The CHAT series invite established professors to talk about either
(1) their math career in general
(2) their theorems or theories, but explained from a personal and historical perspective, like how they came up with the problem, what the Aha! moment was like, how the problem changes from its initial form to the published rigorous form.
The idea is that instead of talking about their latest theorems, the speakers would take a step back and talk about the trajectory of an idea, the path to the discovery of a theorem, the influence of ideas learned through a paper or a chance conversation with a colleague, and the hazards met and overcome along the way.
| Date/Time | Speaker | Title |
|---|---|---|
| Jan 01, 2021 4-5pm PST |
Ken Ribet (UC Berkeley) | Langlands correspondence and geometry
The Langlands program suggests innumerable problems in arithmetic geometry. One class of problems concerns geometric and cohomological relations between algebraic varieties in case such relations are predicted by a Langlands correspondence between spaces of automorphic forms for different algebraic groups. I will describe how my quest for such a relation led me to realize that there can be a link between the behavior of one Shimura variety in one characteristic and the behavior of a second Shimura variety in a second characteristic.
Slides Video |
| Jan 11, 2021 4-5pm PST |
Benedict Gross (UCSD) | The conjectures of Gan, Gross, and Prasad
I will review the conjectures I made with Wee Teck Gan and Dipendra Prasad, which provide a bridge between number theory and representation theory. Besides stating the various conjectures and reviewing the main results that have been obtained in this direction, I'll make some historical remarks on how we came to formulate them.
Slides Video |
| Feb 01, 2021 4-5pm PST |
Michael Harris (Columbia) | Galois representations and torsion cohomology: a series of misunderstandings
In 2013, Peter Scholze announced his proof that Galois representations with finite coefficients could be associated to torsion classes in the cohomology of certain locally symmetric spaces. The existence of such a correspondence had been predicted by a number of mathematicians but for a long time no one had the slightest idea how to construct the Galois representations. In this talk I will review some of the history of the problem, with emphasis on the many false starts and occasional successes, and on my own intermittent involvement with this and related problems.
Slides Video |
| April 05, 2021 4-5pm PST |
Barry Mazur (Harvard) | Thoughts about Primes and Knots
Knots and their exquisitely idiosyncratic properties, are the vital essence of three-dimensional
topology. Primes and their exquisitely idiosyncratic properties, are the vital essence of number
theory. A striking (and extremely useful) analogy between Knots and Primes helped me as I
became as passionate about number theory as I was (and still am) about knots. I’m delighted
to have been asked by Shekhar and Chi-Yun to be part of the ‘experiment’ in this (experimental)
series of talks: CHAT: Career, History and Thoughts to think again about this, and take part
in a Q&A with people in the seminar.
Slides Video |
| May 17, 2021 4-5pm PST |
Peter Sarnak (Princeton) | Automorphic Cuspidal Representations and Maass Forms
The building blocks for automorphic representations on GLn are the cusp forms. Even the existence of Maass cusp forms is subtle and tied to arithmetic. I will describe some of my many encounters with these trascendental objects and speculate about their role in number theory.
Slides Video |
| May 24, 2021 4-5pm PST |
Henri Darmon (McGill) | Modular functions and explicit class field theory: private reminiscences and public confessions
The problem of constructing class fields of number fields from explicit values of modular functions has its roots in the theory of cyclotomic fields and the theory of complex multiplication. The latter theory acquired a renewed currency in the second half of the 20th century through its connections to the arithmetic of elliptic curves, manifested in the work of Coates--Wiles, Rubin, Gross--Zagier, and Kolyvagin.
I will give a personal account of my path towards a (slightly) better understanding of explicit class field theory for real quadratic fields and its applications to elliptic curves, taking advantage of the CHAT format to focus on the misconceptions, false starts, and dead ends that have marked my roundabout and tortuous, but also very enjoyable, mathematical journey so far.
Slides Video |
| Dec 4, 2023 3-4pm PST |
Hélène Esnault (FU Berlin/Harvard/Copenhagen) | Codimension one in Algebraic and Arithmetic Geometry
The notions of weight in complex geometry and in \ell-adic theory in geometry over a finite field have been developed by Deligne and by the Grothendieck school. The analogy between the theories is foundational and led to predictions and theorems on both sides. On the complex Hodge theory side, not only do we have the weight filtration, but we also have the Hodge filtration.
The analogy on the \ell-adic side over a finite field hasn’t really been documented by Deligne.
Thinking of this gave the way to understand the Lang--Manin conjecture according to which smooth projective rationally connected varieties over a finite field possess a rational point.
http://page.mi.fu-berlin.de/esnault/p....
On the other hand, we know the formulation in complex geometry of the Hodge conjecture: on a smooth projective complex variety X, a sub-Hodge structure of H^{2j}(X) of Hodge type (j,j) should be supported on a codimension j cycle. The analog \ell-adic conjecture has been formulated by Tate, even over a number field. Grothendieck’s generalized Hodge conjecture is straightforwardly formulated: a sub-Hodge structure H of H^i(X) of Hodge type (i-1,1), (i-2,2), \ldots, (1,i-1) should be supported on a codimension 1 cycle. Equivalently it should die at the generic point of the variety.
This is difficult to formulate because Hodge structures are complicated to describe. But there is one instance for which we can bypass the Hodge formulation: H=H^i(X) and H^{0,i}=H^i(X, O)(=H^{i,0}=H^0(X, \Omega^i))=0. Then the conjecture descends to the field of definition of X and becomes purely algebraic. It is on the one hand related to the (quite bold) motivic conjectures predicting that H^i(X,O)=0 for all i\neq 0 should be equivalent to the triviality of the Chow group of 0-cycles over a large field (this brings us back to the proof of the Lang--Manin conjecture). On the other hand, as it is purely algebraic, one can try to think of it in the framework of today’s p-adic Hodge theory.
Slides Video |