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.

Unlike their counterparts on bounded domains, semilinear heat equations on $R^N$ admit bounded solutions with very diverse large-time behavior. I will first present several examples of solutions with interesting and sometimes entertaining behavior in compact regions. Then I will discuss a few general results describing the behavior of more specific classes of solutions. Some ideas and techniques of more general interest, such as the Sturmian zero number and the method of spatial trajectories, will also be discussed. 

The triangulation conjecture stated that any n-dimensional
topological manifold is homeomorphic to a simplicial complex. It is
true in dimensions at most 3, but false in dimension 4 by the work of
Casson and Freedman. In this talk I will explain the proof that the
conjecture is also false in higher dimensions. This result is based
on previous work of Galewski-Stern and Matumoto, who reduced the
problem to a question in low dimensions (the existence of elements of
order 2 and Rokhlin invariant one in the 3-dimensional homology
cobordism group). The low-dimensional question can be answered in the
negative using a variant of Floer homology, Pin(2)-equivariant
Seiberg-Witten Floer homology.

A subset of the real line is called a Cantor set if it is compact,
perfect, and nowhere dense. Cantor sets arise in many areas; in this
talk we will discuss their relevance in the spectral theory of
Schr\"odinger operators. We discuss several results showing that the
spectrum of such an operator is a Cantor set, from the discovery of the
first example by Moser to a genericity result by Avila, Bochi, and
Damanik. A Cantor measure is a probability measure on the real line
whose topological support is a Cantor set. A primary example in the
spectral theory context is the density of states measure in situations
where the spectrum is a Cantor set. A conjecture of Simon claims a
strict inequality between the dimensions of the set and the measure for
the Fibonacci potential. If time permits, we will discuss a recent
result of Damanik, Gorodetski, and Yessen, which establishes this
conjecture in full generality.