Analysis and Geometry Seminar

Organizers: Dan-Andrei Geba and Allan Greenleaf.

The seminar will be held usually Fridays, 2-3PM, in Hylan 1104, with the exception of a few dates when it will take place on Thursday at the same time.

Fall 2007 Speakers:

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

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

• 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.

• 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.

• 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.

• Spring 2007 Speakers:

  • Jan. 19: Sung Kim, University of Rochester, "Calderon's conductivity problem for Lipschitz, piecewise linear functions with polyhedral boundaries".

    In this talk, I will show the global uniqueness for the Calderon's problem for the functions that are piecewise linear and Lipschitz along the polyhedral boundaries using the well known complex geometrical optics solutions and the geometry of Faddeev Green's function.

  • Jan. 26: Shannon Starr, University of Rochester, "Berezin-Lieb Inequalities and Quantum Stat.-Mech.".

    Berezin and Lieb independently discovered a pair of inequalities that relate traces of quantum observables to expectations of classical random variables. This is very useful in understanding the behavior of quantum systems on the basis of our understanding of their classical counterparts. Recently with Lincoln Chayes and Marek Biskup, we obtained related inequalities which are useful in the study of phase transitions. I will describe this, starting with the original Berezin-Lieb inequalities.

  • Feb. 2: Carl Mueller, University of Rochester, "The speed of a random traveling wave" (POSTPONED FOR MARCH 9).

  • Feb. 9: Sarada Rajeev, University of Rochester, "Inequalities for the Spectral Gap of Laplace and Schrodinger Operators".

    There is a well-known inequality (Bochner) that bounds the first eigenvalue of the Laplace operator on a finite dimensional Riemannian manifold by its Ricci tensor. I will describe generalizations of this inequality to the Schrodinger operator. This suggests some (not yet rigorously proved) inequalities for the first eigenvalue for Schrodinger operators on some infinite dimensional manifolds. It is hoped that this will provide an approach to understanding the mass gap of Yang-Mills theory, following ideas of I. Singer and R. P. Feynman. I will outline why this is a difficult but important goal for mathematical physics.

  • Feb. 16: Camil Muscalu, Cornell University, "Classical multi-linear harmonic analysis revisited" (POSTPONED FOR APRIL 27).

    We plan to revisit some of the most classical operators of harmonic analysis, namely Calderon's commutators and the Cauchy integral on Lipschitz curves.

  • Feb. 23: Rui Hu, University of Rochester, "Norm estimates for the restrictions of the eigenfuctions of Laplace operator to submanifolds".

    There are many ways to study eigenfunction estimates for self-adjoint elliptic pseudo-differential operators. In this talk, I am going to talk about the L^p norm of the restrictions of eigenfuctions to submanifolds, with the help of the estimates of oscillatory integral operators whose canonical relations have fold type singularities.

  • Mar. 2: Robert Hladky, University of Rochester, "CR manifolds and the Kohn Laplacian".

    The study of Cauchy-Riemann manifolds arises from the desire to understand the properties of real hypersurfaces, in particular the boundaries of open domains, embedded in complex manifolds. Associated to a CR manifold is a natural differential operator and complex, d-bar_b which can be thought of as the restriction of the complex differential operator d-bar. One method for studying the inhomogeneous equation d-bar_b u =f is to consider the Neumann problem for a self-adjoint second order operator, analogous to the Laplacian for the exterior derivative. There are some sharp contrasts between the analytic properties of the classical Neumann problem and the d-bar_b-Neumann problem. We shall discuss both positive and negative results.

  • Mar. 9: Carl Mueller, University of Rochester, "The speed of a random traveling wave".

    We describe joint work with Leonid Mytnik and Jeremy Quastel. The KPP equation is a standard model for the study of traveling waves. A large class of initial conditions yield solutions which converge to a limiting shape, which moves with constant velocity. Adding noise to the equation may give a stationary ensemble of shapes, with an average speed which is different than the speed of the deterministic wave. Brunet and Derrida have conjectured some surprising results about the speed of the wave in the random case, when the noise is small. Conlon and Doering have given an inequality which partially verifies half of the conjecture. We completely prove both halves of the conjecture. In addition, we give some further error terms which partially confirm a more recent conjecture of Brunet and Derrida. Although I talked on this topic before, we have only recently found a complete proof of the conjectures.

  • Mar. 23: Miguel Alonso, The Institute of Optics, "Phase space and uncertainty relations in optics".

    The first part of this talk consists of a review of the definition and properties of some standard quasi-phase-space distributions, especially the Wigner function. The applications of these representations in quantum mechanics, signal analysis, and classical optics, are briefly discussed. Then, a family of generalizations of the Wigner function is proposed that is useful in describing the propagation of the properties (intensity, flux, polarization) of wave fields beyond the paraxial approximation. These new phase space representations are defined over semi-compact phase spaces. The second part of the talk is centered on uncertainty relations for compact spaces, and the fundamental bounds they impose on the focal properties of monochromatic wave fields.

