Math 9400, Topics in Analysis, Fall 2011

MATH 9400
Fall 2011 Topics in Analysis
Course Information

Instructor Cristian E. Gutiérrez
Wachman Hall 432
Phone: 1-7284
email: gutierre@temple.edu

Lectures Tuesdays and Thursdays 2:00 - 3:20 PM, Wachman Hall 527.

References C. E. Gutiérrez. The Monge-Ampère equation, Birkhaüser, Boston, MA, 2001.
C. Villani. Optimal transport, old and new, Grundlehren der mathematischen Wissenschaften, Vol.338, Springer-Verlag, 2009. Available for download at here
M. Born and E. Wolf. Principles of Optics, Electromagnetic theory, propagation, interference and diffraction of light. Cambridge University Press, seventh (expanded), 2006 edition, 1959.
U. Leonhardt and T. Philbin. Geometry and Light, the science of invisibility. Dover Publications, NY, 2010.

Syllabus Fully nonlinear pdes appear in several areas within Mathematics and in applications in broader scientific disciplines such as fluid dynamics, phase transitions, mathematical finance, geometric optics, and image processing in computer science. In the past few decades, there have been many new developments in this area including the understanding of regularity of generalized solutions, the study of singularities and symmetric properties of solutions. A goal in this course is to present some of these important developments including an introduction to Monge-Ampère (MA) type equations and its applications to geometric optics. These are, in general, equations involving the Jacobian determinant of a map, and arise in the mathematical description of numerous optical, acoustic, and electromagnetic applications, in particular, in lens and reflector antenna design. The course will present basic facts about the MA equation such us weak solutions, existence, uniqueness and describe regularity results. We next build up on these ideas to show how to construct generalized solutions to other problems by means of the Minkowski method from convex geometry and also by optimal mass transportation. The physical background underlying some of these problems is related to the Maxwell equations that will described in detail. The course will be useful for graduate students interested in analysis, applied mathematics, physics and engineering.

Homework
Homework 1 (due 9/13/11).
Homework 2 (due 9/29/11.

MATH 9400 Fall 2011	Topics in Analysis Course Information

Instructor	Cristian E. Gutiérrez Wachman Hall 432 Phone: 1-7284 email: gutierre@temple.edu

Lectures	Tuesdays and Thursdays 2:00 - 3:20 PM, Wachman Hall 527.

References	C. E. Gutiérrez. The Monge-Ampère equation, Birkhaüser, Boston, MA, 2001. C. Villani. Optimal transport, old and new, Grundlehren der mathematischen Wissenschaften, Vol.338, Springer-Verlag, 2009. Available for download at here M. Born and E. Wolf. Principles of Optics, Electromagnetic theory, propagation, interference and diffraction of light. Cambridge University Press, seventh (expanded), 2006 edition, 1959. U. Leonhardt and T. Philbin. Geometry and Light, the science of invisibility. Dover Publications, NY, 2010.


Syllabus	Fully nonlinear pdes appear in several areas within Mathematics and in applications in broader scientific disciplines such as fluid dynamics, phase transitions, mathematical finance, geometric optics, and image processing in computer science. In the past few decades, there have been many new developments in this area including the understanding of regularity of generalized solutions, the study of singularities and symmetric properties of solutions. A goal in this course is to present some of these important developments including an introduction to Monge-Ampère (MA) type equations and its applications to geometric optics. These are, in general, equations involving the Jacobian determinant of a map, and arise in the mathematical description of numerous optical, acoustic, and electromagnetic applications, in particular, in lens and reflector antenna design. The course will present basic facts about the MA equation such us weak solutions, existence, uniqueness and describe regularity results. We next build up on these ideas to show how to construct generalized solutions to other problems by means of the Minkowski method from convex geometry and also by optimal mass transportation. The physical background underlying some of these problems is related to the Maxwell equations that will described in detail. The course will be useful for graduate students interested in analysis, applied mathematics, physics and engineering.

Homework	Homework 1 (due 9/13/11). Homework 2 (due 9/29/11.