András Vasy

I am an Associate Professor in the Department of Mathematics of Stanford University. My short CV is here. Until August 2006, I was on the faculty of the Department of Mathematics of the Massachusetts Institute of Technology, and in 2004-2006, I was a Clay Research Fellow at the Clay Mathematics Institute.

My address is: Department of Mathematics, Building 380, Stanford University, 450 Serra Mall, Stanford CA 94305-2125, USA. My fax number is 650-725-4066.

My e-mail address is andras "at" math.stanford.edu.

• Math 220A has a web page from last year. The content and format of the course will be unchanged, and the primary text will be the lecture notes that can be found at the same address. The textbook closest to the course is Strauss' `Partial Differential Equations: An introduction', but the course is at a somewhat higher level. Evans' `Partial Differential Equations' is a more advanced text, and it covers course topics not dealt in Strauss' book. Both of these, as well as John's `Partial Differential Equations' will be on reserve at the Math CS library.
• Math 113 will be similar to the course taught by Ravi Vakil last winter, whose web page is here. One difference is the text. In addition to Professor Katznelson's lecture notes we will use Axler' `Linear Algebra Done Right' as primary texts. Axler' book is somewhat easier to digest, and it will be supplemented by Prof. Katznelson's notes as the material covered requires. Additionally, texts by Hoffman and Kunze, as well as by Lax (all titled `Linear algebra') and Halmos (`Finite dimensional vector spaces') will be on reserve at the Math CS library.

My research area is partial differential equations, more specifically microlocal analysis and geometric scattering theory. This is a link to the web page of the Stanford seminar calendar for the current week, this to the MIT analysis and PDE seminar, and here is the Northwestern math seminar calendar.

In March 2002, I co-organized a conference in honor of Richard Melrose's 25 years at MIT.

These are my lecture notes for my February 5, 2006, talk at the CUNY Geometric Analysis Conference on Scattering theory on symmetric spaces and N-body scattering in postscript; also in pdf. The Berkeley colloquium pdf version and the further improved UW colloquium pdf version are also available in pdf.

I gave a lecture at École Polytechnique in April, 2005, on the propagation of singularities for the wave equation on manifolds with corners. Most of the results stated there are written up in a preprint listed below; the lecture notes are available here.

Propagation of singularities for the wave equation on manifolds with corners.

A more accessible version of this talk was given at the Mathematical Physics meeting in Birmingham, Alabama, and the notes are being published in the Contemporary Mathematics series of AMS in the volume ``Recent Advances in Differential Equations and Mathematical Physics''. The pdf and ps files are available here.

A sequel, on diffraction by edges, concentrating on the strength of the singularity of the reflected wave, was given at Berkeley in February 2006. Its slightly modified version, to be given in Cambridge, is available as overheads or as a computer presentation.

These are the overhead transparencies for my talk at the Perspectives in Inverse Problems meeting in Helsinki in June 2004. An abbreviated version was presented at the AMS meeting in Evanston in October, 2004.

These are my lecture notes from the minicourse I gave at the Université de Nantes in May 2002, which have been published as Geometry and analysis in many-body scattering, Inside out: inverse problems and applications, 333--379, Math. Sci. Res. Inst. Publ., 47, Cambridge Univ. Press, Cambridge, 2003.

These are my lecture notes from the Inverse Problems session of the Pisa AMS-UMI joint meeting in June 2002.

In spring 2001, Vesselin Petkov, Maciej Zworski and I organized a semester-long program in scattering theory at the Erwin Schrödinger Institute in Vienna. Information is available here and from the Erwin Schrödinger Institute web site.

I gave a lecture at École Polytechnique in February, 2001, on a joint project with Andrew Hassell and Richard Melrose. Part of the results stated there are written up in a preprint listed below; the lecture notes are available here.

Spectral and scattering theory for symbolic potentials of order zero.

This is a link to the Geometry, analysis and mathematical physics conference in San Feliu in September, 2000, where I was an invited speaker. I participated in the 1999 Conference on partial differential equations at St. Jean-de-Monts. The lecture notes are available here.

You can find my manuscripts, some joint work with Bernd Ammann, Andrew Hassell, Robert Lauter, Rafe Mazzeo, Richard Melrose, Marius Mitrea, Victor Nistor, Michael Taylor, Gunther Uhlmann, Xue Ping Wang, Jared Wunsch or Maciej Zworski, in postscript format below.

• Scattering poles for negative potentials.

  Published in Communications in PDEs, 21:185-194 (1997).

• Structure of the resolvent for three-body potentials.

  Published in Duke Math. J., 90:379-434 (1997).

• Asymptotic behavior of generalized eigenfunctions in N-body scattering.

  Published in J. Func. Anal , 148:170-184 (1997).

• Propagation of singularities in three-body scattering.

  This is the original version of my PhD thesis. A somewhat modified (improved) paper version is below. The thesis abstract is available separately.

• Propagation of singularities in three-body scattering.

  Published in Astérisque, 262 (2000); paper version of my thesis (see description above) from December 4th, 1997. Its introduction and the list of references are available separately. There is an addendum, Appendix C, that was added in proof (in September, 1999) to fill in details of the positivity estimates in Sections 12 and 14.

• Symbolic functional calculus and N-body resolvent estimates, joint work with Andrew Hassell.

