Both of these talks will be useful preparation for Helmut Hofer's up coming mini-course on polyfold theory on April 4th and 5th
Both of these talks will be useful preparation for Helmut Hofer's up coming mini-course on polyfold theory on April 4th and 5th.
A conjecture of Langlands-Rapoport predicts the structure of the mod p points on a Shimura variety. The conjecture forms part of Langlands' program to understand the zeta function of a Shimura variety in terms of automorphic L-functions.
I will report on progress towards the conjecture in the case of Shimura varieties attached to non-exceptional groups.
The braid group on n strands may be viewed as an infinite analog of the symmetric group on n elements with additional topological phenomena. It appears in several areas of mathematics, physics and computer sciences, including knot theory, algebraic geometry, quantum mechanics, quantum computing and cryptography.
In FT-mollification, one smooths a function while maintaining good quantitative control on high-order derivatives. This is a continuation of my talk from last week, and I will continue to describe this approach and show how it can be used to show that bounded independence fools polynomial threshold functions over various distributions (Gaussian, Bernoulli, and p-stable). I may also touch on other applications in approximation theory and learning theory.
The history of digital computing can be divided into an Old Testament whose prophets, led by Gottfried Wilhelm Leibniz, supplied the logic, and a New Testament whose prophets, led by John von Neumann, built the machines. Alan Turing, whose “On Computable Numbers, with an Application to the Entscheidungsproblem” was published shortly after his arrival in Princeton as a twenty-four-year-old graduate student in October 1936, formed the bridge between the two. In this talk, George Dyson, a Director’s Visitor in 2002–03 and the author of Turing’s Cathedral: The Origins of the Digital Universe (Pantheon, 2012), discusses the role of the Institute's Electronic Computer Project as modern stored-program computers were developed after WWII. Turing’s one-dimensional model of universal computation led directly to von Neumann’s two-dimensional implementation, and the world has never been the same since.
We explain in this talk how Ramanujan graphs can be used to devise optimal cycle codes and review how other graph families related to a construction proposed by Margulis yield interesting families of quantum codes with logarithmic minimum distance. We finish the talk by providing another simple graph theoretic construction with improved minimum distance which grows proportionally to the square root of the quantum code length. (This is joint work with Gilles Zemor.)
My goal in this talk is to survey some of the emerging applications of polynomial methods in both learning and in statistics. I will give two examples from my own work in which the solution to well-studied problems in learning and statistics can be best understood through the language of algebraic geometry.