An exact solution representing black holes in an expanding universe is found. The black holes are maximally charged and the universe is expanding with arbitrary equation of state. It is an exact solution of the Einstein-scalar-Maxwell system, in which we have two Maxwell-type U(1) fields coupled to the scalar field. The potential of the scalar field is an exponential. The solution depends on two parameters, the charge Q and one parameter (the ratio of the energy density of U(1) fields to that of the scalar field). We find a regular horizon, which is static because of the balance on the horizon between gravitational attractive force and U(1) repulsive force acting on the scalar field. We also calculate the black hole temperature. For the case without a potential, we can derive such a solution from a time-dependent intersecting M-brane solution in eleven dimensions by the dimensional reduction.

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.