Research

Summary

I study higher structures in geometry and physics. My research mainly involves ideas from (derived) algebraic geometry and (higher) category theory.  More precisely, I work on the following subjects:

• Stacks and stacky constructions.  I am interested in how stacks appear in geometry and physics (e.g. gauge and functorial field theories). In particular, my Ph.D. thesis presents stacky reformulations for Einstein’s gravity theory. 

• Derived geometry. I am mostly interested in derived algebraic/symplectic geometry and neighboring subjects. In particular, my last postdoc project (at the Uni. of Zurich) leads to introducing derived contact structures and examining possible consequences (e.g. Darboux-type local presentations; the notion of symplectification; constructing examples and further structures).  Currently, I have been working on different aspects of such structures.

• 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).

For details, you may see the Research Statement or Curriculum Vitae tabs. 

Live long and prosper _V/

Publications - Preprints

• K. İ. Berktav, Shifted contact structures Part-III: Legendrian structures in derived geometry, doi: 10.13140/RG.2.2.31642.04808, arXiv:2406.17416

• 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, doi: 10.13140/RG.2.2.21012.88969, arXiv:2401.03334. (submitted & under revision)

• K. İ. Berktav, Shifted contact structures and their local theory, arXiv:2209.09686. (to appear in the Ann. Fac. Sci. Toulouse, Math.)

• K. İ. Berktav, Notes on Derived Geometric Formulations in Physics, arXiv:1904.13331, Int. J. Geom. Methods in Mod. Phy. (2022) Vol. 19, No. 10. 

• 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's Lemma (2021),  arXiv:2202.06628

• K. İ. Berktav, An Introduction to Geometric Quantization and Witten’s Quantum Invariant (2019), arXiv:1902.10813