Genus of abstract modular curves with level \(\ell\) structure

Ana Cadoret
Ecole Polytechnique; Member, School of Mathematics
November 21, 2013
To any bounded family of \(\mathbb F_\ell\)-linear representations of the etale fundamental of a curve \(X\) one can associate families of abstract modular curves which, in this setting, generalize the `usual' modular curves with level \(\ell\) structure (\(Y_0(\ell), Y_1(\ell), Y(\ell)\) etc.). Under mild hypotheses, it is expected that the genus (and even the geometric gonality) of these curves goes to \(\infty\) with \(\ell\). I will sketch a purely algebraic proof of the growth of the genus - working in particular in positive characteristic.

Concert Talks - Lara St. John and Martin Kennedy, Edward T. Cone Concert Series

Lara St. John, Martin Kennedy, Sebastian Currier
November 24, 2013
The Edward T. Cone Concert Series continued on November 22 with Musical Geographies, featuring Lara St. John and Martin Kennedy. Over the years, acclaimed violinist Lara St. John has collected thousands of folk tunes from Eastern Europe and the Middle East, and has commissioned composers to make arrangements of some of the material. In this video Sebastian Currier, Artist-in-Residence, Lara St. John, and Martin Kennedy take the stage to discuss the concert.

Random Cayley Graphs

Noga Alon
Tel Aviv University; Member, School of Mathematics
November 25, 2013
The study of random Cayley graphs of finite groups is related to the investigation of Expanders and to problems in Combinatorial Number Theory and in Information Theory. I will discuss this topic, describing the motivation and focusing on the question of estimating the chromatic number of a random Cayley graph of a given group with a prescribed number of generators.

Toward Better Formula Lower Bounds: An Information Complexity Approach to the KRW Composition Conjecture

Or Meir
Institute for Advanced Study; Member, School of Mathematics
November 26, 2013
One of the major open problems in complexity theory is proving super-polynomial lower bounds for circuits with logarithmic depth (i.e.,\(P \not\subseteq NC_1\) ). This problem is interesting both because it is tightly related to understanding the power of parallel computation and of small-space computation, and because it is one of the first milestones toward proving super-polynomial circuit lower bounds.

Diffusion for the (Markov) Anderson model

Jeffrey Schenker
Michigan State University; Member, Institute for Advanced Study
November 26, 2013
I will discuss the proof by Yang Kang and myself of diffusion for the Markov Anderson model, in which the potential is allowed to fluctuate in time as a Markov process. However, I want to highlight the method of the proof more than the result itself and to speculate a bit about how things might work for the Anderson model itself.

A solution to Weaver's \(KS_2\)

Adam Marcus
Yale University
December 2, 2013
We will outline the proof that gives a positive solution of to Weaver's conjecture \(KS_2\). That is, we will show that any isotropic collection of vectors whose outer products sum to twice the identity can be partitioned into two parts such that each part is a small distance from the identity. The distance will depend on the maximum length of the vectors in the collection but not the dimension (the two requirements necessary for Weaver's reduction to a solution of Kadison Singer).

From Gromov to the Moon

Joel Fish
Massachusetts Institute of Technology; Member, School of Mathematics
December 2, 2013
I will present some recent applications of symplectic geometry to the restricted three body problem. More specifically, I will discuss how Gromov's original study of pseudoholomorphic curves in the complex projective plane has led to the construction of global surfaces of section, and more generally finite energy foliations, below and slightly above the first Lagrange point in the regularized planar circular restricted three body problem. The talk will be accessible to a general mathematical audience.

Multi-party Interactive Coding

Allison Lewko
Columbia University; Member, School of Mathematics
December 3, 2013
We will discuss interactive coding in the setting where there are n parties attempting to compute a joint function of their inputs using error-prone pairwise communication channels. We will present a general protocol that allows one to achieve only a constant multiplicative overhead in communication complexity compared to the error-free case in the presence of adversarial error. We will discuss the implications in other error models as well, and some accompanying lower bounds. This is joint work with Abhishek Jain and Yael Kalai.