Geometric Analysis MATH 742, Fall 2016

- Lectures : Tuesday-Thursday 9:30-10:45pm, MATH 0403.

- Instructor : Pierre-Emmanuel Jabin
- Contact information, Phone : 301 405 5122, Office : 2307 in Math. Bldg and 4149 in CSCAMM, e-mail : pjabin@umd.edu
- Office hours: Tuesday-Thursday 2PM-3PM

- Textbook: There is no required textbook but the class will use materials from Optimal Transport, Old and new, by C. Villani (ISBN 978-3-540-71050-9) and Gradient Flows in metric space and in the space of probability measures, by L. Ambrosio, N. Gigli and G. Savare (ISBN-13: 978-3764387211).
- Material: A more complete description of the material covered week by week and assignmnents can be found at the address http://www2.cscamm.umd.edu/~jabin/Schedule16_742.html
- Homework: There will be several individual projects to be completed.
- Grade: The grade will based on the projects.
- Disabilities : Please inform me as soon as possible if you need accommodations because of a disability.
- Communication : Information about the course will be given during class and through this web page. In specific cases (unexpected class cancellations), students will be contacted by e-mail.
- Academic integrity : The University of Maryland, College Park has a nationally recognized Code of Academic Integrity, administered by the Student Honor Council. This Code sets standards for academic integrity at Maryland for all undergraduate and graduate students. As a student you are responsible for upholding these standards for this course. It is very important for you to be aware of the consequences of cheating, fabrication, facilitation, and plagiarism. For more information on the Code of Academic Integrity or the Student Honor Council, please visit www.shc.umd.edu. To further exhibit your commitment to academic integrity, remember to sign the Honor Pledge on all examinations and assignments: "I pledge on my honor that I have not given or received any unauthorized assistance on this examination (assignment)."
- Some examples about academic integrity : It is all right for students to discuss between them about the projects. It is similarly permitted to look for tips on the internet, from other students or other resources. It is wrong to simply copy from any student, a book, a web page or any other source.

Course Description (Preliminary)

This class introduces some basic notions of analysis and calculus of variation in the context of Riemannian geometry. In particular some of the goals of the class are

- Understand the basic concepts around solving Partial Differential
Equations in a non-Euclidian setting.

- Understand the basic theory of gradient flows on a Riemannian or pseudo-Riemannian manifold.
- Define the necessary pseudo-Riemannian structure
on the space of probability measures.

- Model and study parabolic equations on a manifold using gradient flows and understand their connection to basic stochastic processes.
- Consider some elementary flows in infinite dimension, such as the limit of a Brownian motion when the dimension goes to infinity.

No priori knowledge of stochastic analysis is required for this class, only basic notions of calculus of variation, differential geometry and real analysis.