Department of Mathematics

Details

Boundary rigidity, volume minimality, and minimal surfaces in L_∞: a survey

by Dima Burago | Pennsylvania State
Time: 11:10 — 12:00  (Thursday, Mar. 04, 2010)
Location: BA2195, Bahen Center, 40 St. George St.
Abstract:
A Riemannian manifold with boundary is said to be boundary rigid if its metric is uniquely determined by the boundary distance function, that is the restriction of the distance function to the boundary. Loosely speaking, this means that the Riemannian metric can be recovered from measuring distances between boundary points only. The goal is to show that certain classes of metrics are boundary rigid (and, ideally, to suggest a procedure for recovering the metric).

To visualize that, imagine that one wants to find out what the Earth is made of. More generally, one wants to find out what is inside a solid body made of different materials (in other words, properties of the medium change from point to point). The speed of sound depends on the material. One can "tap" at some points of the surface of the body and "listen when the sound gets to other points". The question is if this information is enough to determine what is inside.

This problem has been extensively studied from PDE viewpoint: the distance between boundary points can be interpreted as a "travel time" for a solution of the wave equation. Hence this becomes a classic Inverse Problem when we have some information about solutions of a certain PDE and want to recover its coefficients. For instance such problems naturally arise in geophysics (when we want to find out what is inside the Earth by sending sound waves), medical imaging etc.

In a joint project with S. Ivanov we suggest an alternative geometric approach to this problem. In our earlier work, using this approach we were able to show boundary rigidity for metrics close to flat ones (in all dimensions), thus giving the first class of boundary rigid metrics of non–constant curvature beyond two dimensions. We were now able to extend this result to include metrics close to a hyperbolic one.

The approach is grew up from another long-term project of studying surface area functionals in normed spaces, which we have been working on it for more than ten years. There are a number of related issues regarding area-minimizing surfaces in Riemannian manifold. The talk gives a non-technical survey of ideas involved. It assumes no background in inverse problems and is supposed to be accessible to a general math audience (in other words, we will not get into any technical details of the proofs).

Dates in this series

· Wednesday, Jan. 13, 2010: A tale of two spaces: locally symmetric spaces and moduli space of curves (Lizhen Ji)

· Wednesday, Jan. 20, 2010: Bernstein-Kushnirenko theorem and its generalizations  (Askold Khovanskii)

· Wednesday, Jan. 27, 2010: The Dynamical Mordell-Lang Conjecture (Dragos Ghioca)

· Monday, Feb. 01, 2010: SPECIAL COLLOQUIUM -- Some recent convergence results for nonconventional ergodic averages (Tim Austin)

· Wednesday, Feb. 10, 2010: Higher-order filling inequalities and nilpotent groups (Robert Young)

· Monday, Feb. 22, 2010: SPECIAL COLLOQUIUM -- Group Representation Patterns in Digital Signal Processing (Shamgar Gurevich)

· Wednesday, Feb. 24, 2010: Interval Exchange Transformations (Jon Chaika)

· Thursday, Mar. 04, 2010: Boundary rigidity, volume minimality, and minimal surfaces in L_∞: a survey (Dima Burago)

· Wednesday, Mar. 10, 2010: Einstein Metrics, Complex Surfaces, and Symplectic 4-Manifolds (Claude LeBrun)

· Wednesday, Mar. 24, 2010: Measure rigidity for dynamical systems on (very) non-compact spaces (Omri Sarig)

· Wednesday, Mar. 31, 2010: Large scale behaviour of the continuum random polymer and KPZ (Jeremy Quastel)

· Wednesday, Jul. 07, 2010: Zeroes of Gaussian entire functions. (Fedor Nazarov)

· Wednesday, Sep. 29, 2010: On bounded-depth Boolean circuits (Mark Braverman)

· Wednesday, Oct. 06, 2010: New Invariants on Algebraic Cycles (James Lewis)

· Wednesday, Oct. 13, 2010: Optimality of Hopf fibrations and Hopf vector fields (Herman Gluck)

· Wednesday, Oct. 27, 2010: Pentagonal Tilings and Other Partition Problems (Frank Morgan)

· Wednesday, Nov. 03, 2010: Particle systems and Landau damping (Cédric Villani)

· Wednesday, Nov. 10, 2010: Poisson varieties and D-modules (Travis Schedler)

· Wednesday, Nov. 17, 2010: Non-archimedean integrals over algebraic varieties and their applications (Victor Batyrev)

· Wednesday, Nov. 24, 2010: Singular numbers of random matrices: the asymptotic non-limiting theory (Nicole Tomczak-Jaegermann)

· Wednesday, Dec. 01, 2010: Quantum information as asymptotic geometry (Patrick Hayden)

· Monday, Jan. 17, 2011: SPECIAL COLLOQUIUM -- Counting Representations of Arithmetic Groups (Nir Avni)

· Tuesday, Jan. 18, 2011: Classical algebraic geometry in the tropical limit (Grisha Mikhalkin)

· Wednesday, Jan. 19, 2011: Topological solitons and dispersion (David Stuart)

· Wednesday, Jan. 19, 2011: SPECIAL COLLOQUIUM -- Spectral gaps for lattices in Lie groups (Dubi Kelmer )

· Wednesday, Jan. 26, 2011: Surface operators in 3d topological field theory (Natalia Saulina)

· Monday, Jan. 31, 2011: Boundedness of algebraic varieties (Chenyang Xu)

· Wednesday, Feb. 02, 2011: Statistical Mechanics on Sparse Random Graphs (Amir Dembo)