  • Mar. 30: Suresh Eswarathasan, University of Rochester, "The Kakeya Problem".

    In 1917, S. Kakeya posed the following question: what is the smallest area that allows a unit line segment to be rotated 360 degrees in the plane? Besicovitch solved this in the 1920's with a rather surprising answer. This problem, although at first glance appears recreational, has lead to some of the most important results in harmonic analysis/PDEs in recent years. In this lecture, I will recreate Besicovitch's construction as well as address some of the more immediate consequences of his work in relation to more analysis-relevant topics.

  • Apr. 13: Suresh Eswarathasan, University of Rochester, "Applications of Kakeya-type techniques in Harmonic Analysis and PDE".

    In this lecture, we will discuss various applications of Kakeya-type techniques to oscillatory integral problems (Bochner-Riesz, Restriction) and to estimates for dispersive PDEs. We will sketch a proof of the Tomas-Stein restriction theorem.

  • Apr. 20: Yonathan Shapir, University of Rochester, "Non-Trivial Scale Invariant Distribution for a Dynamically Fluctuating Random Diffusion-like coefficient in epsilon dimensions".

    We look at a class of linear systems described by a linear equation diffusion-like equation where the diffusion constant D fluctuates independently both in space and in time, according to a given distribution function. In addition to random random walks, it also describes random conductance networks, scalar waves in random media, random scalar elasticity, etc. Using the Renormalization Group approach we look for non-trivial scale-invariant distribution for D . By mapping this system to a model of fluctuating surfaces, we find the Inverse Gaussian distribution to be scale-invariant in epsilon dimensions. At the critical point the diffusion (or the scaling relations) become anomalous.

  • Apr. 27: Camil Muscalu, Cornell University, "Classical multi-linear harmonic analysis revisited".

    We plan to revisit some of the most classical operators of harmonic analysis, namely Calderon's commutators and the Cauchy integral on Lipschitz curves.

  Previous semesters

  Fall 2006 Speakers:

  • Sept. 8: Organizational meeting.

  • Sept. 15: Allan Greenleaf, University of Rochester, "Counterexamples to uniqueness for the Calderon problem".

    I will discuss work with Matti Lassas and Gunther Uhlmann from 2003 concerning the existence of pathological anisotropic conductivity functions which cannot be distinguished from a constant homogeneous isotropic conductivity by boundary measurements, i.e., the Cauchy data of solutions to the conductivity equation. When we did this work, we considered it a curiousity, since one is not likely to encounter naturally occurring objects, e.g., tumors being imaged in electrical impedance tomography, having the needed conductivity properties. This year, one of our constructions was rediscovered by two groups of physicists (Pendry, Schurig and Smith; and Leonhardt) and proposed as a basis for a cloaking device (a la Star Trek or Harry Potter). I make no such claims, but will review the earlier papers and point out some interesting mathematical questions raised by the recent work.

  • Sept. 22: Dan-Andrei Geba, University of Rochester, "Nonlinear wave equations on a curved background".

    I will discuss a local well-posedness result, obtained in joint work with Daniel Tataru, for a semilinear wave equation with variable coefficients in 4+1 dimensions. As part of the proof, I will present a new definition for the X^{s,b} spaces in the variable coefficient case.

  • Sept. 29: Robert Hladky, University of Rochester, "Variational problems in subRiemannian geometry, Part 1".

    This talk is accessible to advanced undergraduates. I'll introduce the concept of a subRiemannian space and discuss how these naturally arise from physical situations. I'll talk about classical results on the geodesic problem and how this differs from the Riemannian case. Then I'll consider the subRiemannian analogue of the minimal surface problem and describe how this relates to models of the visual cortex in neurobiology and problems in digital image completion.

  • Oct. 6: Robert Hladky, University of Rochester, "Variational problems in subRiemannian geometry, Part 2".

    I'll discuss my recent work (joint with Scott Pauls) introducing subRiemannian analogues of the Levi-Cevita connection, frame bundles and second fundamental form. I'll show how these can be used to derive first and second variational formulas for the horizontal perimeter measure of hypersurfaces in general subRiemannian manifolds. I'll briefly indicate how these results relate to the isoperimetric problem in the Heisenberg groups.

  • Oct. 13: Richard Lavine, University of Rochester, "Quantum Times of Decay and the Zeno Effect".

