## Adrian Albert Lectures in Algebra

The Albert Lectures are the oldest of the five lecture series. They are named after Abraham Adrian Albert (1905-1972), who received his Ph.D from Chicago in 1928, under the supervision of L.E. Dickson. Albert later returned to Chicago as a member of the faculty, and served for a time as chair of the department and President of the AMS.

### 2019 Speaker: Akshay Venkatesh (IAS)

#### Lecture 1: Period mappings and Diophantine equations.

Eckhart 206, Tuesday May 21, 4pm to 5pm

Abstract: I will give some friendly examples introducing the period mapping. This is an analytic mapping which controls many aspects of how algebraic varieties change in families. After that I will explain joint work with Brian Lawrence which shows that one can exploit transcendence properties of the period mapping to prove results about Diophantine equations. For example we give another proof of the Mordell conjecture (originally proved by Faltings): there are onlyfinitely many rational points on an algebraic curve over \(Q\) whose genus is at least 2.

#### Lecture 2: The stable homology of symplectic groups over \(\mathbb{Z}\).

Ryerson 251, Wednesday May 22, 4pm to 5pm

Abstract: There are many natural sequences of moduli spaces in algebraic geometry whose homology approaches a "limit" despite the fact that the spaces themselves have growing dimension. If these moduli spaces are defined over a field \(K\), this limiting homology comes with an extra structure -- an action of the Galois group of \(K\) -- and to understand this extra structure is often of arithmetic interest. First of all I'll introduce a few examples of this situation. Then I will specialize to the case of the moduli space of abelian varieties: I will explain the answer, why it is interesting, and some geometric consequences. No familiarity with abelian varieties will be assumed for the talk. This is joint work with Tony Feng and Soren Galatius.

#### Lecture 3: Duality of automorphic periods.

Ryerson 251, Thursday May 23, 4pm to 5pm

Abstract: Given a Lie group \(G\) acting on a manifold \(M\), we can ask: how does \(L^2(M)\) decompose as a representation of \(G\)? For example, when \(G\) is the group of rotations acting on a sphere, spherical harmonics give an explicit answer to this. I will explain how (refined versions of) this question are related to the theory of periods of automorphic forms. (I will explain what is meant by the theory of periods, but some familiarity with automorphic forms will be needed for this part of the talk.) Then I will discuss a duality that switches the question with a similar question for the dual group. The duality is conjectural but there is a lot of experimental evidence for it. Joint work, in progress, with David Ben-Zvi and Yiannis Sakellaridis.

Past Albert Lecturers include: Nathan Jacobson, Michael Atiyah, John Milnor, Jürgen Moser, Enrico Bombieri, Shing-Shen Chern, Dennis Sullivan, H. Jerome Keisler, Barry Mazur, John Griggs Thompson, William Fulton, Armand Borel, Joe Harris, Benedict Gross, J.P. Serre, Andrei Suslin, Efim Zelmanov, Karl Rubin, Phillip Griffiths, Jacques Tits, Richard Swan, Michael Artin, Jeremy Rickard, Carlos Simpson, Maxim Kontsevich, Richard Taylor, Michel Broué, Don Zagier, Alexander Merkurjev, Andrei Okounkov, Claire Voisin, Raphaël Rouquier, Jacob Lurie, Peter Sarnak, Gerd Faltings, and Spencer Bloch.