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 proved that the corresponding energy hypersurfaces are contact for energies below and slightly above the first critical value.
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 more informal elements. This can be difficult, requiring an array of techniques closer to software engineering than formal logic, but it is essential to obtaining formal proofs of graduate-level mathematics, and can give new insight as well.
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 study the distribution of nodal length of the random Laplace eigenfunctions for high eigenvalues ("high energy limit").
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. When n is even, the compatibility is known up to semisimplification or when the representation satisfies an additional regularity condition. I will extend the compatibility to Frobenius semisimplification by identifying the monodromy operators.
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, this problem plays the role of a simple approximation. We shall discuss the Cartan curvature and some of its relations with indices in the three-body problem. This is joint work with Kai Cieliebak and Urs Frauenfelder.
In this talk we will discuss recent progresses meant as a contribution to the GLS-project, the second generation proof of the Classification of Finite Simple Groups (jointly with R. Lyons, R. Solomon, Ch. Parker).
Boolean Threshold Functions (BTF) arise in many contexts, ranging from computer science and learning theory to theoretical neurobiology. In this talk, I will present non-rigorous approaches developed in the statistical physics of disordered systems to characterize BTF in a quantitative way , with an emphasis on computational and geometrical aspects. These techniques will be illustrated on two particular cases: the celebrated perceptron (Linear Threshold Function) , and the more realistic tempotron model of a neuron [3,4].