HColl_2015-2016

We will discuss and give an overview of the geometric inverse problem which consists in determining a Riemannian metric on a manifold with boundary from boundary measurements. More precisely, one aims to see if the lengths of all geodesics relating boundary points and/or their tangent vectors at the endpoints allow to determine the metric inside the manifold up to gauge invariance.

Gero Friesecke: Identifying atomic structure via diffraction of standard and twisted X-rays

Diffracting X-rays from crystallized samples is the central method for the determination of atomic structure in materials science and molecular biology. It relies on the spectacular phenomenon (which was discovered in 1912 by Von Laue and which I will of course explain mathematically in the talk) that the diffraction pattern consists of DISCRETE spots. This phenomenon has nothing to do with atomistic discreteness of the sample. Diffraction from highly regular but non-periodic, say helical, molecules yields a CONTINUOUS diffraction image (even when the atoms are idealized as point particles), from which the detailed atomic structure CANNOT be inferred in practice.
In recent work with Dominik Juestel (TUM) and Richard D. James (University of Minnesota) we asked ourselves the question ''why crystals''.
The answer (arXiv 1506.04240) is ''because the incoming X-rays used by the experimentalists are plane waves''. If you have, say, a helical, molecule, you should use a new family of incoming X-rays (mathematically: new exact solutions to Maxwell's equations) which call twisted X-rays; these will give a discrete diffraction pattern. In computer simulations we showed that atomic structure can now be inferred, just as in the crystal case.
Our construction ''symmetric structure --> incoming X-ray waveform'' is related to the abstract mathematical map ''symmetry group --> character group'' and the novel insight that the latter has a natural embedding as a solution manifold to Maxwell's equations. The discreteness of the diffraction pattern is a consequence of A. Weil's generalization of Poisson summation to Abelian groups, a result which has not hitherto been thought of as relevant to applied science.
In my talk I will not assume any prior knowledge of X-ray diffraction, and begin by explaining how longstanding physical challenges (e.g., infer atomic structure from an experimental X-ray diffraction pattern) translate to mathematical problems (e.g., the phase problem for the Fourier transform).

Robert Haslhofer: Weak solutions for the Ricci flow

The Ricci flow is a geometric heat equation designed to evolve Riemannian manifolds towards optimal ones. Its most famous topological applications are Perelman’s proof of the Poincare and geometrization conjecture, and the Brendle-Schoen proof of the differentiable sphere theorem. The key task in order to understand the flow in more general situations is to analyze the formation of singularities and to find ways to continue the flow beyond the first singular time. In particular, it has been a longstanding open problem to find a notion of weak (generalized) solutions for the Ricci flow. In this lecture, I’ll first give a general survey of the field and then describe a new class of estimates from my recent work with Naber. Our estimates are strong enough to characterize solutions of the Ricci flow. Based on our estimates we develop a theory of weak solutions.

Tere M. Seara: Oscillatory orbits in the restricted planar three body problem

In this talk we will show the existence of some special motions in the restricted planar three body problem, which models the motion of a massless body under the Newtonian gravitational force of two bodies evolving in Keplerian ellipses.

The possible motions the massless body can perform were already known by Chazy (1922), who gave a complete classification of all possible states that the body q(t) can approach as time tends to infinity. The possible final states are reduced to four: Hyperbolic, Parabolic, Bounded and Oscillatory.

Examples of all these types of motion, except the oscillatory ones, were already known by Chazy.

In this talk, we prove the existence of oscillatory motions for any value of the masses of the primaries assuming they move in ellipses whose exectricity is small enough.

The key idea is to show that, in a suitable system of coordinates due to McGehee, the "infinity" can be seen as a topologically invariant normally Hyperbolic manifold filled by periodic orbits with tansversal heteroclinic intersections. Then we prove that there are some orbits which follow closely these heteroclinic connexions.

The main achievement is to rigorously prove the existence of these orbits without assuming the mass ratio between the primaries small since then this transversality can not be checked by means of classical perturbation theory.

This is a joint work with M. Guardia and P. Martin.

Valentino Tosatti: The Ricci flow on compact Kähler manifolds

The behavior of the Ricci flow on compact Kähler manifolds is intimately related to the complex structure of the manifold. In particular on projective manifolds it has direct connections with the minimal model program in algebraic geometry. It is known that the maximal existence time of the flow can be computed from simple cohomological data. In the case when this is finite, I will give a geometric description of the set where the singularities occur. When the maximal existence time is infinite, I will discuss what is known about metric behavior as time goes to infinity.