Title: What is an automorphic representation? Abstract: Andrew Wiles' famous proof of Fermat's last theorem was crucially based on constructing a mysterious connection between two classes of mathematical objects: "modular forms" and "elliptic curves". However, if you try looking up what a modular form actually is, you'll find an elementary but very unmotivated definition involving functions on […]
Title: Boundary behavior of the Allen-Cahn equation and the construction of free boundary minimal hypersurfacesAbstract: In the late 70s, the work of Modica, Mortola, De Giorgi, and many others established deep connections between the Allen-Cahn equation, a semi-linear elliptic equation arising in the van der Waals-Cahn-Hilliard theory of phase transitions, and minimal hypersurfaces, i.e. critical […]
Title: Liquid & gaseous modules
Title: Isotrivial fibrations of hyper-Kähler manifolds.Abstract: A Lagrangian fibration of a projective hyper-Kähler manifold is called isotrivial if its smooth (abelian variety) fibers are all isomorphic to each other. Given an isotrivial fibration of a HK manifold f : X -> B with at least one rational section, we prove the following four descriptions of the fibration: […]
Title: Twisted Gan-Gross-Prasad conjecture and arithmetic fundamental lemma Abstract: The twisted Gan-Gross-Prasad (GGP) conjecture opens a way of studying (a twist of) central Asai L-values via descents and period integrals. Firstly, I will prove new cases of twisted GGP conjecture (joint work with Weixiao Lu and Danielle Wang), based on the relative trace formula approach […]
Title and abstract TBA
Abstract Plancherel formula & Harish-Chandra Schwartz space
A Brief Introduction to Gödel's Incompleteness Theorems In 1931 Kurt Gödel proved a pair of landmark results that limited the strength of formal theories of mathematics. These "incompleteness theorems" are central to modern understandings of logic, and also to numerous misunderstandings. So today, we'll explore these theorems - how they work, how they were developed […]
Title: Soap films, Plateau's laws, and the Allen-Cahn equation Abstract: Plateau's problem of minimizing area among surfaces with a common boundary is the basic model for soap films and leads to the theory of minimal surfaces. In this talk we will discuss a modification of Plateau's problem in which surfaces are replaced with regions of small […]