• This is the research area I am most interested in. It promises a refined language to describe symmetries, which are crucial in the study of Quantum Field Theory (QFT) and String Theory (ST), and since these two are extremely important in describing physical phenomenon in diverse areas, the generalized symmetries caught a significant attention from physicist in the previous decade. I believe it has a huge potential in the fields related to the phases of gauge theories (the problem of confinement), the phases of matter (topological phases ...), non-perturbative aspects of QFT and  ST, dualities of supersymmetric gauge theories, and holography. 

• Non-perturbative aspects of QFTs are fascinating yet unreachable in most cases. The richness of supersymmetric gauge theories allows for interesting non-perturbative results. For example, N=2 SYM is the first realization of the confinement mechanism on a 4-dimensional theory (Polyakov's realization is in 3d Georgi-Glashow model). I am especially curious about the dualities that give these theories a rich structure. In this regard, that the electric-magnetic duality has a  close relation with the existence of a 6d N=(2,0) SCFT is a statement I wish to understand as deep as it gets. Not to forget N=4 SYM, and its topological twist which has deep relations with the Geometric Langlands duality, as shown by Kapustin and Witten.

• Like in the case of supersymmetric gauge theories, String Theory is full of interesting dualities. The notion of M-theory, how it relates to different superstring theories, and how it is related to matrix theories (BFSS) are very inspiring questions.

• Condensed matter systems have the remarkable ability to show us topological stuff in the lab! Studying the phases of gapped systems and finding that they are related to topology, and expected to be related to Topological Quantum Field Theory (TQFT) makes it very attractive. The application of the perspective from generalized symmetries undoubtedly contains many new surprises, to which I would be happy to try to contribute!

• A problem that is more than 50 years old, which still motivates many research fields. Although lattice gauge theory simulations verify confinement, an understanding of it in Yang-Mills theory remains inaccessible. In 4-dimensional N=2 supersymmetric gauge theory, Seiberg and Witten found a confinement mechanism (1994). And, Ünsal gave a realization of confinement in a non-supersymmetric setup (2007). However, confinement is still not completely understood. I believe the non-perturbative nature of the problem makes it very interesting as any step towards solving confinement is a step towards understanding non-perturbative aspects.

• Another outstanding problem that has deep consequences for quantum gravity, which again motivates many research fields such as holography. I am particularly interested in the use of quantum information ideas and von Neumann algebras in this problem. I do not understand the current status of this research field but I would like to learn about the important ideas.

• I am also very interested in the approach that uses the correspondence between (2+1)d gravity and Chern-Simons gauge theory, which then has some relations with Conformal Field Theories. By studying aspects of CS gauge theory and CFT, many important lessons can be learned about gravity. I am particularly curious about what extended objects with the new perspective of generalized symmetries can teach us about gravity. The study of generalized symmetries in gravity in arbitrary dimensions is likewise very exciting, and it would be fascinating if it were to have consequences for thermodynamic/quantum informatic aspects of black holes.

• The Yang-Mills instantons are remarkably very influential in the study of 4-dimensional topology (other 4d gauge theories are also extremely influential in 4-manifold topology). There is a huge literature on this due to people like Atiyah, Donaldson, Freed, Uhlenbeck, Ward, Witten, and others whose names I do not write because I do not know the literature very well (sorry about that). 

• In addition to having a huge influence on 4-manifold topology, instanton equations have the remarkable property that some subset of solutions obeying certain symmetry conditions comprise solutions of lower-dimensional integrable systems. Based on this observation, Ward conjectured* that this is a general phenomenon and many (perhaps all?) integrable equations can be obtained by reductions of self-duality equations and their generalizations. This is also very curious.

*R. S. Ward. Integrable and solvable systems, and relations among them. Phil. Trans. Roy. Soc. Lond. A, 315:451–457, 1985.

In the non-perturbative understanding of QFT, conformal field theories play a very interesting role. Apparently, QFTs at long distances (IR) generically become scale-invariant. Sometimes this invariance is stronger and there is a conformal group of transformations under which the IR theory is invariant, which is a strong restriction on the IR theory. Bootstrap has the ambitious goal of providing a non-perturbative understanding of QFT*, and has been one of the popular tools in many areas. I would like to learn this powerful tool and apply it to various problems when possible.

Of course, there is also the special case of 2d CFT, with its infinite-dimensional Virasoro algebra and rich structure. I am familiar with some of the constructions in this case as can be seen from my project on RCFT and Verlinde operators, but there's always more to learn!

*David Simmons-Duffin. The Conformal Bootstrap. In Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings, pages 1–74, 2017.