Research

Research interest and experience

My main interest is the construction of various geometric structures on in the framework of algebroids. Algebroids are certain vector bundles in which one can generalize the usual notions of differential geometric structures. These new structures naturally appear in string and M theories, where the extended objects enjoy stringy duality symmetries.

In the past, we constructed metric-connection geometries on Leibniz algebroids. For certain special (admissible) connections, we proved many desirable properties including Cartan structure equations, Bianchi identities and the Schouten decomposition. Furthermore, we introduced statistical, conjugate connection and Hessian structures, and proved some analogous results to the ones in the manifold setting. In order to extend a central result in generalized geometry, known as the Severa classification of exact Courant algebroids, we introduced metric-Bourbaki algebroids, where we generalized the Cartan calculus of Lie derivative, exterior derivative and interior product. Later this lead us to a further abstraction, a calculus framework on algebroids, where we introduced the concept of bialgebroids and their Drinfeld doubles, extending Lie bialgebroids and matched pairs of Lie algebroids. Since (Poisson-Lie) T-duality can be captured in the framework of Lie bialgebroids, our framework sustains a useful generalization for U-duality. Furthermore, by including generalizations of the (H- and R-)twists, we also captured the twistful case with the notion of protobialgebroids by extending the proto Lie bialgebroids. 

Currently, I am the principal investigator in the TÜBİTAK 3501 project "Stringy Geometries in the Framework of Algebroids". One of the main aims of this project is a better understanding of duality symmetries. Exceptional Drinfeld algebras have been introduced as custom-tailored generalizations of Drinfeld doubles of Lie bialgebras for U-duality. We are able to consider these algebraic objects in our calculus framework in a rigorous manner; and we aim to fully analyze the topic in order to have a closer look on duality symmetries. In the twistful case, we are interested in Jacobi structures, Jacobi-Lie T-plurality and their possible extensions. Second aim of the project is to generalize certain geometric structures to the algebroid setting. In particular, we aim to extend our previous work on admissible connections for finding other special connections for various geometric structures. We plan to focus on conformal, projective and symplectic structures, where for the first two we have some preliminary results. The third goal of the project is to be able to express AKSZ sigma model construction in our calculus framework on algebroids with the hope of translating the AKSZ dictionary to a bosonic language, as we did for protobialgebroids. 

Apart from this project, I am interested in frameworks against pointillisme in physics. We are currently reviewing such frameworks where the notion of points do not naturally arise. This ongoing study includes a wide range of physical, mathematical and philosophical topics from various generalizations of geometries to spacetime substantivalism/relationalism debate.

You can find my profiles for Research gate, arXiv, Inspire, ORCID, Linkedin.