We will meet on Thursday 11:10 am - 12:30 pm via zoom. Please contact the organizers to get the Zoom link.

• 1/28 Stanley Xiao, On the Lawrence-Venkatesh method

Abstract: In this talk, I will give an overview of the proof of Mordell's conjecture by Faltings, from which Lawrence and Venkatesh draw their inspiration. From here we will see how Lawrence and Venkatesh prove Mordell's conjecture, especially without Faltings' height.

• 2/4 Heejong Lee, Representation of p-adic groups from the arithmetic point of view

Abstract: The main subject of this talk is the representation theory of p-adic groups, and the motivation is local Langlands correspondence. We first introduce the Galois side of local Langlands program and discuss how the information from the Galois side translates into the representations of p-adic groups. In particular, we discuss Bernstein decomposition and Bernstein center. As a final remark, I'll mention some of the recent advancements in local Langlands.

• 2/11 Stefan Dawydiak, The Plancherel formula for p-adic groups

Abstract: The Plancherel formula for R^n relates the value of an appropriate function at 1 with the integral of its Fourier transform. We will discuss how a representation of a p-adic (or real) group provides a notion of Fourier transform in direct analogy with the classical case, and try to give a precise statement of the Plancherel theorem in this case, which is due to Harish-Chandra. While intricate, it can be made explicit and computationally tractable, and we will conclude with some sample calculations for GLn for small n, as well as some words about how the Plancherel theorem relates to the Bernstein decomposition and Bernstein centre, time permitting.

• 2/18 Malors Espinosa, Buildings and Integrals

Abstract: We usually think of buildings as objects associated to algebraic groups over local fields. Nevertheless, the theory of buildings is very broad and has been extended in many directions beyond its applications in algebraic group theory and is now a huge area in its own merit and full of open problems and nice results.

 While learning about orbital integrals, and more generally about p-adic harmonic analysis, facts around buildings appear frequently and sometimes in mysterious fashion when one wants to make both meet ends: the way the modern theory exists independent of the theory of algebraic groups and the way they are used in p-adic harmonic analysis. 

 The point of my talk will be two fold: first, I want to introduce what buildings are from their own world, without reference to algebraic groups, and explain how we move towards the theory of algebraic groups and their structure. We will not start from the classical construction of a group acting in some sort of lattices, but rather see how this is just a particular situation in the whole modern theory of buildings, where the group G acting on a building appears much later than when the building is defined.

 Second, I want to show how the knowledge of this huge structure, the building, allows us to do computations and organize structures that are relevant for the theory of groups. I will try to mention some examples of this around the theory of orbital integrals and Shalika Germs.

• 2/25 Matthew Sunohara, The local Langlands correspondence for real groups

Abstract: This talk is an introduction to the local Langlands correspondence. We will motivate the problem of formulating and constructing the local Langlands correspondence from the principle of functoriality and then give an overview of its construction, due to Langlands, in the archimedean case.

• 3/4 Feodor Kogan, Arithmetic Noncommutative Geometry

Abstract: Alain Connes has initiated an operator algebra framework, inspired by statistical quantum mechanics, to study the Riemann hypothesis and related problems in number theory. The key idea is to associate to a number field a dynamical system which consists of a C*-algebra and a family of automorphisms, indexed by real numbers. To such a system one can associate a partition function, which will coincide with the Dedekind zeta function of the number field, while certain equilibrium states will carry an action of the quotient of the idele class group by its connected component of the identity. I will try to describe the construction, when the number field is just the rational numbers, explaining the construction of the dynamical system and how it relates to various number theory phenomena. I will do my best to make this talk as accessible as possible, explaining both the number theory and the operator algebra that pops up along the way.

• 3/11 Sina Zabanfahm, Rank 2 Higgs bundles

Abstract: In the first half of this talk I will give a short introduction on the moduli-space of Higgs bundles. For the second half of the talk I will discuss some arithmetic properties of rank 2 Higgs bundles over a discrete valuation ring. 

