100 Years of General Relativity

Robbert Dijkgraaf
Director of the Institute for Advanced Study and Leon Levy Professor
October 14, 2015
The general theory of relativity, which Albert Einstein formulated 100 years ago, is one of the great pillars of modern physics. It is based on profound and elegant principles that connect the physics of motion and mass to the geometry of space and time. With Einstein’s equations, even the universe itself became an object of study. Only now, after a century of calculations and observations, the full power of this theory has become visible, from black holes and gravitational lenses to the practical use of GPS devices.

Adjoint Selmer groups for polarized automorphic Galois representations

Patrick Allen
University of Illinois, Urbana-Champaign
October 15, 2015
Given the $p$-adic Galois representation associated to a regular algebraic polarized cuspidal automorphic representation, one naturally obtains a pure weight zero representation called its adjoint representation. Because it has weight zero, a conjecture of Bloch and Kato says that the only de Rham extension of the trivial representation by this adjoint representation is the split extension. We will discuss a proof of this case of their conjecture, under an assumption on the residual representation.

Non-orientable knot genus and the Jones polynomial

Efstratia Kalfagianni
Michigan State University
October 20, 2015
The non-orientable genus (a.k.a crosscap number) of a knot is the smallest genus over all non-orientable surfaces spanned by the knot. In this talk, I’ll describe joint work with Christine Lee, in which we obtain two-sided linear bound of the crosscap number of alternating link in terms of the Jones link polynomial. The bounds are often exact and they allow us to compute the crosscap numbers of infinite families of alternating knots as well as the crosscap number of 283 knots with up to twelve crossings that were previously unknown.

Remembering Patricia Crone (1945–2015)

Diana Frank, Thomas Frank, Michael Cook, Judith Herrin, Carol Bakhos, Emma Gannagé, Carmela V. Franklin, Robbert Dijkgraaf, Nicola Di Cosmo
October 24, 2015

Patricia Crone, Professor Emerita in the School of Historical Studies, helped to establish the Institute as a recognized center for the pursuit of Islamic culture and history. Crone’s insightful work shed important new light on the critical importance of the Near East—in particular on the cultural, religious, and intellectual history of Islam—in historical studies.

Quantum Ergodicity for the uninitiated

Zeev Rudnick
Tel Aviv University; Member, School of Mathematics
October 26, 2015
A key result in spectral theory linking classical and quantum mechanics is the Quantum Ergodicity theorem, which states that in a system in which the classical is ergodic, almost all of the Laplace eigenfunctions become uniformly distributed in phase space. There are similar statements which are valid for some integrable and pseudo-integrable systems, such as flat tori and rational polygons. I will give an introduction to these notions, including explanations of the undefined terms above, and describe some connections with number theory.

Algorithmic proof of the Lovasz Local Lemma via resampling oracles

Jan Vondrák
IBM Almaden Research Center; Member, School of Mathematics
October 27, 2015
For a collection of events on a probability space with specified dependencies, the Lovasz Local Lemma ("LLL") gives a sufficient condition for the existence of a point avoiding all the events. Following Moser's discovery of an efficient algorithm for many applications of the Lovasz Local Lemma, there has been extensive research on various extensions of this result. We present a unifying algorithmic proof of the Lovasz Local Lemma (and its stronger variants such as Shearer's Lemma), independent of the underlying probability space.