School of Mathematical Sciences

Rough CAT(0) is a condition which generalizes both CAT(0) and Gromov hyperbolicity, and which implies bolicity in the sense of Kasparov and Skandalis. We give an overview of the interior and boundary geometry of rough CAT(0) spaces. In particular, we see that there is an associated "bouquet boundary" which generalizes both the ideal boundary of a complete CAT(0) space and the Gromov boundary of a Gromov hyperbolic space.

Given the size of an integer polynomial P at a real number x criteria will be given to determine whether the closest root of P to x is real or non-real. If the root is real then bounds will be obtained for the distance of this root from x.

The relationship between Gromov-Witten invariants and birational geometry is a very important subject in Gromov-Witten theory. This provides the natural property of the theory. In this talk we will present the method on how genus zero Gromov-Witten invariants change under orbifold flops, which are crucial types of birational transformations. Let Y → Y' be a simple orbifold flop satisfying the so called Hard Lefschetz condition. We show that there is a classical correspondence F on the orbifold cohomology of Y and Y' preserving the orbifold degree. We further prove that F preserves the orbifold quantum cohomology.

I will review joint work with Stuart Hall, where we investigate the linear stability of Perelman's nu-functional for a closed Kahler-Ricci soliton (M,g). The essential problem is to understand whether, after a small perturbation of g, the Ricci flow will flow back to g (in which case g is stable), or flows away (unstability) . We will see that generically Kahler-Ricci solitons are unstable, but when one restricts to perturbations which are natural with respect to complex geometry, the issue is more subtle.

We consider the problem of determining whether a given ODE is geometric, i.e. of Picard-Fuchs type, and solve it in the simplest setting. Namely, we study a D3 equation. This is a Fuchsian differential equation of 3rd order with polynomial coefficients which depends on 6 parameters. Such equations arise in mirror symmetry for Fano threefolds with one-dimensional Picard lattice. We determine those values of parameters for which D3 equations are geometric.