Georgia Institute of Technology College of Sciences

An electron interacting with the vibrational modes of a polar crystal is called a polaron. Polarons are the simplest Quantum Field Theory models, yet their most basic features such as the effective mass, ground-state energy and wave function cannot be evaluated explicitly. And while several successful theories have been proposed over the years to approximate the energy and effective mass of various polarons, they are built entirely on unjustified, even questionable, Ansätze for the wave function. 

In this talk I shall provide the first explicit description of the ground-state wave function of a polaron in an asymptotic regime: For the Fröhlich polaron localized in a Coulomb potential and exposed to a homogeneous magnetic field of strength $B$ it will be shown that the ground-state electron density in the direction of the magnetic field converges pointwise and in a weak sense as $B\rightarrow\infty$ to the square of a hyperbolic secant function--a sharp contrast to the Gaussian wave functions suggested in the physics literature. 

New and proposed missions for approaching moons, and particularly icy moons, increasingly require the design of trajectories within challenging multi-body environments that stress or exceed the capabilities of the two-body design methodologies typically used over the last several decades. These current methods encounter difficulties because they often require appreciable user interaction, result in trajectories that require significant amounts of propellant, or miss potential mission-enabling options. The use of dynamical systems methods applied to three-body and multi-body models provides a pathway to obtain a fuller theoretical understanding of the problem that can then result in significant improvements to trajectory design in each of these areas. The search for approach trajectories within highly nonlinear, chaotic regimes where multi-body effects dominate becomes increasingly complex, especially when landing, orbiting, or flyby scenarios must be considered in the analysis. In the case of icy moons, approach trajectories must also be tied into the broader tour which includes flybys of other moons. The tour endgame typically includes the last several flybys, or resonances, before the final approach to the moon, and these resonances further constrain the type of approach that may be used.

In this seminar, new methods for approaching moons by traversing the chaotic regions near the Lagrange point gateways will be discussed for several examples. The emphasis will be on landing trajectories approaching Europa including a global analysis of trajectories approaching any point on the surface and analyses for specific landing scenarios across a range of different energies. The constraints on the approach from the tour within the context of the endgame strategy will be given for a variety of different moons and scenarios. Specific approaches using quasiperiodic or Lissajous orbits will be shown, and general landing and orbit insertion trajectories will be placed into context relative to the invariant manifolds of unstable periodic and quasiperiodic orbits. These methods will be discussed and applied for the specific example of the Europa Lander mission concept. The Europa Lander mission concept is particularly challenging in that it requires the redesign of the approach scenario after the spacecraft has launched to accommodate landing at a wide range of potential locations on the surface. The final location would be selected based on reconnaissance from the Europa Clipper data once Europa Lander is in route. Taken as a whole, these methods will provide avenues to find both fundamentally new approach pathways and reduce cost to enable new missions.

I will talk about the structure of large square random matrices with centered i.i.d. heavy-tailed entries (only two finite moments are assumed). In our previous work with R. Vershynin we have shown that the operator norm of such matrix A can be reduced to the optimal sqrt(n)-order with high probability by zeroing out a small submatrix of A, but did not describe the structure of this "bad" submatrix, nor provide a constructive way to find it. Now we can give a very simple description of this small "bad" subset: it is enough to zero out a small fraction of the rows and columns of A with largest L2 norms to bring its operator norm to the almost optimal sqrt(loglog(n)*n)-order, under additional assumption that the entries of A are symmetrically distributed. As a corollary, one can also obtain a constructive procedure to find a small submatrix of A that one can zero out to achieve the same regularization.

I am planning to discuss some details of the proof, the main component of which is the development of techniques that extend constructive regularization approaches known for the Bernoulli matrices (from the works of Feige and Ofek, and Le, Levina and Vershynin) to the considerably broader class of heavy-tailed random matrices.

During the last 30 years there has been much interest in random graph processes, i.e., random graphs which grow by adding edges (or vertices) step-by-step in some random way. Part of the motivation stems from more realistic modeling, since many real world networks such as Facebook evolve over time. Further motivation stems from extremal combinatorics, where these processes lead to some of the best known bounds in Ramsey and Turan Theory (that go beyond textbook applications of the probabilistic method). I will review several random graph processes of interest, and (if time permits) illustrate one of the main proof techniques using a simple toy example.