    The times of events governed by quantum mechanics are often measured, and are agreed to be random. But quantum mechanics provides no prescription for their probability distributions, as it does for quantities measured at a specific moment. If one attempts to use this to determine the time of decay of an initial state, by repeatedly testing whether it is still undecayed, the quantum Zeno effect typically occurs: the decay is retarded, and even prevented in the limit as the frequency of tests goes to infinity. A detector does not repeatedly probe the system; it simply waits for the decay. We attempt to determine probability distributions by modelling a detector interacting with a quantum system. We find that the Zeno effect still occurs when the detector is too responsive, but reasonable results can be obtained when the initial state has close to exponential decay. There will be a brief summary of quantum mechanics. The mathematics is fairly simple: Hilbert space and Fourier analysis.

  • Oct. 20: James Colliander, University of Toronto, "Concentration properties of blowup solutions of nonlinear Schrodinger equations".

    I will survey some aspects of finite time blowup solutions of the nonlinear Schrodinger equation. A new result, obtained in joint work with S. Roudenko, shows the rate of explosion of a Strichartz norm is linked with the rate of shrinking of the concentration set.

  • Oct. 26 (Special day: Thursday): Michael Greenblatt, SUNY Buffalo, "An elementary coordinate-dependent resolution of singularities".

    A geometric resolution of singularities algorithm is described. This method is elementary in its statement and proof, using explicit coordinate systems as much as possible. Each coordinate change used in the resolution procedure is one-to-one on its domain, and is of one of a few explicit canonical forms. As an application of these methods, we give a general theorem regarding the existence of critical integrability exponents. Also, a new proof of a well-known inequality of Lojasiewicz is given.

  • Oct. 31 (Special day: Tuesday): Daniel Knopf, University of Texas at Austin, "Elements of the proof of the Poincare' Conjecture".

    We will survey key elements in the path to resolving the Poincare' Conjecture, especially contributions of Richard Hamilton and Grisha Perelman, 2006 Fields Medalist. The talk is geared for a general audience.

  • Nov. 2 (Special day: Thursday) joint with the Colloquium series: Christopher Sogge, Johns Hopkins University, "Estimates for eigenfunctions of the Laplacian".

    I shall present various L^p estimates for eigenfunctions of the Laplacian in manifolds with and without boundary. A common theme is to try to see how the underlying geometry, as measured by the long-term dynamics of the geodesic flow, does or does not lead to blowup of L^p norms. In the case where p=infinity we shall also discuss blowup and non-blowup results for quasimodes. This is joint work with Hart Smith, and joint work with John Toth and Steve Zelditch.

  • Nov. 10: Ben Lichtin, "Estimates for Fourier transforms of surface measures via multivariable asymptotic analysis".

    Certain problems can be reduced to a uniform bound of the Fourier transform of a surface measure. For example, the "discrepancy" between the number of lattice points and volume of an expanding family of domains can be reduced to such an estimate (work of Randol, Hlawka ...). Sometimes it is useful to "damp" the measure by an additional factor. This introduces a family of operators whose L^p boundedness can be estimated by combining the decay of the damped Fourier integrals with analytic interpolation of operators (work of Stein and his students...). Techniques to study such questions have typically exploited either geometric properties (e.g. curvature of the surface) or analytic properties (e.g. estimates for the L^p integrability of associated functions). Our approach to such problems uses a relatively new technique of "multivariable asymptotic analysis", a natural (but not necessarily evident) extension of the standard one variable technique. We illustrate this method by bounding either type of oscillatory integral for an interesting class of nonconvex surfaces in R^3.

  • Nov. 17: Michael Gage, University of Rochester, "Remarks on the Santalo-Bonnesen isoperimetric inequality".

    Rated PG: This talk should be suitable for undergraduates. The isoperimetric inequality for plane figures states that the length of the boundary squared is greater than 4 pi times the enclosed area: L^2-4 pi A >= 0. A Bonnesen type inequality strengthens this to L^2-4 pi A >= B where B is some non-negative quantity determined from the plane figure which is zero only for the circle. There are different versions of the Bonnesen inequality and most of them have several genuinely different proofs. I'll give a description of a beautiful proof by Santalo of one Bonnesen inequality using integral geometry. I'll then show how to extend this proof to a stronger result for convex figures and give some hints about how to extend the result to planar Finsler geometries (also known as Minkowski geometries).

  • Dec. 1: Xiaobo Liu, University of Notre Dame, "Mean curvature flow for isoparametric submanifolds".

    This is a joint work with Chuu-Lian Terng. A submanifold of a Euclidean space is isoparametric if its normal bundle is flat and principal curvates along any parallel normal vector fields are constant. We explicitly solved the mean curvature flow equations for such submanfolds and proved that the mean curvature flows always converge to a focal submanifold in finite time.

  • Dec. 6 (Special day: Wednesday): Colleen Robles, Texas A&M University, "Rigidity and flexibility of homogeneous varieties".

    I'll discuss recent results and current work on the rigidity of subvarieties of projective space. The basic question is: when do two sub-varieties differ by a projective linear transformation?