Zygmund-Calderón Lectures in Analysis

Visibility and Invisibility

Monday April 14, 4:00 pm, Room 202 Eckhart Hall, 1118 E 58th Street

We give a general introduction to visibility and invisibility in relation to Calderón's inverse problem. Visibility refers to finding the electrical conductivity of a medium by making voltages and current measurements at the boundary. This problem arose in geophysical prospection and it has been proposed as a diagnostic tool in medical imaging, particular early breast cancer detection. Invisibility refers to finding non- homogeneous conductivities that give rise to the same boundary measurements as an homogeneous conductivity. The conductivity equation has transformation laws that allow for design of conductivities that steer the currents around a hidden region, returning it to its original path on the far side.


Travel Time Tomography

Tuesday April 15, 4:30 pm, Room 206 Eckhart Hall, 1118 E 58th Street

In this lecture we will describe a surprising connection between Calderón's inverse problem and travel time tomography. This latter problem consists in determining the index of refraction (sound speed) of a medium by measuring the travel times of sound waves going though the medium. This problem arises in determining the inner structure of the Earth by measuring the travel times of seismic waves as well as in ultrasound imaging.


Invisibility

Wednesday April 16, 4:00pm, Room 202 Eckhart Hall, 1118 E 58th Street

We describe recent theoretical and experimental progress on making objects invisible to detection by electromagnetic waves, acoustic waves and quantum waves. Similar to the conductivity equation, Maxwell's equations and Helmholtz equations have transformation laws that allow for design of electromagnetic material parameters that steer light around a hidden region, returning it to its original path on the far side. Not only would observers be unaware of the contents of the hidden region, they would not even be aware that something was being hidden. The object, which would have no shadow, is said to be cloaked. We recount the recent history of the subject and discuss some of the mathematical and physical issues involved.