Title: Introduction to the trace formula.Abstract: This will be an introductory lecture to a very informal learning seminar that will run this semester. Topics will vary, but many of the talks will be related to the trace formula, which is one of the most important tools in the theory of automorphic representations. I will start […]
Title: Canonical representations of surface groups.Abstract: For $Sigma_{g,n}$ a genus $g$ surface with $n$ punctures, we study the character variety parameterizing representations of $pi_1(Sigma_{g,n})$. This character variety has a natural action of the fundamental group of the moduli space of curves. In joint work with Josh Lam and Daniel Litt, we aim to describe the points with […]
Title: Twisted GGP conjecture for unramified quadratic extensions Abstract: The twisted Gan--Gross--Prasad conjectures consider the restriction of representations from GL_n to a unitary group over a quadratic extension E/F. In this talk, I will explain the relative trace formula approach to the global twisted GGP conjecture. In particular, I will discuss some differences between the […]
Title: Introduction to the trace formula. (Cont.)Abstract: In the first talk I presented some useful notions and translations in the theory of automorphic representations (Hecke operators and p-adic groups, locally symmetric spaces and adelic quotients, etc.). In this talk, I will give a general introduction to trace formulas, and some of the questions that we […]
Title: A reintroduction to proofs Abstract: An introduction to proofs course aims to teach how to write proofs informally in the language of set theory and classical logic. In this talk, I'll explore the alternate possibility of learning instead to write proofs informally in the language of dependent type theory. I'll argue that the intuitions […]
Title: Maximal Subellipticity Abstract: The theory of elliptic PDE stands apart from many other areas of PDE because sharp results are known for very general linear and fully nonlinear elliptic PDE. Many of the classical techniques from harmonic analysis were first developed to prove these sharp results; and the study of elliptic PDE leans heavily on […]
Title: Pyknotic setsAbstract: In this talk, I will introduce the idea of Pyknotic/Condensed sets and how they can potentially be used to solve the problem of the category of topological spaces not being abelian. I will also go over the prerequisite concepts in category theory such as the notion of Grothendieck topology, pro-object in a […]
Title: On the cone conjecture for log Calabi-Yau threefolds.Abstract: Let $Y$ be a smooth projective threefold admitting a $K3$ fibration $f: Y rightarrow mathbb{P}^{1}$ with $-K_{Y} = f^{ast} mathcal{O}(1)$. Then the extremal rays of the cone of curves of $Y$ in the region $K_{Y} < 0$ are of two types: the blowup of a smooth curve (Type […]
Title: Modularity of trianguline Galois representations Abstract: The Fontaine-Mazur conjecture (proved by Kisin and Emerton) says that (under certain technical hypotheses) a Galois representation rho: Gal_Q -> GL2(Qp) is modular if it is unramified outside finitely many places and de Rham at p. I will talk about what this means, and I will discuss an […]
Title: Speculations on Computing with the Relative Trace FormulaAbstract: I will discuss and motivate the problem of computing statistics of automorphic representations. Many statistics have been computed previously using "explicit" forms of Arthur's trace formula. I will present a very speculative plan to develop similar "explicit" versions of relative trace formulas and mention some new specific […]
Title: A Generalization of the Geroch Conjecture with Arbitrary EndsThe Geroch conjecture (proven by Schoen-Yau and Gromov-Lawson) says that the torus T^n does not admit a metric of positive scalar curvature. In this talk, I will explain how to generalize it to some non-compact settings using μ-bubbles. In particular, I will talk about why the […]