of Ring Spectra”,
now Assistant Professor, Radboud University Nijmegen, Netherlands

Constanze Roitzheim (2007): “Rigidity and Exotic Models for the K-local Stable Homotopy Category”,
now Lecturer, University of Kent, England, UK

Moritz Groth (2011): “On the theory of derivators”,
now Postdoc, University of Bonn

Lennart Meier (2012): “United elliptic homology”,
now Postdoc, University of Bonn

Irakli Patchkoria (2013): “Rigidity in equivariant stable homotopy theory”,
now Postdoc, University of Bonn

Karol Szumilo (2014): “Two models for the homotopy theory of cocomplete homotopy theories”,
now Postdoc, University of Western Ontario, Canada

Markus Hausmann (2016): “Symmetric products, subgroup lattices and filtrations of global K-theory”,
now Postdoc, University of Copenhagen, Denmark

Supervised Theses

  • Master theses: 9, currently 5
  • Diplom theses: 15
  • PhD theses: 13, currently 1
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