We study mathematical theories related to quantum gravity, conformal field theory, topological state of matter and other physical problems, and applies them to physics. The contents include but are not limited to holographic duality, AdS/CFT duality, classification of topological order, category symmetry, two-dimensional and high-dimensional conformal field theory, string theory, black holes, lattice models, and random walk.
String theory, a subject that is over fifty years old, is still at the center of efforts by majority of theoretical physicists to find a unified fundamental theory of nature. Nowadays "String Theory" is a broad subject encompassing most of non-phenomenological fundamental Theoretical Physics. Besides String Theory in the strict sense (and its non-perturbative completions 11d M-theory and 12d F-theory) it dwells with non-perturbative aspects of Quantum Field Theory, with special reference to: (1) powerful techniques such as the AdS/CFT correspondence, (2) geometrical methods to solve QFTs, (3) the construction/solution/classification of interacting QFTs in more than 4 dimensions, (4) etc.
String theory provides a fully consistent framework to unify all interactions, including Quantum Gravity. It a prime theoretical laboratory to test ideas and proposed physical principles for Quantum Gravity in general and in particular in its black hole sectors.
Studying string theory and specially black holes in string theory last two decades has also led to the discovery of deep and surprising connections between black holes and many declines in pure mathematics.
Recent developments in string theory reveals that the string theory landscape of false vacua is vast, so it is natural to ask whether the landscape is as immense as allowed by consistent effective field theories.
This leads to the swampland program to characterize the physical theories which admit a UV-completion/non-perturbative-completion to a consistent Quantum Gravity. Last but not least, string theory is a very powerful/suggestive tool for mathematics with particular reference to the geometry of moduli spaces of Calabi-Yau manifolds of various dimensions and enumerative Algebraic Geometry. The main directions of research actively pursued are: (1) the swampland program, (2) geometry of Calabi-Yau moduli spaces, (3) geometrical techniques for the non-perturbative solutions of interacting QFTs in various dimensions (4) quantum aspects of black holes and (5) scattering amplitude.