We will discuss a simple proof that the symplectic mapping class groups of many K3s are infinitely generated, extending a recent result of Sheridan and Smith. The argument will be based on some basic family Seiberg-Witten theory and algebraic geometry.

According to the Yau-Tian-Donaldson conjecture, the existence of a constant scalar curvature Kähler (cscK) metric in the cohomology class of an ample line bundle $L$ on a compact complex manifold $X$ should be equivalent to an algebro-geometric "stability condition" satisfied by the pair $(X,L)$. The cscK metrics are the critical points of Mabuchi's K-energy functional $M$, defined on the space of Kähler potentials, and an important result of Chen-Cheng shows that cscK metrics exist iff $M$ satisfies a standard growth condition (coercivity/properness). Recently the speaker has shown that the K-energy is indeed proper if and only if the polarized manifold is stable. The stability condition is closely related to the classical notion of Hilbert-Mumford stability. The speaker will give a non-technical account of the many areas of mathematics that are involved in the proof. In particular, he hopes to discuss the surprising role played by arithmetic geometry in the spirit of Arakelov, Faltings, and Bismut-Gillet-Soule.

According to the Yau-Tian-Donaldson conjecture, the existence of a constant scalar curvature Kähler (cscK) metric in the cohomology class of an ample line bundle $L$ on a compact complex manifold $X$ should be equivalent to an algebro-geometric "stability condition" satisfied by the pair $(X,L)$. The cscK metrics are the critical points of Mabuchi's K-energy functional $M$, defined on the space of Kähler potentials, and an important result of Chen-Cheng shows that cscK metrics exist iff $M$ satisfies a standard growth condition (coercivity/properness). Recently the speaker has shown that the K-energy is indeed proper if and only if the polarized manifold is stable. The stability condition is closely related to the classical notion of Hilbert-Mumford stability. The speaker will give a non-technical account of the many areas of mathematics that are involved in the proof. In particular, he hopes to discuss the surprising role played by arithmetic geometry in the spirit of Arakelov, Faltings, and Bismut-Gillet-Soule.

The Yamabe invariant is a real-valued diffeomorphism invariant coming from Riemannian geometry. Using Seiberg-Witten theory, LeBrun showed that the sign of the Yamabe invariant of a Kähler surface is determined by its Kodaira dimension. We consider the extent to which this remains true when the Kähler hypothesis is removed. This is joint work with Claude LeBrun.

Gromov-Witten invariants of a given Kahler target space are defined as suitable intersection numbers in moduli spaces of stable maps of complex curves into the target space. Their K-theoretic analogues are defined as holomorphic Euler characteristics of suitable vector bundles over these moduli spaces.

We will describe how the Kawasaki-Riemann-Roch theorem expressing holomorphic Euler characteristics in cohomological terms leads to the adelic formulas for generating functions encoding K-theoretic Gromov-Witten invariants.