# UW RTG IPDE Summer School 2013

## Topic

## Details

**Overview**

A three-week summer school giving advanced undergraduates and beginning graduate students the opportunity to study with the University of Washington's integrated Inverse Problems/PDE group. Students will attend lectures in the morning and problem sessions in small groups with mentors in the afternoon. Advance your career by learning from experts in these fields while visiting Seattle during summer 2013, when the Pacific Northwest is at its best.

**Microlocal Analysis and Inverse Problems**

In inverse problem one attempts to determine the internal properties of a medium by making measurements outside the medium. In many instances this is not possible to do and one would like to recover the singularities of the medium parameters. For this, microlocal analysis (MA) has proven to be particularly useful. MA is, roughly speaking, local analysis in phase space, which arose as a natural development of the methods of geometrical optics. Hörmander introduced in the early 70's the concept of wave front set of a distribution and developed a calculus of Fourier integral operators (FIO). These were very important developments that led to many results in the study of singularities of solutions of partial differential equations and have also had important applications in other fields.

We will concentrate in two applications of MA to inverse problems:

Generalized Radon transforms: The Radon transform and the X-ray transform integrate a function along planes or lines. We will consider generalizations of these transforms to integrate functions (or vector fields or tensors) along more general surfaces and curves. These arise naturally, for instance, in seismic imaging.

Coupled-Physics Inverse Problems: Coupled-Physics, also called Hybrid, inverse problems are imaging modalities that have received a lot of attention in recent years due to the great promises they hold for medical imaging and other fields. By combining two or three different types of waves (or physical fields) these methods overcome limitations of classical tomography techniques and deliver otherwise unavailable, potentially life-saving diagnostic information. Among these methods are the Thermoacoustic Tomography (TAT), Photo-Acoustic Tomography (PAT), Ultrasound Modulated Optical and Impedance Tomographies (UMOT, UMEIT), Magneto-Acousto-Electric Tomography (MAET) and several other modalities combining magnetic fields with ultrasound scanning of the tissue. Closely related to these methods are so-called combined physics modalities such as Current Density Imaging and Elastography. Besides medical imaging there has been also recent interest on coupled-physics inverse methods in oil exploration in particular on the Seismo-electric effect.

Problem sessions and computer labs will be scheduled during the afternoons.

## Additional Information

**Prerequisites**

The course is aimed at graduate students, but strong advanced undergraduate students with the appropriate background might find it suitable. We assume familiarity with real analysis at the level of a first year graduate course, and the Fourier transform. Basic knowledge of linear partial differential equations and Sobolev spaces will be useful. A good reference for the background topics are the books by L.C. Evans: Partial Differential Equations or Folland: Introduction to PDE, in particular the chapters in these books which deal with linear second order partial differential equations. No previous knowledge of MA is required.

Program details

Open to advanced undergraduates and beginning graduate students.

On-campus accommodation and meals will be provided, plus a travel allowance of up to $600.

Must be a U.S. citizen or permanent resident.

Note: Applications from international students may be considered, but international students must provide their own support for travel, accommodation, and meals. In particular, small travel grants may be available for Canadian students, who should contact PIMS Deputy Director George M. Homsy for possible support.**To Apply:**

Submit by **April 1** via the online application:

Personal statement about why you would like to attend the IPDE Summer School at the University of Washington.

Names of two people whom you will ask to submit letters of recommendation.

Recommendations should be submitted using the online recommendation form.

- Mark Anastasio (Washington University, St. Louis)
- Guillaume Bal (Columbia)
- Francois Monard (UW)
- Plamen Stefanov (Purdue)
- Gunther Uhlmann (UW)

**Scientific, Summer School**

**July 8–26, 2013**

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