L4

I will explain how to obtain local L^\infty estimates for optimal transport problems. Considering entropic optimal transport and optimal transport with p-cost, I will show how such estimates, in combination with a geometric linearisation argument, can be used in order to obtain ε-regularity statements. This is based on recent work in collaboration with M. Goldman (École Polytechnique) and R. Gvalani (ETH Zurich).

In the 1990s, physicists introduced an ideal way to count curves inside a Calabi-Yau 3-fold, called the Gopakumar-Vafa (GV) theory. Building on several previous attempts, Maulik-Toda recently gave a mathematical rigorous definition of the GV invariants. We expect that the GV invariants and the Gromov-Witten (GW) invariants are related by an explicit formula, but this stands as a challenging open problem. In this talk, I will explain recent mathematical developments on the GV theory, especially for local curves, including the cohomological chi-independence theorem and the GV/GW correspondence in a special case.

I will speak about my work with Helge Ruddat on how to construct explicitly log structures and morphisms. I will also discuss some motivation. I will try to stay informal and assume no prior knowledge of log structures.

Modern approaches to infinitesimal deformations of algebro-geometric objects (like varieties) use the setting of formal moduli problems, from derived geometry. It allows to prove that all kinds of deformations are governed by a tangent complex equipped with a derived Lie algebra structure. I will use this framework to study equivariant deformations of varieties with respect to the action of an algebraic group. Then, I will explain how this theory of equivariant deformations allows us to prove a dichotomous behaviour for almost all varieties that are homogeneous under a reductive group : either they deform to characteristic 0, or they admit no deformation to any ring of characteristic greater than p.

Let $X_0 $ be a rational surface with a cyclic quotient singularity $(1,a)/r$.  Kawamata constructed a remarkable vector bundle  $F_0$  on $X_0$ such that the finite-dimensional algebra End$(F_0)$ "absorbs'' the singularity of $X_0$ in a categorical sense. If we deform over an irreducible component of the versal deformation space of $X_0$ (as described by Kollár and Shepherd-Barron), the vector bundle $F_0$ also deforms to a vector bundle $F$. These results were established using abstract methods of birational geometry, making the explicit computation of the family of algebras challenging. We will utilise homological mirror symmetry to compute End$(F)$ explicitly in a certain bulk-deformed Fukaya category. In the case of a $Q$-Gorenstein smoothing, this algebra End$(F)$ is a matrix order over $k[t]$ and "absorbs" the singularity of the corresponding terminal 3-fold singularity. This is based on joint work with Jenia Tevelev.

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.