Analysis Seminar

Organizers: Dan-Andrei Geba and Allan Greenleaf.

The seminar will usually be held on Fridays, 2-3PM, in Hylan 1106A.

Fall 2008 Speakers:

Previous semesters

Spring 2008 Speakers:

  • April 25: Joachim Krieger, UPenn ; "On singularity formation for certain geometric wave equations"

    Abstract: We dicuss recent results, obtained jointly with W. Schlag and D. Tataru, on a new kind of singularity formation for certain critical nonlinear geometric wave equations.

  • April 18: Allan Greenleaf, Rochester: " A brief survey of the Erd\"os Distance Problem"

  • April 11: Rob Hladky, Rochester: "Fredholm and regularity theorems for the \bar{\partial}_b-Neumann problem"

    Abstract: The inhomogeneous equation $\bar{\partial}_b u =f$ for the tangential Cauchy-Riemann operator on domains with boundary is an important largely unsolved problem in the theory of CR geometry. One approach is to consider the Neumann problem for the associated Laplacian. Ideally one would like to establish a priori estimates, Fredholm theorems and sharp regularity results for this locally subelliptic operator. Unfortunately, many of the classical techniques fail in this instance. We shall outline the methods required to establish priori estimates, look at cases where Fredholm theorems can be established and provide examples for when they fail. Our sufficient Fredholm criteria is surprisingly geometric in nature and is based on some deep results in topology and geometry.

  • April 3: COLLOQUIUM: Steve Zelditch, Johns Hopkins; "Ergodicity, complex numbers and nodal lines "

    Abstract: Since Chladni in 1800, mathematicians and physicists have tried to understand the shape and size of nodal lines, i.e. the patterns that sand forms when sprinkled on a vibrating drum. They are the zeros of the eigenfunctions describing the modes of vibration. My talk will explain how complex analysis can be used to determine the shapes in the high frequency limit when the billiiards on the drum are ergodic.

  • March 28: Suresh Eswarathasan, Rochester; "L^2 Estimates for Averaging Operators with k-folds."

    Abstract: We discuss a 1997 paper of Cuccagna's regarding sharp Sobolev estimates for FIOs whose canonical relation is not a canonical graph. Some motivating examples and history will be given before presenting an outline of his proof.

  • March 21: Jared Wunsch, Northwestern; "Diffraction of waves on singular spaces"

    Abstract: I will talk about work (joint with Richard Melrose and Andras Vasy) about the behavior of solutions to the wave equation in the presence of geometric singularities such as corners, edges, and cones. These singularities influence the qualitative structure of the fundamental solution, owing to diffractive effects.

  • Feb. 8: Allan Greenleaf, Rochester; "Acoustic Cloaking in 3D - Then and Now"

  • Feb. 1: Jeremy Marzuola, Columbia; "Stability of Minimal Mass Soliton Solutions for Saturated Nonlinear Schr\"odinger Equations"

  • Jan. 18: Emanuel Palsu-Andriescu, Rochester; "On Colombeau algebras of generalized functions"

  • Jan. 17: (Special day, time and place.) Allan Greenleaf, Rochester; "The Cheshire cat's NGRIN: nontunnelling for highly singular potentials", joint with Math. Phys. Seminar in Mandel Room, Bausch and Lomb 372, 3:45 pm.

Fall 2007 Speakers:

  • Dec. 6: Dan-Andrei Geba, Rochester; "A set of problems in nonlinear wave equations"

  • Nov. 30: NO SEMINAR

  • Nov. 23: THANKSGIVING BREAK

  • Nov. 16: Matthew Blair, Rochester; "Strichartz estimates on manifolds with boundary, Part 2"

  • Nov. 2: Monica Visan, Inst. for Adv. Study; "The mass-critical nonlinear Schrodinger equation"

    Abstract: I will survey recent results on nonlinear Schrodinger equations at the critical regularity, with particular emphasis on the case of finite-mass (i.e. L^2) initial data. This includes joint work with Rowan Killip, Terry Tao, and Xiaoyi Zhang.

  • Oct. 26: Matthew Blair, Rochester; "Strichartz estimates on manifolds with boundary"

    Abstract: We consider local Strichartz estimates for the wave equation on a Riemannian manifold with boundary. These are a family of space-time integrability estimates that rely on the dispersive effect of the solution map. Such inequalities have applications in the study of nonlinear equations. When the boundary of the manifold is empty, such estimates are well-established. Otherwise, when boundary conditions are present, much less is known in the general case. We discuss new results in the area, widening the range of Lebesgue exponents for which the estimates hold. This is a joint work with H. Smith and C. Sogge.

  • Oct. 19: Philip Gressman, Yale; "Uniform estimates for cubic oscillatory integrals"

    Abstract: I will discuss the problem of proving uniform, optimal asymptotic estimates for scalar oscillatory integrals with a phase function which satisfies an appropriate third-order nondegeneracy condition. The proof relies on the construction of a nontrivial symmetric space structure adapted to the geometry of the phase.

  • THURSDAY, Oct. 18: Dept. Colloquium, Bob Strichartz, Cornell; "Differential equations and quantum mechanics on fractals"

    NOTE: Talk is in Lander Auditorium, 3:30 pm.