A number of remarkable techniques arising from particular gauge theories in physics have long been incarnated into different branches of mathematics. They have been notably employed to study low dimensional topology and geometry in rather sophisticated ways: Donaldson's theory on four-manifolds [12], the work of Floer on the topology of 3-manifolds and Yang-Mills Instantons [13, 14, 21], and Witten's knot invariants [3].

My research motivation in this phase is, in fact, to study different interactions between low-dimensional topology/geometry and certain quantum field theories.

Topological Quantum Field Theory and Quantum Invariants. My main research of interest was first essentially about TQFTs (in the sense of Atiyah [5]) in various dimensions. For almost two years, I have studied Witten's work on QFT and the Jones polynomial [3] and its possible consequences together with the notion of geometric quantization [9, 10, 25]. Indeed, this was my starting point to investigate the notion of TQFT (manifested in the language of category theory), especially in the case of the three-dimensional Chern-Simons gauge theory. 

As a consequence and application of Witten's interpretation of knot invariants, I first investigated (i) Witten's construction of two-dimensional integrable lattice models [4] by means of such a three-dimensional description of knot invariants; and next (ii) I studied the Morse theoretic interpretation of the usual 2+1 topological Chern-Simons gauge theory [13, 14, 21]. One of the outputs of this phase of my research journey, on the other hand, is as follows:

• An Introduction to Geometric Quantization and Witten's Quantum Invariant (2019). arXiv:1902.10813.

This phase of my research journey and all possible outputs can be collected under the title of Derived Geometry and Physics, which includes the following topics: (derived) stacks and moduli problems, higher categories, derived algebraic geometry, shifted geometric structures, and formulations of classical/quantum field theories in the context of derived algebraic geometry, and the structure of observables in those theories via factorization algebras. 

The main goals of our program, in particular, are to provide the reformulation of Einstein's theory of gravity and to capture certain aspects of the theory in view of derived objects and the homotopy theory of stacks [6, 16]. 

Derived geometry, moduli problems, and physics. Derived algebraic geometry (DAG) essentially provides a new setup to deal with non-generic situations in geometry (e.g. non-transversal intersections and ``bad" quotients). To this end, it combines higher categorical objects and homotopy theory with many tools from homological algebra. Hence, roughly speaking, it can be considered as a higher categorical/homotopy theoretical refinement of classical algebraic geometry. In that respect, it offers a new way of organizing information for various purposes. Therefore, it has many interactions with other mathematical domains. For a survey of some directions, we refer to [26, 27].

Regarding physics-related problems, for example, [7, 8] studies gauge theories and factorization algebras in the context of DAG. In [6], a stacky formulation of Yang-Mills fields on Lorentzian manifolds is presented. [31] examines higher structures in algebraic quantum eld theory. [32, 33] focus on geometric functorial eld theories.

Inspired by the aforementioned formulations, in this phase, we present a similar type of analysis in the case of certain Einstein gravities and investigate its possible consequences. For instance, we study 3D Einstein-Cartan-Palatini gravity theory and 3D quantum gravity using the language of stacks. Our constructions, in fact, employ some techniques from Hollander's homotopy theory of stacks [16].

In brief, this phase can be considered as a first naïve step toward the understanding of "stacky" formulations for Einstein’s gravity theory. In that respect, this phase consists of the outputs

• Stacks in Einstein Gravity: Turk J. of Math. (2024) Vol. 48: No. 4, Article 5.

• On moduli theory, stacks, and 2-Yoneda's Lemma, arXiv:2202.06628

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

Shifted geometric structures. In the context of DAG, it is possible to work with familiar geometric structures, but in more general forms. For instance, k-shifted versions of Symplectic and Poisson geometries have already been described and studied in [19, 30]. In this regard, [20, 28, 29] offer some applications and local constructions. We, on the other hand, provide a similar framework for the case of contact geometry. In what follows, we explain our current works and future/ongoing projects.

In this regard,  we propose a shifted version of the classical contact geometry in the following manuscript:

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

In this paper, the goal is to establish the shifted version of the classical connection between symplectic and contact geometries. In brief, we formally define the concept of a k-shifted contact structure on derived stacks and study its local properties in the context of derived algebraic geometry. In this regard, for k-shifted contact derived K-schemes, we present a Darboux-like theorem and formulate the notion of symplectification.

In the sequel(s), our goals will be to extend the main results of this paper from derived schemes to the more general case of derived Artin stacks and to discuss more on the local structure theory of shifted contact derived spaces. Some relevant results will be available online soon.

• 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.

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

I am also very curious about the possible applications and interesting examples of shifted contact structures.

Derived symplectic geometry and the AKSZ-type constructions. I am also interested in the algebraic AKSZ construction and its consequences.

Extended topological/geometric/homotopy functorial field theories. Why not? They look fun...