Summer Graduate School

L 2 Spectra of K for apertures π 15 , · · · 14π 15 , π

The field of Integral Equations has a long and distinguished history, being the driving force behind many fundamental developments in various areas of mathematics including Harmonic Analysis, Partial Differential Equations, Potential Theory, Scattering Theory, Functional Analysis, Complex Analysis, Operator Theory, Mathematical Physics and Numerical Analysis.

This school will:

1. introduce graduate students to the systematic study of integral equations;
2. present some of the latest theoretical advancements in the field and open problems; and
3. involve participants in a hands-on discovery lab focused on deriving results about integral operators in two dimensions relevant for both the theoretical and numerical treatment of Integral Equations in two dimensions. The curriculum of this program will be accessible and will have a broad appeal to graduate students from a variety of mathematical areas (both theoretical and applied).

School Structure
The school will offer the following mini-courses:

1. Calderon-Zygmund Theory of Singular Integral Operators, taught by Dorina Mitrea and Irina Mitrea
2. Boundary and Volume Integral Equations for Elliptic PDEs with Applications to Inverse Problems, taught by Fioralba Cakoni and Shari Moskow

Problem sessions will be held daily, led by the TAs.  The goal of these sessions is to allow for interactive exploration on the themes covered in the mini-courses.
An interactive discovery lab will be held each afternoon.  The discovery lab will be focused on a series of hands-on projects aimed at establishing spectral properties of singular integral operators acting on Lebesgue spaces on the boundary of polygonal domains in two dimensions.  Lecturers and TA's will oversee and guide the projects in the discovery lab.
Professional development activities will also be offered throughout the two weeks.

Suggested Prerequisites
Real Analysis (Lebesgue’s integration theory), basic Functional Analysis, and an introductory course in Partial Differential Equations.  Suggested references include:

• Lawrence C. Evans’ Partial Differential Equations
• Gerard B. Folland’s Real Analysis, Modern Techniques and Their Applications
• Dorina Mitrea’s Distributions, Partial Differential Equations, and Harmonic Analysis
• Walter Rudin’s Principles of Mathematical Analysis
• Walter Rudin’s Functional Analysis