This talk will give a lightning review on $K3$ theories, including some newer developments. In particular, we discuss a procedure recently devised in joint work with Anne Taormina in the context of Mathieu Moonshine. This procedure, which we call reflection, allows to transform certain superconformal field theories into super vertex operator algebras and their admissible modules, thus building a bridge between the two worlds.

We study rotating attractor solutions from the point of view of a Riemann-Hilbert problem associated with the Breitenlohner-Maison linear system. We describe an explicit vectorial Riemann-Hilbert factorization method which we use to show that the near-horizon limit of these extremal solutions can be constructed by Riemann-Hilbert factorization of monodromy matrices with poles of second order.

We explore the question of which shape a manifold is compelled to take when immersed in another one, provided it must be the extremum of some functional. We consider a family of functionals which depend quadratically on the extrinsic curvatures and on projections of the ambient curvatures. These functionals capture a number of physical setups ranging from holography to the study of membranes and elastica.

In this talk, I will present some recent developments in computing the exact entropy of dyonic $1/4$-BPS black holes in four-dimensional $N=4$ supergravity theories originating from Type IIB string theory compactified on $K3 \times T_2$. The exact entropy is obtained in the Quantum Entropy Function formalism by means of supersymmetric localization techniques. The result can then be compared to the degeneracy of the brane/momentum system making up the black hole in the string theory picture. Such degeneracies are given by the Fourier coefficients of so-called mock Jacobi forms, a concept I will review. An exact formula for the coefficients can be obtained via a suitable generalization of the Hardy-Ramanujan-Rademacher circle method which takes into account the mock character of the counting functions. After presenting these results, I will outline some discrepancies (at sub-leading order in the charges) between the supergravity result for the exact entropy and the degeneracies of the brane/momentum system, and point to some aspects of the supergravity calculations which should be examined in more detail if one hopes to get a complete matching.

The singlet sectors of $O(N)$ vector field theories in three dimensions have been conjectured to be dual to theories of higher spins in $AdS_4$, in their large $N$ limit. We use the bilocal description of this invariant sector to obtain a constructive description of this correspondence.

We consider a specific type II string theory model with $N=2$ local supersymmetry, the so-called STU model. This is a model with exact duality symmetries. Using holomorphy and duality, we obtain exact results for this model that go beyond the perturbative formulation of topological string theory.

We introduce a notion of attractors and attractive geometry in the context of static four-dimensional non-extremal black holes, and we establish constraints on the flow of scalars in these black hole backgrounds to ultimately derive area laws pertaining to horizons in these backgrounds that were hitherto known empirically.

Persistent homology studies which homological features of a topological space persist over a long range of scales. I will discuss two applications of this formalism to the study of vacua in string theory. In the first application, I will discuss how to adapt such techniques to address the presence/absence of structure in a series of string compactifications (for example flux vacua in type IIB or heterotic vacua). In the second application, I will address the problem of studying vacua of certain two-dimensional Landau-Ginzburg models and see what information we can get about their algebraic structures.

I will discuss a string theoretic approach to integrable lattice models. This approach provides a unified perspective on various important notions in lattice models, and relates these notions to four-dimensional $N =1$ supersymmetric field theories and their surface operators. I will also explain Nekrasov-Shatashvili correspondence.