W 11:00 Yoon-Joo Kim (Columbia University): The delta-regularity and Neron mapping property of a Lagrangian fibration
A Lagrangian fibration is a special morphism whose general fiber is an abelian variety. Examples include Hitchin fibrations and arbitrary fibrations from a compact hyper-Kahler manifold. I will talk about some recent progress on the general properties of Lagrangian fibrations. The first property is a delta-regularity, which has been known for many special cases of Hitchin fibrations but was unknown for compact symplectic varieties. The second property is the Neron mapping property (in the sense of Neron and Raynaud in the 1960s), which is new even for the Hitchin fibrations. Together, they explain some old and new phenomena of Lagrangian fibrations. This is partially joint work with Mark de Cataldo and Christian Schnell.
W 15:00 Younghan Bae (University of Michigan): Curves, line bundles and abelian varieties
For a family of smooth compact algebraic curves, the relative Jacobian forms a family of abelian varieties. When the curves degenerate to singular ones, the relative Jacobian is no longer compact, and one considers compactified Jacobians instead. Although the group structure is lost in the compactified setting, some structures persist--most notably the Fourier transform--and understanding them leads to many interesting questions.
In this talk, I will relate the Fourier transform on compactified Jacobians over the moduli space of stable curves to logarithmic Abel-Jacobi theory. As an application, I will calculate the pushforward of monomials of divisors on compactified Jacobians using the twisted double ramification cycle formula. This is based on joint work with Samouil Molcho and Aaron Pixton.
W 16:30 Sung Gi Park (Princeton and the IAS): Hodge symmetries of singular varieties
The Hodge diamond of a smooth projective complex variety exhibits fundamental symmetries, arising from Poincaré duality and the purity of Hodge structures. In the case of a singular projective variety, the complexity of the singularities is closely related to the symmetries of the analogous Hodge-Du Bois diamond. For example, the failure of the first nontrivial Poincaré duality is reflected in the defect of factoriality. Based on joint work with Mihnea Popa, I will discuss how local and global conditions on singularities influence the topology of algebraic varieties.
Th 13:00 Federico Pasqualotto (UC San Diego): From instability to singularity formation in incompressible fluids
In this talk, I will first review the singularity formation problem in incompressible fluid dynamics, describing how particle transport poses the main challenge in constructing blow-up solutions for incompressible fluids. I will then outline a new mechanism that allows us to overcome the effects of particle transport, leveraging the instability seen in the classical Taylor--Couette experiment. Using this mechanism, we construct the first swirl-driven singularity for the incompressible Euler equations in R^3. This is joint work with Tarek Elgindi (Duke University).
Th 14:30 Heejong Lee (HCMC): p-adic Galois representations and modularity
Galois representations naturally arise from arithmetic geometry and automorphic representations. For the first half of the talk, I will give a brief overview of p-adic Galois representations and its relationship with automorphic representations, including one example that any high-schooler can understand. For the second half of the talk, I will introduce a result about (families of) p-adic local Galois representations and how to prove it using algebro-geometric techniques, which also have a slight dynamical flavor. This is based on a joint work with Daniel Le, Bao Le Hung, Brandon Levin, and Stefano Morra.