Location: Minard 304A (Seminar Room) Time: Wednesdays, 4:00-4:50 PM Organizer: Dr. Maria Alfonseca-Cubero

Fall 2009 Schedule August 24 Organizational Meeting Monday, August 24, 10:00 AM Mathematics Department Seminar Room September 09 Speaker: Rob Hladky, NDSU Title: CR manifolds and the tangential Cauchy-Riemann equations Abstract: The study of CR manifolds lies in the intersection of subRiemannian geometry and several complex variables. We'll look from a geometric perspective at what it means for a manifold to carry a complex structure and see how much of this complex structure survives when instead you look at submanifolds. We shall then define CR manifolds as a natural abstraction of this concept and look at the equivalent notions of holomorphic functions and forms; solutions to the tangential Cauchy-Riemann equations. Next we study the Kohn Laplacian, a natural analogue of the standard Laplacian, and see how the classical elliptic theory can be modified to study this sub-elliptic Laplacian and the tangential CR-equations. September 16 Speaker: Rob Hladky, NDSU Title: CR manifolds and the tangential Cauchy-Riemann equations (PART II) Abstract: The study of CR manifolds lies in the intersection of subRiemannian geometry and several complex variables. We'll look from a geometric perspective at what it means for a manifold to carry a complex structure and see how much of this complex structure survives when instead you look at submanifolds. We shall then define CR manifolds as a natural abstraction of this concept and look at the equivalent notions of holomorphic functions and forms; solutions to the tangential Cauchy-Riemann equations. Next we study the Kohn Laplacian, a natural analogue of the standard Laplacian, and see how the classical elliptic theory can be modified to study this sub-elliptic Laplacian and the tangential CR-equations. September 23 Speaker: Rob Hladky, NDSU Title: CR manifolds and the tangential Cauchy-Riemann equations (PART III) Abstract: The study of CR manifolds lies in the intersection of subRiemannian geometry and several complex variables. We'll look from a geometric perspective at what it means for a manifold to carry a complex structure and see how much of this complex structure survives when instead you look at submanifolds. We shall then define CR manifolds as a natural abstraction of this concept and look at the equivalent notions of holomorphic functions and forms; solutions to the tangential Cauchy-Riemann equations. Next we study the Kohn Laplacian, a natural analogue of the standard Laplacian, and see how the classical elliptic theory can be modified to study this sub-elliptic Laplacian and the tangential CR-equations. September 30 No meeting October 07 Speaker: Marian Bocea, NDSU Title: An introduction to Young measures Abstract: The notoriously poor behavior of weak convergence with respect to nonlinear operations is a source of many difficulties in Nonlinear Analysis. Originally introduced by L.C. Young to study nonconvex problems in optimal control theory, Young measures (or parametrized probability measures) have been efficiently used in recent years to understand certain oscillatory phenomena in a more general Calculus of Variations and PDE framework. I will give an introduction to this concept outlining its main properties as well as some of its drawbacks. October 14 Speaker: Marian Bocea, NDSU Title: An introduction to Young measures (Part II) Abstract: The notoriously poor behavior of weak convergence with respect to nonlinear operations is a source of many difficulties in Nonlinear Analysis. Originally introduced by L.C. Young to study nonconvex problems in optimal control theory, Young measures (or parametrized probability measures) have been efficiently used in recent years to understand certain oscillatory phenomena in a more general Calculus of Variations and PDE framework. I will give an introduction to this concept outlining its main properties as well as some of its drawbacks. October 21 Speaker: Marian Bocea, NDSU Title: An introduction to Young measures (Part III) Abstract: The notoriously poor behavior of weak convergence with respect to nonlinear operations is a source of many difficulties in Nonlinear Analysis. Originally introduced by L.C. Young to study nonconvex problems in optimal control theory, Young measures (or parametrized probability measures) have been efficiently used in recent years to understand certain oscillatory phenomena in a more general Calculus of Variations and PDE framework. I will give an introduction to this concept outlining its main properties as well as some of its drawbacks. October 28 Speaker: Cristina Popovici, NDSU Title: A Decomposition Result for Sequences of Gradients Abstract: We will discuss a decomposition result for sequences of gradients of Sobolev functions which plays an important role in the proofs of a number of key results in the Calculus of Variations, including the lower semicontinuity result of Acerbi and Fusco, Kinderlehrer and Pedregal's characterization of gradient Young measures, and various relaxation results for nonconvex integrands. The proof uses L^p estimates for maximal functions, Lipschitz extensions of Sobolev functions, and Young measures. Spring 2009 schedule