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## Adrian Albert Lectures in Algebra

The Albert Lectures are the oldest of the four 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.

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, and Peter Sarnak.

### 2014 Speaker: Spencer Bloch (University of Chicago)

#### Lecture 1: Algebraic cycles

Friday February 28, 2014, 4:00PM--5:20PM, Ryerson 251Abstract: A rapid overflight of the great peaks in the algebraic cycle range, including Abel's theorem, the Riemann Roch theorem, enumerative geometry, higher K-theory, motivic cohomology, and the Hodge conjecture.

#### Lecture 2: Periods associated to cycles

Monday March 3, 2014, 4:00PM--5:20PM, Ryerson 251Abstract: Extensions associated to cycles, Beilinson's conjectures, Feynman amplitudes, Nahm's conjecture.

#### Lecture 3: Recent work related to the Hodge conjecture

Tuesday March 4, 2014, 4:30PM--5:20PM, Ryerson 251Abstract: I will discuss joint work with H. Esnault and M. Kerz. We use Thomason's descent theory for K-theory of singular schemes to show in both mixed characteristic and characteristic zero that the Hodge conjecture is true infinitesimally; algebraic cycles lift infinitesimally if and only if the crystalline or horizontal lifts of their cycle classes in cohomology are Hodge.

### 2012 Speaker: Peter Sarnak (Princeton University, IAS) on Randomness in Number Theory

#### Lecture 1: Thin matrix groups and Diophantine problems

Wednesday November 7, 4-5 PM, Eckhart 206Abstract: The general Ramanujan Conjectures for congruence subgroups of arithmetic groups, and approximations that have been proven towards them, are central to many diophantine applications. Recently analogous results have been established for quite general subgroups of GL(n,Z). We will describe these and review some of their applications and the ubiquity of thin groups.

#### Lecture 2: Mobius randomness and horocycle flows at prime times

Thursday November 8, 3-4 PM, Eckhart 206Abstract: The Mobius Function mu(n) is minus one to the number of prime factors of n, if n has no square factors and is zero otherwise. Understanding the randomness (often referred as the Mobius randomness principle) in this function is a fundamental and very difficult problem. We will explain a precise dynamical formulation of this randomness principle and report on recent advances in establishing it and its applications especially in connection with a prime number type theorem for horocycle flows at prime times.

#### Lecture 3: Symmetry types for families of automorphic L-functions

Friday November 9, 4-5 PM, Eckhart 206Abstract: A theory for the local distribution of the zeros for families of automorphic L-functions has been developed over the last 15 years. We will describe some of the basics of this theory, specifically the apparent 4-fold symmetry types from random matrix ensembles that arise, as well as some recent advances.

### 2011 Speaker: Jacob Lurie (Harvard)

#### Tamagawa Numbers via Nonabelian Poincaré Duality

May 13 at 4pm in Ry 251May 16 at 4pm in E 206

May 17 at 4:30pm in E 206

Abstract: Let L be a positive definite lattice. There are only finitely many positive definite lattices L' which are isomorphic to L modulo N for every N > 0. In fact, there is a formula for the number of such lattices (counted with appropriate multiplicities), called the Siegel mass formula. In the first lecture, I will review the Siegel mass formula and how it can be deduced from a conjecture of Weil on volumes of adelic points of algebraic groups. This conjecture was proven over number fields by Kottwitz, building on earlier work by Langlands and Lai. In the third lecture, I will describe some joint work with Dennis Gaitsgory which provides an approach to proving the analogue of Weil's conjecture over function fields. This approach is inspired by a ``nonabelian'' version of Poincare duality, which I will describe in the second lecture.