Markus Zetto

Welcome to my Homepage! Since May 2023 I am a PhD Student at the University of Hamburg with David Reutter. My research is about applying Higher Category Theory and Homotopy Theory to construct and classify (derived) Extended Topological Field Theories.

Enriched ∞-categories and Extended TFTs

While (weak) n-categories and (∞,n)-categories are of central importance in the study of extended topological field theories, they are notoriously hard to even define, let only work with. However, their description via iterated weak enrichment can be used to prove a surprising amount of statements from general principles. Currently, I am using the machinery of enriched ∞-categories to construct and characterize lax idempotents, also known as condensations, in them. This and related constructions should prove useful to construct and classify (derived) Extended TFTs.

Hermitian K-Theory and Applications

Non-degenerate (anti-)symmetric bilinear forms, as well as quadratic forms, appear in many areas of mathematics - often in a derived context, consider for example the intersection form on a manifold (or Witt stratified space) inducing Poincaré duality, or shifted symplectic forms in field theory. After dividing by a bordism relation, they assemble into the algebraic L-groups, the study of which is important in contexts which ask about the existence or classification of the above structures. For instance, I want to develop the L-groups of Verdier self-dual constructible sheaves as useful, calculable invariants of a good stratified space.

Classical Field Theory

Interestingly, the BV formalism that associates to a field theory its BV-complex, i.e. perturbative phase space, has a neat interpretation in derived geomtry. The covariant phase space associated to an action will generally be a derived stack (possibly equipped with smooth/ functional analytic information), which can be made explicit for AKSZ-theories. I am thinking about the extension of these construction to higher codimension, i.e. the relationship between extended field theories with values in Lagriangian corresponences of derived stacks, simplicial BV-theories and the (global, or state-summed) BV-BFV complex.

Further Interests

Higher Category Theory and Higher Algebra, Derived & Spectral Algebraic Geometry, Factorization Algebras, Stratified Homotopy Theory and Topology of Stratified Manifolds. I want to learn more about Fusion Categories & State Sums, Algebraic K-Theory, Exodromy & Étale homotopy theory, Chromatic Homotopy Theory.

Visited Conferences

Time Name Place
23.-27.09.2024 Scottish Talbot On Algebra and Topology - "Higher tensor categories and their extensions" Glenmore, Scotland
09.-13.09.2024 Workshop on "Dualisable Categories & Continuous K-theory" Bonn, Germany
10.-14.06.2024 Masterclass: Continuous K-theory, dualizable and rigid categories Copenhagen, Danemark
15.-19.04.2024 (∞,n)-Categories and their applications Utrecht, Netherlands
08.-09.04.2024 Bavarian Geometry & Topology Meeting XII Regensburg, Germany
19.01.2024 16th Seminar on Conformal Field Theory Darmstadt, Germany
18.-22.09.2023 Conference on Characteristic Classes and Singular Spaces Kiel, Germany
14.-18.08.2023 Higher Structures in Functorial Field Theory Regensburg, Germany
19.-23.06.2023 Hausdorff School: TQFTs and their connections to representation theory and mathematical physics Bonn, Germany
22.-24.03.2023 Higher categorical methods in algebra and geometry Hamburg, Germany
18.-22.07.2022 Domoschool 2022: Gauge Theory and the Langlands Program Domodossola, Italy
04.-08.10.2021 Pure Spinors, Superalgebras, and Holomorphic Twists Heidelberg, Germany
12.-16.07.2021 Junior Researchers Conference: Higher Structures in QFT and String Theory Online

Contact

If you are interested in, or want to talk about anything of the above, do not hesitate to send me a mail!