I am a PhD student in mathematics (since October 2021) at the University of Hamburg funded by the Cluster of Excellence Quantum Universe. My advisor is Tobias Dyckerhoff.
My research concerns Higher Segal Spaces, Categorified Homological Algebra, and their connections to Topological Field Theories.
This is my CV. For a letter of reference, please contact Tobias Dyckerhoff, Paul Wedrich and Julian Holstein.
Higher Segal Spaces are simplicial spaces that satisfy a generalization of the famous Segal conditions. Algebraically they encode generalized convolution algebras structures that encode algebraic structures of Hall and Hecke type.
Abstract: Algebra objects in ∞-categories of spans admit a description in terms of 2-Segal objects. We introduce a notion of span between 2-Segal objects and extend this correspondence to an equivalence of ∞-categories. Additionally, for every ∞-category with finite limits C, we introduce a notion of a birelative 2-Segal object in C and establish a similar equivalence with the ∞-category of bimodule objects in spans. Examples of these concepts arise from algebraic and hermitian K-theory through the corresponding Waldhausen S-construction. Apart from their categorical relevance, these concepts can be used to construct homotopy coherent representations of Hall algebras.
Categorified Homological Algebra is a rapidly developing new are of mathematics. It aims to categorify concepts from homological algebra and simplicial homotopy theory. The categorification is performed by replacing complexes (resp. simplicial) abelian groups by complexes (resp. simplicial) stable ∞-categories.
Outline:
Derived Topological Field Theories describe a new approach to construct TFT's from non-semi-simple input data using ∞-categories. In particular, I currently study derived versions of Turaev-Viro theories and their interactions with categorified homological algebra and Higher Segal spaces.
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Abstract: We illustrate the advantage of derived TFT's over classical TFT's on the example of Quantum Mechanics. Therefore, we first explicitly describe Quantum Mechanics as an ordinary $1$-dimensional TFT with values in vector spaces. After introducing basic terminology from homological algebra, we then extend this theory to a derived 1-dimensional TFT.