After discussing a few examples of herding and flocking, we review the mean field game paradigm as introduced by Lasry and Lions. Using a probabilistic reformulation of the problem, we demonstrate how the solutions of these models can be identified with solutions of forward - backward stochastic differential equations (FBSDEs) of McKean-Vlasov type. We give existence and uniqueness results for a large class of these FBSDEs and if time permits, we discuss the similarities and differences with the solutions of the optimal control of McKean-Vlasov stochastic differential equations

We consider the two-dimensional water wave problem in the case where the free interface of the fluid meets a vertical wall at a possibly non-trivial angle; our problem also covers interfaces with angled crests. We assume that the fluid is inviscid, incompressible, and irrotational, with no surface tension and with air density zero. We construct a low-regularity energy and prove a closed energy estimate for this problem, and we show that the two-dimensional water wave problem is solvable locally in time in this framework. Our work differs from earlier work in that, in our case, only a degenerate Taylor stability criterion holds, with $-\frac{\partial P}{\partial \bold{n}} \ge 0$, instead of the strong Taylor stability criterion $-\frac{\partial P}{\partial \bold{n}} \ge c > 0$. This work is partially joint with Rafe Kinsey.

In many imaging problems such as X-ray crystallography, detectors can only record the intensity or magnitude of a diffracted wave as opposed to measuring its phase.  Phase retrieval concerns the recovery of an image from such phaseless information.  Although this problem is in general combinatorially hard, it is of great importance because it arises in many applications ranging from astronomical imaging to speech analysis. This talk discusses novel acquisition strategies and novel convex and non-convex algorithms which are provably exact, thereby allowing perfect phase recovery from a minimal number of noiseless and intensity-only measurements. More importantly, we also demonstrate that our noise-aware algorithms are stable in the sense that the reconstruction degrades gracefully as the signal-to-noise ratio decreases. This may be of special contemporary interest because phase retrieval is at the center of spectacular current research efforts collectively known under the name of coherent diffraction imaging aimed, among other things, at determining the 3D structure of large protein complexes.  

The origin of diffraction limit in wave physics, and the way to overcome it, can be revisited using the time-reversal mirror concept. According to time-reversal symmetry, a broadband wave can be focused both in time and space regardless of the complexity of a scattering medium. In a complex environment a time-reversal mirror acts as an antenna that uses complex environments to appear wider than it is, resulting in a refocusing quality that does not depend on the time-reversal antenna aperture.  The broadband nature of time-reversed waves distinguishes them from continuous phase-conjugated waves and allows revisiting the origin of diffraction limits, suggesting new ways to obtained subwavelength focusing for broadband waves.

One approach consists in introducing the initial source inside a micro structured medium made of subwavelength resonators with a mean distance smaller than the used wavelengths. It will be shown that, for a broadband source located inside such structure, a time-reversal mirror located in the far field radiated a time-reversed wave that interacts with the medium (random or periodic) to regenerate not only the propagating but also the evanescent waves required to refocus below the diffraction limit. This focusing process is very different from the one developed with superlenses made of negative index material only valid for narrowband signals.  We will emphasize the role of the frequency diversity in time-reversal focusing and a modal description of the spatiotemporal focusing will be presented. It shows the super-resolution properties obtained with acoustic and electromagnetic waves suggesting for the future also new kind of metamaterials for optical waves.

Another approach is related to the concept of a perfect time-reversal experiment that needs, not only to time-reverse the wavefield but also to time-reverse the source. It is the concept of an acoustic or electromagnetic “sink” or drain that is related to the perfect absorber theory. Is it possible to build a blackbody of infinitively small size? 

Frank Bäuerle and Tony Tromba from UC Santa Cruz, will describe UC's Calculus Online, now available to all UC students through our new cross campus enrollment system as well as to all non matriculated students including foreign nationals. Calculus I for Science and Engineering Students has been successfully running for over a year and Calculus II since the Spring. Calculus III and IV are currently in development.
The courses have many components, from introductory welcome lectures, historical enrichment video lectures, online lecture videos ( all synchronized with an online interactive e-text originally developed for print by UCLA Professor Jon Rogawski), to an online discussion forum platform all accessible via UC's Canvas Learning Management System. We would very much welcome questions and suggestions.