

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 


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 twodimensional instance of this proposal using our theory of cyclic 2Segal spaces. With Brav, I am working on relative variants of various notions of a noncommutative CalabiYau 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 twodimensional 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 KRtheory 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.


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 finitedimensional algebras, algebraic Ktheory, and mathematical physics. Among the results already established is a categorified variant of the DoldKan correspondence which forms a decisive step towards a general theory. 


[ 1] T. Dyckerhoff, M. Kapranov
Crossed simplicial groups and structured surfaces Stacks and categories in geometry, topology, and algebra of Contemp. Math. : 37110 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): 12491311 2013 DOI: 10.1215/001270942142641[ 3] Tobias Dyckerhoff, Daniel Murfet
The KapustinLi formula revisited Adv. Math. , 231: (34): 18581885 2012 DOI: 10.1016/j.aim.2012.07.021[ 4] Tobias Dyckerhoff
Compact generators in categories of matrix factorizations Duke Math. J. , 159: (2): 223274 2011 DOI: 10.1215/001270941415869




• Higher Structures (since 2016)


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 


 Master theses: 2, currently 2
 PhD theses: 2, currently 2


Download Profile 