L4

When doing arithmetic geometry, it is helpful to have invariants of the objects which we are studying that see both the arithmetic and the geometry. Motivic homotopy theory allows us to produce new invariants which generalise classical topological invariants, such as the Euler characteristic of a variety. These motivic invariants not only recover the classical topological ones, but also provide arithmetic information. In this talk, I'll review the construction of a motivic Euler characteristic, then study its arithmetic properties, and mention some applications. I'll then talk about work in progress with Ran Azouri, Stephen McKean and Anubhav Nanavaty which studies a "higher Euler characteristic", allowing us to produce an invariant of automorphisms valued in an arithmetically interesting group. I'll then talk about how to relate part of this invariant to a more classical invariant of quadratic forms.

I will discuss the Brill-Noether theory of a general elliptic 𝐾3 surface using wall-crossing with respect to Bridgeland stability conditions. As an application, I will provide an example of a general 𝑘-gonal curve from the perspective of Hurwitz-Brill-Noether theory. This is joint work with Gavril Farkas and Andrés Rojas.

As discovered by Perelman, the study of ancient Ricci flows which are $\kappa$-noncollapsed is a crucial prerequisite to understanding the singularity behaviour of more general Ricci flows. In dimension three, these so-called "$\kappa$-solutions" have been fully classified through the groundbreaking work of Brendle, Daskalopoulos, and Šešum. Their classification result can be extended to higher dimensions, but only for those Ricci flows that have uniformly positive isotropic curvature (PIC), as well as weakly-positive isotropic curvature of the second type (PIC2); it appears the classification result fails with only minor modifications to the curvature assumption. Indeed, with the alternative assumption of non-negative curvature operator, a rich variety of new examples emerge, as recently constructed by Buttsworth, Lai, and Haslhofer; Haslhofer himself has conjectured that this list of non-negatively curved $\kappa$-solutions is now exhaustive in dimension four. In this talk, we will discuss some recent progress towards resolving Haslhofer's conjecture, including a compactness result for non-negatively curved $\kappa$-solutions in dimension four, and a symmetry improvement result for bubble-sheet regions. This is joint work with Anusha Krishnan and Timothy Buttsworth.