• 3/18 Ali Cheraghi, Colmez Conjecture

Abstract: Colmez conjecture relates Faltings height of a CM abelian variety to a value of the logarithmic derivative of an Artin L-function. In this talk, I will talk about Colmez conjecture and known cases of it.

• 3/25 Soheil Memariansorkhabi, Rarity of Rational Points on Hyperbolic Varieties

Abstract: In 1983, Faltings proved that a smooth projective curve of genus >1 over Q has only finitely many rational points. Lang stated a series of conjectures on generalization of Faltings’ theorem to higher-dimensional hyperbolic varieties. These conjectures have surprising consequences on uniform boundedness of the number of rational points on curves of genus> 1.  In this talk, I’ll discuss what Lang’s conjectures predict about the qualitative structure of rational points for various notions of hyperbolic varieties, particularly for varieties of general type and varieties with ample cotangent bundle.

• 4/1 Kenneth Chiu, A brief introduction to infinity categories

Abstract: We will discuss simplicial sets, infinity groupoids, quasicategories, and their motivations written in Lurie's book Higher Topos Theory.

• 4/8 Hyungseop Kim, Snaith's theorem

Abstract: Snaith's theorem ('79) interprets the topological K-theory as a localization of a suspension spectrum at the Bott element. We review a chromatic proof of the theorem by Hopkins and Mathew.  

We will meet on Tuesday 12:10 pm via zoom. Please contact the organizers to get the Zoom link.

• 10/6 Heejong Lee, Galois representation and its deformation

Abstract: In this talk, I'll try to convince you why Galois representations are interesting and explain some of the ways to study them. In particular, I will introduce the Galois deformation, the theory to study different Galois representations with the same reduction. One of its main application was the proof of Fermat's last theorem by Andrew Wiles and Richard Taylor. If time permits, I will discuss relevant results on Galois representations that are used to compute Galois deformation rings.

After talk: Further topics in this direction can be Grothendieck's monodromy theorem/Weil-Deligne representation (can be used to study Galois deformation for l≠p), p-adic Hodge theory (to study Galois deformation for l=p), and the use of Galois cohomology in Galois deformation theory is also interesting(to compute dimension of Galois deformation ring, present global deformation ring over local deformation rings). I would be happy to discuss these if one is interested.

• 10/20 Debanjana Kundu, Overview of Iwasawa Theory

Abstract:  In this introductory talk, we will give an overview of classical Iwasawa theory (starting around the late 1950's). We will explain some of the guiding questions and conjectures in this area. Then we will move on to the Iwasawa theory of Elliptic Curves (introduced by Mazur in the 1970's). We will touch upon some work of Greenberg and Bloch-Kato. Finally, we will state the Iwasawa Main Conjecture (which is now a theorem!) and if time permits, point out the key ideas in Kato's proof of one direction of the Main Conjecture.

Prerequisites: I will not assume that everyone in the audience is familiar with the notion of number fields/ class groups/ elliptic curves/ Selmer groups. But, if you are familiar with (some or all of) these terms it will be easier to follow. I will be using terminology from Galois theory; I will try to ensure that even if you don't think about Galois groups everyday, you will not feel lost :) There will also be a short section which will be advanced to (hopefully) satisfy the mature members of the audience :)

• 10/27 Matthew Sunohara, An introduction to the Langlands program

Abstract: This talk is an introduction to the Langlands program with emphasis on its historical development. First, we will follow some of the main events leading up to Langlands’s 1967 letter to Weil in which he first communicated one of the two central conjectures of the Langlands program: the principle of functoriality. Then, we will briefly touch upon some of the developments in the years after the letter to Weil, including refinements to the principle of functoriality and the second central conjecture of the Langlands program: the reciprocity conjecture.

