Homotopy theory, Category theory, Knot theory, Applications to mathematical physics, distributed algorithms, and neuroscience.

Number theory and arithmetic geometry: elliptic curves and the Birch and Swinnerton-Dyer conjecture, Euler systems and Selmer groups, special cycles on Shimura varieties; mathematical foundations of cryptology and applied cryptography: curve-based cryptography, complexity of computing discrete logarithms, hard-to-compute bits, secure multiparty computation and applications.

Pure Mathematics. Emphasis on Geometry, Group Theory, Analysis, Dynamical systems.

Combinatorial and Algorithmic Geometry, Extremal Theory of Geometric Graphs.

The main focus of the Chair of Algebraic Geometry is the classification theory of higher dimensional algebraic varieties, including exploring its connections to adjacent fields such as commutative algebra, arithmetic geometry and complex geometry.

Geometry, differential geometry, geometric analysis, and analysis on metric spaces.