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| 2010 | PhD in Mathematics, University of Pennsylvania, Philadelphia, PA, USA | | 2010 - 2013 | Simons Postdoctoral Fellow, Yale University, New Haven, CT, USA | | 2013 - 2014 | Titchmarsh Fellow, University of Oxford, England, UK | | Since 2014 | Bonn Junior Fellow, University of Bonn |
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My current work adresses various questions arising in derived noncommutative geometry described via differential graded categories. I am mainly interested in developing techniques which can be applied in the context of topological Fukaya categories. This is a circle of ideas inspired by a proposal of Kontsevich to provide a purely combinatorial description of Fukaya categories of Stein manifolds. With Kapranov, I have developed a systematic realization of a two-dimensional instance of this proposal using our theory of cyclic 2-Segal spaces. With Brav, I am working on relative variants of various notions of a noncommutative Calabi-Yau structures which are amenable to gluing procedures for topological Fukaya categories of surfaces. In work in progress, we are analyzing the relation to shifted symplectic geometry. In another work, I have calculated  -homotopy invariants of two-dimensional topological Fukaya categories.
My future research plans include a systematic realization of Kontsevich's proposal in higher dimensions based on the theory of structured higher Segal spaces. With Kapranov, Schechtman, and Soibelman, we have are working on a theory of topological Fukaya categories with coefficients for surfaces. In future work, we hope to apply this theory to provide new descriptions of various interesting categories arising in representation theory and mirror symmetry. With Brown and Blanc, we are developing an approach to topological KR-theory for real differential graded categories. One of our main goals are interesting applications in noncommutative singularity theory. Further, I hope to investigate the relevance of the theory of higher Segal spaces in the context of Hall algebras.
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Research Area F*
My current main long term activity concerns the development of categorified homology theory providing the foundations for a topological approach to Fukaya categories. The theory is based on an application of the principle of categorification to homological algebra, replacing complexes of abelian groups by complexes of stable  -categories. A major role is played by certain simplicial structures which bridge between various subjects such as representation theory of finite-dimensional algebras, algebraic K-theory, and mathematical physics. Among the results already established is a categorified variant of the Dold-Kan correspondence which forms a decisive step towards a general theory. |
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[ 1] T. Dyckerhoff, M. Kapranov
Crossed simplicial groups and structured surfaces
Stacks and categories in geometry, topology, and algebra
of Contemp. Math. : 37--110
Publisher: Amer. Math. Soc., Providence, RI
2015
DOI: 10.1090/conm/643/12896[ 2] Tobias Dyckerhoff, Daniel Murfet
Pushing forward matrix factorizations
Duke Math. J. , 162: (7): 1249--1311
2013
DOI: 10.1215/00127094-2142641[ 3] Tobias Dyckerhoff, Daniel Murfet
The Kapustin-Li formula revisited
Adv. Math. , 231: (3-4): 1858--1885
2012
DOI: 10.1016/j.aim.2012.07.021[ 4] Tobias Dyckerhoff
Compact generators in categories of matrix factorizations
Duke Math. J. , 159: (2): 223--274
2011
DOI: 10.1215/00127094-1415869[ 5] Tobias Dyckerhoff
Isolated hypersurface singularities as noncommutative spaces
Thesis (Ph.D.)--University of Pennsylvania
Publisher: ProQuest LLC, Ann Arbor, MI
2010
ISBN: 978-1124-32510-1 |
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• Higher Structures (since 2016)
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| 2011 | Steklov Institute, Moscow, Russia | | 2012 | PASI, Guanajuato, Mexico | | 2013 | Topology Seminar, Stanford University, CA, USA | | 2013 | AGNES Workshop, Yale, CT, USA | | 2014 | ICM Satellite Conference on Homological mirror symmetry and symplectic topology, POSTEC, Korea | | 2015 | Kolloquium, Münster | | 2015 | Kolloquium, Hamburg |
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- Master theses: 2, currently 2
- PhD theses: 2, currently 2
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