**Philip T. Gressman**

Department of Mathematics

University of Pennsylvania

David Rittenhouse Lab

209 South 33rd Street

Philadelphia PA 19104

Email:

Phone: (215) 898-7845

Office: DRL 3E5C

**Office Hours Fall 2014**

Tuesday 1:00-2:00

Thursday 4:30-5:30

or by appointment

I am a professor in the

Department of Mathematics at the

Unversity
of Pennsylvania. I am also affiliated with the

Applied
Mathematics and Computational Science (AMCS) program. My research interests lie at the intersection of harmonic analysis and geometry, including
the study of geometric averaging operators (generalizing the Radon transform), oscillatory integral operators, sublevel set estimates, the Fourier
restriction problem, and related objects and applications. Recently I have also worked on
applications of harmonic analysis to PDEs, specifically the Boltzmann equation and the Gross-Pitaevskii
Hierarchy. I am currently
supported by an Alfred P. Sloan Reasearch Fellowship and NSF grant DMS-1361697.

**Spring 2014 Courses** Both courses will have websites hosted by

Canvas.

**Math 584: Mathematics of Medical Imaging and
Measurement.**
In the last 25 years there has been a revolution in image reconstruction techniques in fields from astrophysics to electron microscopy and most notably in medical imaging. In each of these fields one would like to have a precise picture of a 2 or 3 dimensional object which cannot be obtained directly. The data which is accessible is typically some collection of averages. The problem of image reconstruction is to build an object out of the averaged data and then estimate how close the reconstruction is to the actual object. In this course we introduce the mathematical techniques used to model measurements and reconstruct images. As a simple representative case we study transmission X-ray tomography (CT). In this context we cover the basic principles of mathematical analysis, the Fourier transform, interpolation and approximation of functions, sampling theory, digital filtering and noise analysis.

Meets TTh 10:30-12:00 in DRL 4C6. See the hand-out sheet. Students with limited programming experience should contact me.

**Math/AMCS 609: Real Analysis** Construction and properties of Lebesgue measures in Euclidean space, Borel measures and convergence theorems. Elementary function spaces. Some general measure theory, including the Caratheodory construction of measures from outer measures, the Radon-Nikodym theorem, the Fubini theorem, and Hausdorff measure. Stone Weierstrass theorem. Elements of classical Harmonic analysis: the Fourier transform on basic function spaces, the Hilbert and Cauchy transforms.

Meets TTh 1:30-3:00 in DRL 3C2. See the hand-out sheet.

**Fall 2012 Courses**
**Math 241: Calculus IV Section 001.** Sturm-Liouville problems, orthogonal functions, Fourier series, and partial differential equations including solutions of the wave, heat and Laplace equations, Fourier transforms. Introduction to complex analysis. Use of symbolic manipulation and graphics software.

Meets TTh
3:00-4:30 in DRL A2 with recitations on Wednesdays and Fridays at either
9 or 10. See the department calculus page for general
information and the section 001 blackboard page for more details. Also don't forget about Piazza.

**Math 584: Mathematics of Medical Imaging and
Measurement.**
In the last 25 years there has been a revolution in image reconstruction techniques in fields from astrophysics to electron microscopy and most notably in medical imaging. In each of these fields one would like to have a precise picture of a 2 or 3 dimensional object which cannot be obtained directly. The data which is accessible is typically some collection of averages. The problem of image reconstruction is to build an object out of the averaged data and then estimate how close the reconstruction is to the actual object. In this course we introduce the mathematical techniques used to model measurements and reconstruct images. As a simple representative case we study transmission X-ray tomography (CT). In this context we cover the basic principles of mathematical analysis, the Fourier transform, interpolation and approximation of functions, sampling theory, digital filtering and noise analysis.

Meets TTh 12:00-1:30 in DRL 4C8. See the hand-out sheet. Students with limited programming experience should let me know.