

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 


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 Ktheory.
Future goals include a better understanding of the universal properties of global Ktheory (both algebraic and topological) and global equivariant bordism. Alongside, we want to further exploit naturally occuring global structures for new computations.


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
HIMTrimester, 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”
ViceCoordinator (since 2017) and Principal Investigator


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 ‘order’, a numerical invariant of triangulated categories that measures ‘how strongly’ objects of the form (a mapping cone of the times the identity map) are annihilated by , 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]. 


[ 1] Fernando Muro, Stefan Schwede, Neil Strickland
Triangulated categories without models Invent. Math. , 170: (2): 231241 2007 DOI: 10.1007/s0022200700612[ 2] Stefan Schwede
Algebraic versus topological triangulated categories Triangulated categories of London Math. Soc. Lecture Note Ser. : 389407 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): 13131344 2008 DOI: 10.2140/gt.2008.12.1313[ 4] Stefan Schwede
The stable homotopy category is rigid Ann. of Math. (2) , 166: (3): 837863 2007 DOI: 10.4007/annals.2007.166.837[ 5] Stefan Schwede
Equivariant properties of symmetric products J. Amer. Math. Soc. , 30: (3): 673711 2017 DOI: 10.1090/jams/879[ 6] Stefan Schwede
The norder of algebraic triangulated categories J. Topol. , 6: (4): 857867 2013 DOI: 10.1112/jtopol/jtt014[ 7] Stefan Schwede
The porder of topological triangulated categories J. Topol. , 6: (4): 868914 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): 441512 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): 491511 2000 DOI: 10.1112/S002461150001220X[ 10] Stefan Schwede, Brooke Shipley
Stable model categories are categories of modules Topology , 42: (1): 103153 2003 DOI: 10.1016/S00409383(02)00006X[ 11] Stefan Schwede
An exact sequence interpretation of the Lie bracket in Hochschild cohomology J. Reine Angew. Math. , 498: : 153172 1998 DOI: 10.1515/crll.1998.048



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


2007  Plenary talk, Joint International Meeting UMI  DMV, Perugia, Italy  2008  Plenary talk, European Mathematical Society  Joint Mathematical Weekend Copenhagen, Denmark 


Christian Ausoni (2008), now Professor, University of Paris 13, France
Gerald Gaudens (2010)
Steffen Sagave (2013), now Assistant Professor, Radboud University Nijmegen, Netherlands


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 Klocal 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 Ktheory”,
now Postdoc, University of Copenhagen, Denmark


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


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