This is a one year Research Training Group (RTG) type of project in Analysis and Differential Equations at the Department of Mathematics of the University of Chicago. The project funds graduate students and visitors and the organization of a summer course aimed towards advanced undergraduates, graduate students and young postdocs in mathematics, engineering and sciences. Analysis and differential equations are central, deep and rich fields of mathematics. From an applied perspective, differential equations are the most fundamental mathematical models in all of science and engineering. They describe phenomena from quantum mechanics to weather prediction and climate and are the basic theoretical tools in science and technology. The exponential improvement in the capabilities of modern computers allows for the practical use of increasingly complex systems of differential equations. Although recent results have solved numerous outstanding problems, many still remain open. The understanding of the mathematical properties of these models is essential. The need in both academia and industry for young, well-trained mathematicians in these areas is ever growing.
The PIs along with others in the mathematical community believe that there is a severe shortage of US PhDs trained in (applied) analysis and differential equations. The goal of the proposed one year project is to provide the seed towards developing a well-rounded and modern educational program to increase the number of American researchers working in the applied analysis and differential equations and to improve the quality of the training of future mathematical scientists (working in either academia or industry). Having support for graduate students working in this general area will increase the number of students willing to work in applied analysis. The expectation is that the proposed summer course will serve as a model of the type of educational activity needed to attract more researchers to differential equations as well as to educate student and postdocs in engineering and sciences.
This grant was used in order to create activities to enhance awareness for students and postdocs in problems in analysis and pde. The major activitywas a two week summer school organized by Carlos Kenig and Panagiotis Souganidis on homogenization. The school, which was attended by approximately 40 graduate students and postdocs, was very successful. The lecturers were Claude Le Bris (Cermics, Paris), Andrea Braides (University Roma 2), Guilemme Bal (Columbia University), Zhongwei Shen (University of Kentucky, Carlos Kenig (University of Chicago) and Panagiotis Souganidis (Univeristy of Chicago). The topics covered are: Homogenized and stochastic models for equation with highly oscillatory coefficients. Nonperiodic homogenization of elliptic equations: stochastic and determinitisc approaches. Homogenization of lattice systems. Periodic homogenization of elliptic problems. Stochastic homogenization for first and second order nonlinear partial differential equations. The remaining funds were used by the PIs to assist in the organization of a four week summer school that was organized in the summer of 2014. The school attracted a total of approximately 80 students (40 undergraduate and beginning graduates students during the first two weeks and 40 advanced graduated students and psotdocs the second two week period). The first two weeks were of introducatory nature. The coursses offered were in partial differential equations (Sylvetre and Souganidis), Harmonic Analysis (Kenig and Schlag), Probability (Lalley and Lawler), Geometric Measure Theory (Chorney) and Dynamical Systems (Wilkinson). The advanced program consisted of courses in imicrolocal analysis of PDEs (Wunsch), local regularity of nonlocal equations (Kassmann), Active scalars in fluids and mathematical biology (Kisilev), topics in Probability (Zeitouni), Mean Field Games (Achdou), Subadditivity in stochastic homogenization (Smart), Geometry of measures description (Torro) and Nonperiodic multiscale problems ILe Bris).
