Modern Optimization Meets Physics: Recent Progress on the Phase Retrieval Problem

Speaker: 

Emmanuel Candes

Institution: 

Stanford University

Time: 

Thursday, November 6, 2014 - 4:00pm to 5:00pm

Location: 

Natural Sciences II Room 1201

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.  

 

Estimates for elliptic systems from composite materials

Speaker: 

Yanyan Li

Institution: 

Rutgers University

Time: 

Thursday, April 16, 2015 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

The mathematical problem of gradient estimates for solutions of divergence form elliptic systems with piece-wise smooth coefficients arises in studying composite materials in applied science.

We will start with ideas in joint works with Vogelius (2000) and Nirenberg (2003) about a decade ago, in particular an open problem in the paper with Nirenberg, then discuss recent progress in closely related topics, such as gradient estimates for solutions of the Lame system with partially infinite coefficients (Arch. Rational Mech. Anal. (2015), joint with JiGuang Bao and HaiGang Li).

This is an expository lecture accessible to first year graduate students.

Controlling waves at subwavelength scales in space and time through complex media

Speaker: 

Mathias Fink

Institution: 

Institut Langevin, Ecole Supérieure de Physique et de Chimie Industrielles de la Ville de Pari

Time: 

Thursday, October 30, 2014 - 3:30pm to 4:30pm

Host: 

Location: 

RH 101

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? 

UC Calculus Online

Speaker: 

Frank Bauerle and Tony Tromba

Institution: 

University of California, Santa Cruz

Time: 

Thursday, October 16, 2014 - 4:00pm to 5:15pm

Host: 

Location: 

Natural Sciences 2 (NS2) Room 2201

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.

Dynamics of non-archimedean Polish groups

Speaker: 

Alexander Kechris

Institution: 

Caltech

Time: 

Thursday, April 17, 2014 - 4:00pm

Host: 

Location: 

RH306

Recently there has been considerable activity in the study of the dynamics of these groups and this work has led to interesting interactions between logic, finite combinatorics, group theory (both in the topological and algebraic context), topological dynamics, ergodic theory and representation theory. In this lecture I will give a survey of some of the main directions in this area of research.
 

Large-time behavior of bounded solutions of semilinear heat equations on the entire space

Speaker: 

Peter Polacik

Institution: 

University of Minnesota

Time: 

Thursday, May 8, 2014 - 4:00pm

Location: 

RH 306

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. 

Rational points on elliptic and hyperelliptic curves

Speaker: 

Manjul Bhargava

Institution: 

Princeton University

Time: 

Thursday, February 27, 2014 - 4:00pm

Host: 

Location: 

RH306

Given a random elliptic or hyperelliptic curve of genus g over Q, how many rational points do we expect the curve to have? Equivalently, how often do we expect a random polynomial of degree n to take a square value over the rational numbers?  In this talk, we give an overview of recent conjectures and theorems giving some answers and partial answers to this question.

The triangulation conjecture

Speaker: 

Ciprian Manolescu

Institution: 

UCLA

Time: 

Thursday, March 6, 2014 - 4:00pm

Host: 

Location: 

RH306

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.

Tales of Our Forefathers

Speaker: 

Barry Simon

Institution: 

Caltech

Time: 

Thursday, May 1, 2014 - 4:00pm

Location: 

NS 1201

This is not a mathematics talk but it is a talk for mathematicians. Too
often, we think of historical mathematicians as only names assigned to theorems.
With vignettes and anecdotes, I'll convince you they were also human beings and that,
as the Chinese say,"May you live in interesting times" really is a curse.

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