## The Gravity of Kepler Systems

Eric Agol

University of Washington

September 15, 2015

Eric Agol

University of Washington

September 15, 2015

Valentin Blomer

Georg-August-Universität Göttingen; von Neumann Fellow, School of Mathematics

September 17, 2015

Starting from the Poisson summation formula, I discuss spectral summation formulae on GL(2) and GL(3) and present a variety of applications to automorphic forms, analytic number theory, and arithmetic.

Ziyang Gao

Member, School of Mathematics

September 21, 2015

Steven Frankel

Member, School of Mathematics

September 21, 2015

Ana Caraiani

Princeton University; Veblen Research Instructor, School of Mathematics

September 21, 2015

Benedikt Ahrens

Member, School of Mathematics

September 21, 2015

Eshan Chattopadhyay

University of Texas at Austin

September 21, 2015

We explicitly construct an extractor for two independent sources on $n$ bits, each with min-entropy at least $\log^C n$ for a large enough constant $C$. Our extractor outputs one bit and has error $n^{-\Omega(1)}$. The best previous extractor, by Bourgain, required each source to have min-entropy $.499n$. A key ingredient in our construction is an explicit construction of a monotone, almost-balanced boolean function on n bits that is resilient to coalitions of size $n^{1-\delta}$, for any $\delta > 0$.

Andrey Kravtsov

University of Chicago

September 22, 2015

June Huh

Princeton University; Veblen Fellow, School of Mathematics

September 22, 2015

Robert Hough

Member, School of Mathematics

September 22, 2015