# Video Lectures

### The Cartan Geometry of the Rotating Kepler Problem

Otto van Koert

GEOMETRY/DYNAMICAL SYSTEMS

In this talk we shall discuss the Cartan geometry of the rotating Kepler problem. The rotating Kepler problem appears as the limit of the restricted planar three-body body when one of the masses goes to zero. As such...

### Local-Global Compatibility and Monodromy

Given a cuspidal automorphic representation of GL(n) which is regular algebraic and conjugate self-dual, one can associate to it a Galois representation. This Galois representation is known in most cases to be compatible with local Langlands...

### Fluctuations of the Nodal Line Length of Laplace eigenfunctions on the Arithmetic torus

Igor Wigman

ANALYSIS AND MATHEMATICAL PHYSICS SEMINAR

Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspace with Gaussian probability measure. This induces a notion of a random Gaussian Laplace eigenfunctions on the torus. We...

### Mechanizing the Odd Order Theorem: Local Analysis

Georges Gonthier

Abstract: In addition to formal definitions and theorems, mathematical theories also contain clever, context-sensitive notations, usage conventions, and proof methods. To mechanize advanced mathematical results it is essential to capture these...

### Contacting the Moon

Urs Frauenfelder

GEOMETRY/DYNAMICAL SYSTEMS

The restricted 3-body problem has an intriguing dynamics. A deep observation of Jacobi is that in rotating coordinates the problem admits an integral. In joint work with P. Albers, G. Paternain and O. van Koert, we...

### Weak Stability Boundary and Capture in the Three-Body Problem

Edward Belbruno

GEOMETRY/DYNAMICAL SYSTEMS

The problem of capture in the planar restricted three-body problem is addressed. In particular, weak capture is described, which occurs at a complicated region called the weak stability boundary, where the motion is...

### Efficiently Learning Mixtures of Gaussians

Given data drawn from a mixture of multivariate Gaussians, a basic problem is to accurately estimate the mixture parameters. We provide a polynomial-time algorithm for this problem for any fixed number ($k$) of Gaussians in $n$ dimensions (even...

### Cross-Validation and Mean-Square Stability

Sergei Vassilvitskii

A popular practical method of obtaining a good estimate of the error rate of a learning algorithm is k-fold cross-validation. Here, the set of examples is first partitioned into k equal-sized folds. Each fold acts as a test set for evaluating...

### Moment-Angle Complexes, Spaces of Hard-Disks and Their Associated Stable Decompositions

Fred Cohen

Topological spaces given by either (1) complements of coordinate planes in Euclidean space or (2) spaces of non-overlapping hard-disks in a fixed disk have several features in common. The main results, in joint work with many people, give...

### Universality and Chaos in Two-Dimensional Classical Ising Spin Glasses

David Huse

ANALYSIS/MATHEMATICAL PHYSICS SEMINAR

We develop the droplet scaling theory for the low temperature critical behavior of two-dimensional Ising spin glasses. The models with integer bond energies vs. continuously-distributed bond energies are in the...