Thomas Schick

Schick studied mathematics and physics at the Johannes Gutenberg University Mainz, where he received in 1994 his Diplom in mathematics and in 1996 his PhD (Promotion) under the supervision of Wolfgang Lück with thesis Analysis on Manifolds of Bounded Geometry, Hodge-deRham Isomorphism and $L^2$-Index Theorem.{{MathGenealogy|id=27755}} As a postdoc he was from 1996 to 1998 at the University of Münster and from 1998 to 2000 an assistant professor at Pennsylvania State University, where he worked with Nigel Higson and John Roe. Schick received his habilitation in 2000 from the University of Münster and is since 2001 a professor for pure mathematics at the University of Göttingen.

His research deals with topological invariants, e.g. $L^2$-invariants and those invariants which result from the K-theory of operator algebras. Such invariants arise in generalizations of the Atiyah-Singer index theorem.

Schick, with Wolfgang Lück, introduced the strong Atiyah conjecture. Given a discrete group G, the Atiyah conjecture states that the $L^2$-Betti numbers of a finite CW-complex that has fundamental group G are integers, provided that G is torsion-free; furthermore, in the general case, the $L^2$-Betti numbers are rational numbers with denominators determined by the finite subgroups of G. In 2007 Schick, with Peter Linnell, proved a theorem which established conditions under which the Atiyah conjecture for a torsion-free group G implies the Atiyah conjecture for every finite extension of G; furthermore, they proved that the conditions are satisfied for a certain class of groups.{{cite journal|author=Schick, T.|author2=Linnell, P.|title=Finite group extensions and the Atiyah conjecture|journal=Journal of the American Mathematical Society|volume=20|year=2007|issue=4|pages=1003–1061|url=http://www.uni-math.gwdg.de/schick/publ/extAtiyah.html|arxiv=math/0403229|doi=10.1090/S0894-0347-07-00561-9|bibcode=2007JAMS...20.1003L |s2cid=12160184}} In 2000 Schick proved the Atiyah conjecture for a large class of special cases.{{cite journal|author=Schick, T.|title=Integrality of $L^2$ Betti numbers|journal=Mathematische Annalen|volume=317|issue=4|year=2000|pages=727–750|doi=10.1007/PL00004421|arxiv=math/0001101|s2cid=59127019}} In 2007 he presented a method which proved the Baum-Connes conjecture for the full braid groups, and for other classes of groups which arise as (finite) extensions for which the Baum-Connes conjecture is known to be true.{{cite journal|author=Schick, T.|title=Finite group extensions and the Baum-Connes conjecture|journal=Geometry and Topology|volume=11|year=2007|issue=3|pages=1767–1775| arxiv=math/0209165 | doi=10.2140/gt.2007.11.1767|doi-access=free}} [https://arxiv.org/abs/math.KT/0209165 arXiv preprint]{{cite web|title=Thomas Schick: Finite group extensions and the Baum-Connes conjecture|website=Schick's website at the University of Gôttingen (uni-math.gwdg.de)|url=http://www.uni-math.gwdg.de/schick/publ/BC_finite_ext.html}}

In the 1990s there were proofs of many special cases of the Gromov-Lawson-Rosenberg conjecture concerning criteria for the existence of a metric with positive scalar curvature; in 1997 Schick published the first counterexample.{{cite journal|author=Schick, T.|title=A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture|journal=Topology|volume=37|year=1998|issue=6|pages=1165–1168|arxiv=math/0403063| doi=10.1016/s0040-9383(97)00082-7|doi-access=free}} [https://arxiv.org/abs/math/0403063 arXiv preprint]

He is the coordinator of the Courant Research Center's Strukturen höherer Ordnung in der Mathematik (Structures of Higher Order in Mathematics) at the University of Göttingen.{{cite web|website=Göttinger Tageblatt|title=Neuartige Probleme der Mathematik lösen, Göttingen, Courant Forschungszentrum|date=19 May 2009|url=http://www.goettinger-tageblatt.de/Nachrichten/Wissen/Wissen-vor-Ort/Neuartige-Probleme-der-Mathematik-loesen}} A major goal of the research center is the investigation of mathematical structures that could play a role in modern theoretical physics, especially string theory and quantum gravity.

He was the managing editor for Mathematische Annalen. In 2014 he was an invited speaker with talk The topology of scalar curvature at the International Congress of Mathematicians in Seoul. In 2016 he became a full member of the Göttingen Academy of Sciences and Humanities.

• [https://arxiv.org/abs/1405.4220 Topology of scalar curvature.] Proc. ICM 2014, Seoul.
• [https://arxiv.org/abs/math/0209164 Operator algebras and topology.] ICTP Summer School, Triest 2001.
• with Ulrich Bunke: [https://arxiv.org/abs/1011.6663 Differential K-Theory.]
• with Ulrich Bunke: [https://arxiv.org/abs/0707.0046 Smooth K-Theory.] In: Astérisque. No. 328 (2009), 45–135 (2010). {{ISBN|978-2-85629-289-1}}.
• with Bernhard Hanke and Wolfgang Steimle: The space of metrics of positive scalar curvature. Publications Mathématiques de l'IHÉS 120 (2014), 335–367. {{doi|10.1007/s10240-014-0062-9}}
• with Bernhard Hanke: Enlargeability and index theory. Journal of Differential Geometry 74 (2006), no. 2, 293–320. [https://arxiv.org/abs/math/0403257 Arxiv]
• with Józef Dodziuk, Peter Linnell, Varghese Mathai, Stuart Yates: Approximating L²-invariants and the Atiyah conjecture. Dedicated to the memory of Jürgen K. Moser. Communications on Pure and Applied Mathematics 56 (2003), no. 7, 839–873. {{doi|10.1002/cpa.10076}}
• with Rostislav Grigorchuk, Peter Linnell, Andrzej Żuk: On a question of Atiyah. Comptes rendus de l'Académie des Sciences 331 (2000), no. 9, 663–668. [https://arxiv.org/abs/math/0009175 Arxiv]
• with Wolfgang Lück: $L^2$ torsion of hyperbolic manifolds of finite volume. In: Geometric and Functional Analysis. vol. 9, 1999, pp. 518–567, [https://arxiv.org/abs/dg-ga/9707009 Arxiv.]
• Integrality of $L^2$ Betti numbers, Mathematische Annalen vol. 317, 2000, pp. 727–750, [https://arxiv.org/abs/math/0001101 Arxiv.]
• $L^2$-index theorem for elliptic boundary problems, Pacific Journal of Mathematics vol. 197, 2001, pp. 423–439, [https://arxiv.org/abs/math/9810133 Arxiv.]

• [http://www.uni-math.gwdg.de/schick/ Homepage]
• {{cite web|title=Presseinformation: Antrittsvorlesung des Mathematikers Prof. Dr. Thomas Schick|date=15 January 2002|website=Georg-August-Universität Göttingen|url=http://www.uni-goettingen.de/de/3240.html?cid=601}}

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Category:German topologists

Category:Differential geometers

Category:20th-century German mathematicians

Category:21st-century German mathematicians

Category:Johannes Gutenberg University Mainz alumni

Category:Academic staff of the University of Göttingen

Category:1969 births

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Category:German academic journal editors

Category:Pennsylvania State University faculty

Category:People from Alzey