• 11/3 Kenneth Chiu, Functional transcendence and o-minimal geometry

Abstract: The Ax-Lindemann-Weierstrass theorem (for exponential functions) is a functional analogue of the Lindemann-Weierstrass theorem in transcendental number theory. After introducing the statement, we will discuss the two important o-minimal geometry ingredients, the Pila-Wilkie theorem and the definable Chow theorem, that were used in Tsimerman's proof of this theorem. O-minimal geometry bridges analytic geometry and algebraic geometry. We will give many examples of o-minimal sets in the Euclidean space.

• 11/10 George Papas, The Pila-Zannier method and the Manin-Mumford Conjecture

Abstract: The Manin-Mumford Conjecture, now a Theorem first proved by Raynaud in the 80s, is a result describing the intersection of a subvariety of an abelian variety with the set of torsion points of the abelian variety. Though it was not the first proof of the Manin-Mumford Conjecture, Pila and Zannier's paper has been extremely influential in developments in the field of Arithmetic Geometry in the past decade. The aim of this talk will be to provide a summary of their paper, focusing on its general idea. Time permitting, we hope to also talk about some further developments in the field that have followed their method.

• 11/17 Nic Fellini, An introduction to L-functions and converse theorems

Abstract: This talk is an introduction to L-functions with a focus on their analytic properties. First, we will see a variety of L-functions taking a fairly general approach and how many of the well known classical L-functions fit into this framework.  Once acquainted with these L-functions, we will see three examples of so-called "converse theorems" and how the analytic properties of L-functions can characterize the objects they are associated to. We conclude with some more modern applications of converse theorems. 

• 11/24 Stefan Dawydiak, The Iwahori-Hecke algebra and a glimpse at constructible-coherent equivalences

Abstract: You may (or may not) have heard that there is a philosophical correspondence between harmonic analysis on a p-adic group G and the complex geometry of its Langlands dual G^\check. In this talk, we'll see the archetypal incarnation of this philosophy. We'll first define the Iwahori-Hecke algebra of a p-adic group and motivate some of its usefulness starting even from finite groups, before defining some algebro-geometric objects for the dual group, namely, the Steinberg variety of triples. We'll see it's vaguely plausible that its K-theory is isomorphic to the Iwahori-Hecke algebra, and very briefly mention a categorical upgrade of this statement.

• 12/1 Malors Emilio Espinosa Lara, A Fundamental Lemma on Fundamental Theory it is

Abstract: Probably most of us heard of the Langlands Program from our very early undergrad years, or even before, as the theory that connects everything and that a central part of it was a statement that goes by the name The Fundamental Lemma. 

One of the reasons this lemma gained notoriety is because the way it evolved: originally it was expected to be a relatively easy claim to prove and it turned out to be a very hard problem that required way more mathematical tools than those that were expected. At the same time, without it the theory would not hold. It was finally proven by Bao Châu Ngô in 2008. 

As we know, the Langlands Program has as central conjecture The Principle of Functoriality, which has been proven in several instances and has had already great number of important consequences, via a method that is called Endoscopy and where the Fundamental Lemma plays a central role. Yet there is just so much of Functoriality that you can prove with endoscopy and to address this issue in the early 2000’s Langlands proposed a new project: Beyond Endoscopy. It is a new strategy which requires new tools, new ideas and is full of open problems whose main aim is to prove the whole principle of Functoriality. 

One of those new problems in his proposal is what we might call the Beyond Endoscopy Fundamental Lemma, which plays a similar role to Beyond Endoscopy than the original one did for Endoscopy. 

It will be the purpose of this talk to do a “tourist version” of several of the things I have said above with the aim of trying to explain, if not deeply at least convincingly, what is Beyond Endoscopy and where is that this "Beyond Endoscopy Fundamental Lemma" appears. 

• 12/8 Ali Cheraghi, Serre's conjecture

• 12/15 Artane Siad, Homological stability in number theory

Abstract: I’ll talk about a couple of examples where homological stability, a phenomenon in algebraic topology, helps us understand questions in number theory over function fields.