· Wednesday, Feb. 09, 2011: Systems of quadratic inequalities (Antonio Lerario)

· Wednesday, Mar. 02, 2011: Integrable systems and geometry of Riemann surfaces (Igor Krichever)

· Wednesday, Mar. 09, 2011: On the stability of filament flows and Schrodinger maps (Didier Smets)

· Wednesday, Mar. 23, 2011: Dunkl operators: from analysis to algebra and back (Raphaël Rouquier)

· Wednesday, Mar. 30, 2011: The Erdos distinct distance problem (Larry Guth)

· Wednesday, Apr. 06, 2011: On the formation of singularities in geometric evolution equations (Igor Rodnianski)

· Wednesday, Apr. 13, 2011: Banach-valued Holomorphic Functions on the Maximal Ideal Space of H^\infty and a Problem of Sz. Nagy (Alex Brudnyi)

· Wednesday, Oct. 12, 2011: Eigenvalues, interlacing inequalities and tropical calculus (Anton Alekseev)

· Wednesday, Oct. 19, 2011: Fusion categories and subfactors (Scott Morrison)

· Wednesday, Nov. 09, 2011: Lieb-Thirring Inequalities at Positive Density (Robert Seiringer)

· Wednesday, Nov. 23, 2011: Convex bodies in algebraic geometry and representation theory (Kiumars Kaveh)

· Monday, Nov. 28, 2011: What is K-theory and what is it good for? (Paul Baum)

· Wednesday, Nov. 30, 2011: Bilipschitz embeddings of metric spaces in Banach spaces (Bruce Kleiner)

· Wednesday, Dec. 07, 2011: Quantifying simple connectivity: an introduction to the Dehn function (Robert Young)

· Monday, Jan. 16, 2012: SPECIAL COLLOQUIUM -- Interval exchange transformations (Jon Chaika)

· Wednesday, Jan. 18, 2012: Partial Frobenius integrability and its applications (Yum-Tong Siu)

· Friday, Jan. 20, 2012: SPECIAL COLLOQUIUM -- Arnold diffusion: a combined geometrical and variational approach  (Ke Zhang)

· Monday, Jan. 23, 2012: SPECIAL COLLOQUIUM -- Probabilistic Techniques in Mathematical Phylogenetics: Relating Combinatorial and Variational Distances Between Trees (Sebastien Roch)

· Wednesday, Jan. 25, 2012: SPECIAL COLLOQUIUM -- Counting closed loops in a stratum of quadratic differentials (Kasra Rafi)

· Friday, Jan. 27, 2012: SPECIAL COLLOQUIUM -- Equivariant Schroedinger Maps in 2D with large data (Ioan Bejenaru)

· Monday, Jan. 30, 2012: SPECIAL COLLOQUIUM -- Symplectic resolutions and Poisson-de Rham homology (Travis Schedler)

· Wednesday, Feb. 01, 2012: SPECIAL COLLOQUIUM -- Semisimple algebras and subfactors (Noah Snyder)

· Wednesday, Feb. 15, 2012: Monotonicity formulas revisited and applications (Toby Colding)

· Wednesday, Mar. 07, 2012: Martin's Maximum (Justin Moore)

· Wednesday, Mar. 14, 2012: Perfectoid Spaces (Peter Scholze)

· Wednesday, Mar. 28, 2012: Beyond the Künneth formula (Jan Nekovář)

· Wednesday, Sep. 05, 2012: The Kollár Problem (Charles Fefferman)

· Thursday, Sep. 20, 2012: PUBLIC LECTURE: Counting stationary modes: a discrete view of geometry and dynamics (Stéphane Nonnenmacher)

· Wednesday, Sep. 26, 2012: Microlocal analysis: some old results and new trends (Gilles Lebeau)

· Wednesday, Oct. 03, 2012: The Hilbert 16th problem and its infinitesimal version (Gal Binyamini)

· Wednesday, Oct. 10, 2012: On the bounded $L^2$ curvature conjecture (Sergiu Klainerman)

· Wednesday, Oct. 17, 2012: Animation, Teeth and Skeletons (Ingrid Daubechies)

· Wednesday, Oct. 24, 2012: Integral Apollonian Gaskets (Alex Kontorovich)

· Wednesday, Nov. 07, 2012: Integrable Probability (Ivan Corwin)

· Wednesday, Nov. 14, 2012: Controlled Active Vision/Image Processing with Applications to Medical Image Computing (Allen Tannenbaum)

· Wednesday, Nov. 21, 2012: Billiards and curves in moduli space (Curtis McMullen)

· Wednesday, Nov. 28, 2012: Rotation of invariant tensors and geometric representation theory (Joel Kamnitzer)

· Wednesday, Dec. 05, 2012: Winning sets for Schmidt games, badly approximable real numbers, and rational billiards (Howard Masur)

· Wednesday, Jan. 23, 2013: Unexpected applications of polynomials in combinatorics (Larry Guth)

· Wednesday, Jan. 30, 2013: Operator limits of random matrices (Balint Virag)

· Tuesday, Feb. 05, 2013: SPECIAL DEPARTMENTAL COLLOQUIUM -- An introduction to unlikely intersections (Jacob Tsimerman)

· Wednesday, Feb. 06, 2013: Universal spaces for birational invariants (Yuri Tschinkel)

· Wednesday, Feb. 27, 2013: From Nash to Onsager: funny coincidences across differential geometry and the theory of turbulence (Camillo De Lellis)