A) Higher Segal Spaces

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.

Preprints and Publications

1. An ∞-category of 2-Segal spaces: Arxiv
   Jonte Gödicke

   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.

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B) Categorified Homological Algebra

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.

Projects

1. Lax Horn fillers and homotopy groups of simplical stable

   Outline:

C) Topological Field Theories

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.

Preprints and Publications

1. Fusion categories from 2-Segal spaces: In preparation.
   Jonte Gödicke

   Abstract:

2. Simons Lectures on Categorical Symmetries: Arxiv
   Davi Costa, Clay Córdova, Michele Del Zotto, Dan Freed, Jonte Gödicke, Aaron Hofer, David Jordan, Davide Morgante, Robert Moscrop, Kantaro Ohmori, Elias Riedel Gårding, Claudia Scheimbauer, Anja Švraka

   Abstract: Global Categorical Symmetries are a powerful new tool for analyzing quantum field theories. This volume compiles lecture notes from the 2022 and 2023 summer schools on Global Categorical Symmetries, held at the Perimeter Institute for Theoretical Physics and at the Swiss Map Research Station in Les Diableret. Specifically, this volume collects the lectures:
   * An introduction to symmetries in quantum field theory, Kantaro Ohmori
   * Introduction to anomalies in quantum field theory, Clay Córdova
   * Symmetry Categories 101, Michele Del Zotto
   * Applied Cobordism Hypothesis, David Jordan
   * Finite symmetry in QFT, Daniel S. Freed
   These volumes are devoted to interested newcomers: we only assume (basic) knowledge of quantum field theory (QFT) and some relevant maths. We try to give appropriate references for non-standard materials that are not covered. Our aim in this first volume is to illustrate some of the main questions and ideas together with some of the methods and the techniques necessary to begin exploring global categorical symmetries of QFTs.

Notes and others

1. Poster, Derived TQFT's: Quantum Mechanics: Poster Quantum Universe Graduate School

   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.

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Teaching

• Summer 2024: No teaching
• Winter 2023: No teaching
• Summer 2023: No teaching
• Winter 2022: Tutor Linear Algebra 1
• Summer 2022: Tutor Algebraic Topology
• Winter 2021: Mathematics for Physicists 3

Talks

• Higher Structures in QFT Seminar
  Title: Rigid Hall monoidal structures
  Munich, Germany,May 2024
• Algebra and Topology Seminar,
  Title: Which 2-Segal objects are rigid?
  Kopenhagen, DenmarkMay 2024
• Workshop Higher Segal Spaces and their applications to Algebraic K-Theory, Hall Algebras, and Combinatorics;
  Title: Rigid 2-Segal Spaces
  Banff International Research StationJanuary 2024
• Higher Strucuture Seminar,
  Title: Rigid 2-Segal Spaces?
  Hamburg, GermanyNovember 2023
• Higher Strucuture Seminar,
  Title: Introduction to the Cobordism Hypothesis
  Hamburg, GermanyNovember 2023
• Global Categorical Symmetries Summer School, Gong-Show Talk
  Title: Towards Derived TV-theory
  Les Diableret,SwitzerlandSeptember 2023
• Summer School: Donaldson Thomas Invariants in Derived Symplectic Geometry,
  Title: Motivic DT-invariants after Kontsevich Soibelmann with Shivang Jindal
  Aussois, France Oktober 2022
• ZMP Seminar,
  Title: Categorification of Cluster Algebras with Merlin Christ
  Hamburg, Germany January 2022

Conferences

• Conference Higher Structures in Functorial Field Theories
  Regensburg, Germany August 2023
• Hausdorff School: TQFT and their connection to Representation theory and Mathematical Physics,
  Bonn, Germany, June 2023
• School: Finiteness Conjecture for Skein Modules
  Matemale, France May 2022