In the first part of the talk I will discuss the meaning of "geometry" in contemporary mathematics and some general developments. In the second part I will focus on a few particular problems which I know more about, such as questions involving topology and geometric structures on manifolds. The talk will in part be built around an imaginary conversation with Hilbert.

We establish a correspondence between generalized quiver gauge theories in four-dimensions and congruence subgroups of the modular group, hinging upon the trivalent graphs which arise in both. The gauge theories and the graphs are enumerated and their numbers are compared. The correspondence is particularly striking for genus zero torsion-free congruence subgroups which are crucial to Moonshine. We analyze in detail the case of index 24, where modular elliptic K3 surfaces emerge: here, the elliptic j-invariants can be recast as dessins d'enfant which dictate the Seiberg-Witten curves.

I will describe a how to construct the supergravity solution for a pair of NS5-branes, intersecting on R^{1,3} and localised in all directions except a single transverse circle. Such solutions are rare, and yet may have many potential applications in gauge-gravity duality and model building. The construction invovles developing an explicit map between the conifold metric and the geometry of two intersecting NS5-branes. I will also describe additional intersecting brane solutions corresponding to smoothing out the intersection by, for example, separating branes or putting a metric on the intersection.

We've known since childhood that symmetry can be a very powerful tool in solving partial differential equations. With a little symmetry, one can reduce the number of independent variables, whereas with some more symmetry one can usually separate variables and reduce the problem to solving ordinary differential equations. Given enough symmetry, though, partial differential equations become algebraic. A large body of current research in our field requires finding solutions to the (super)gravity field equations and in this talk I will motivate the search for homogeneous supergravity backgrounds and mention some recent results in this area.

I will start by discussing the 4d N=4 supersymmetric Yang-Mills theory on two spaces with a boundary - a half-line (times three infinite dimensions) and four dimensional anti-de Sitter space. I will review in detail the possible choices of boundary conditions for this theory, and then exhibit the solutions of string theory (supergravity) that are holographically dual to the N=4 SYM theory on these spaces. The same methods used to obtain these duals can also be used to find duals for 3d N=4 and 4d N=2 superconformal field theories that appear in brane configurations, and I will discuss these solutions as well.