I study higher structures in geometry and physics. My research mainly involves ideas from (derived) algebraic geometry and (higher) category theory. More precisely, I focus on the following subjects:
Stacky constructions. I am interested in how stacks appear in geometry and physics (e.g., moduli spaces, gauge and functorial field theories). In particular, my Ph.D. thesis presents stacky constructions for certain gravity theories.
Derived geometry. I mainly focus on derived algebraic/symplectic geometry and neighboring subjects. In particular, my last postdoc project (at the Uni. of Zurich) introduces shifted contact structures on derived stacks and examines possible consequences (e.g., Darboux-type local presentations, the notion of symplectification, constructing examples and further structures, etc.), leading to the development of derived contact geometry. Currently, I have been working on different aspects of such structures. The results obtained so far can be found below. We call the corresponding three papers the Foundational Trilogy.
Mathematical physics. I am curious about the constructions/classifications of (un)extended functorial field theories (Topological, Homotopical, Geometrical).
Some keywords: Algebraic/differential geometry, derived algebraic/symplectic geometry, higher categorical structures, stacks, and mathematical physics (low-dimensional topology and gauge theory; TQFTs and more general functorial field theories; gravity theories and quantization).
Live long and prosper _V/
K. İ. Berktav, Introduction to derived contact geometry (2025), submitted contribution to the Springer-INdAM volume "Poisson Geometry and Mathematical Physics"
K. İ. Berktav, Shifted contact structures Part-III: Legendrian structures in derived geometry, arXiv:2406.17416 (submitted)
K. İ. Berktav, Stacks in Einstein Gravity, Turk J. of Math. (2024) Vol. 48: No. 4, Article 5.
K. İ. Berktav, Shifted contact structures Part-II: On shifted contact derived Artin stacks, arXiv:2401.03334. (To appear in Higher Structures)
K. İ. Berktav, Shifted contact structures and their local theory (Part-I), Ann. Fac. Sci. Toulouse, Math., Serie 6, Vol. 33 (2024) No. 4, pp. 1019-1057.
The latest version: arXiv:2209.09686 + Addendum (2025).
K. İ. Berktav, Derived Geometric Formulations in Physics, Int. J. Geom. Methods in Mod. Phy. (2022) Vol. 19, No. 10. arXiv:1904.13331,
K. İ. Berktav, F. Özbudak, Euclidean Polynomials for Certain Arithmetic Progressions and the Multiplicative Group of F_p^2, Quaest. Math. (2022) 1-10.
K. İ. Berktav, Moduli theory, stacks and 2-Yoneda Lemma (2021), arXiv:2202.06628
K. İ. Berktav, An Introduction to Geometric Quantization and Witten’s Quantum Invariant (2019), arXiv:1902.10813
Stacks in Mathematical Physics (2024), @ GitHub
"What is DAG?" (2024), @ GitHub-repository.
Notes on Serre's formula (2024), see GitHub-repository
Shifted Geometric Structures (2023), @GitHub