  Published in J. Func. Anal , 173:257-283 (2000).

• Scattering matrices in many-body scattering.

  Published in Commun. Math. Phys. 200:105-124 (1999).

• Geometric scattering theory for long-range potentials and metrics.

  Published in Int. Math. Res. Notices (IMRN), 1998, no. 6, 285-315 (1998).

• Propagation of singularities in many-body scattering.

  Published in Annales Scientifiques de l'École Normale Supérieure (4), 34:313-402 (2001); original version December 15, 1998, revised May 2, 2000. Its introduction and the list of references are available separately.

• The spectral projections and the resolvent for scattering metrics, joint work with Andrew Hassell.

  Published in Journal d'Analyse Mathématique, 79:241-298 (1999).

• Propagation of singularities in many-body scattering in the presence of bound states.

  Published in Journal of Functional Analysis , 184:177-272 (2001). Its introduction and the detailed statement of results and the list of references are available separately. The proof provided in the published version has a minor gap, that is easily fixed, in that Proposition 7.1 is not stated (and proved) in the strongest possible form. The corrected version is available here.

• Semiclassical estimates in asymptotically Euclidean scattering, joint work with Maciej Zworski.

  Published in Commun. Math. Phys., 212:205-217 (2000).

• The resolvent for Laplace-type operators on asymptotically conic spaces, joint work with Andrew Hassell.

  Published in Annales de l'Institut Fourier, 51:1299-1346 (2001).

• Resolvents and Martin boundaries of product spaces, joint work with Rafe Mazzeo.

  Published in Geometric and Functional Analysis, 12:1018-1079 (2002).

• Intersecting Legendrians and blow-ups, joint work with Andrew Hassell.

  Published in Math. Res. Lett. 8:413--428 (2001).

• Smoothness and high energy asymptotics of the spectral shift function in many-body scattering, joint work with Xue Ping Wang.

  Published in Communications in PDEs 27:2139-2186 (2002).

• Low energy inverse problems in three-body scattering, joint work with Gunther Uhlmann.

  Published in Inverse Problems 18:719-736 (2002).

• Fixed energy inverse problem for exponentially decreasing potentials, joint work with Gunther Uhlmann.

  There is also an addendum to include an omitted reference to a paper of Eskin and Ralston.

  Published in Methods and Applications of Analysis 9:239-248 (2002).

• Spectral and scattering theory for symbolic potentials of order zero, joint work with Andrew Hassell and Richard Melrose.

  Published in Advances in Mathematics, 181:1-87 (2004).

• Exponential decay of eigenfunctions in many-body type scattering with second order perturbations.

  Published in Journal of Functional Analysis, 209:468-492 (2004).

• Scattering theory on SL(3)/SO(3): connections with quantum 3-body scattering, joint work with Rafe Mazzeo. Also in pdf.

  To appear in Proc. London Math. Soc. Revised version from 2005 is here, and the original version, from 2002, is here.

• Analytic continuation of the resolvent of the Laplacian on SL(3)/SO(3), joint work with Rafe Mazzeo.

  Published in American Journal of Mathematics, 126:821-844 (2004).

• Complex powers and non-compact manifolds, joint work with Bernd Ammann, Robert Lauter and Victor Nistor.

  Published in Commun. in PDEs, 29:671-705 (2004).

• Inverse problems in N-body scattering, joint work with Gunther Uhlmann.

  Published in Inverse Problems and Spectral Theory (ed. H. Isozaki), Contemporary Mathematics, American Matematical Society (2004).

• Analytic continuation of the resolvent of the Laplacian on symmetric spaces of noncompact type, joint work with Rafe Mazzeo.

  Published in J. Func. Anal. 228:311-368 (2005). Version of November 24, 2003. This version has added references to papers of B. Simon and Hunziker, missing from the September 7, 2003, version. Previously, the September 7 version added a second proof of the analytic continuation, using a more `classical' approach. The preceeding version, of August 13, 2003, is here. The Aug. 13 version in turn has added some references to the original, August 6 version.

• Inverse scattering with fixed energy for dilation-analytic potentials, joint work with Xue Ping Wang.

  Published in Inverse Problems, 20:1349-1354 (2004).

• Propagation of singularities for the wave equation on manifolds with corners.

  To appear in Annals of Mathematics. Previously posted versions include the revised version from September, 2005; the final version only has minor changes in the proof of Lemma 4.2. The original version is here.

• Lipschitz domains, domains with corners and the Hodge Laplacian, joint work with Marius Mitrea and Michael Taylor.

  Published in Commun. in PDEs, 30:1445-1462 (2005). The original version is here.

• Absence of super-exponentially decaying eigenfunctions on Riemannian manifolds with pinched negative curvature, joint work with Jared Wunsch.

  Published in Math. Res. Letters, 12:673-684 (2005). The original, October 2004 version is here.

• Scattering for symbolic potentials of order zero and microlocal propagation near radial points, joint work with Andrew Hassell and Richard Melrose.

  Preprint, 2005.

Together with Richard Melrose and Maciej Zworski, I gave a series of lectures at the beginning of November, 1998, at the Aarhus workshop on Geometric scattering. My lectures focused on propagation estimates (as in propagation of `singularities' for generalized eigenfunctions) and its consequences (structure of S-matrices) in many-body scattering. Here is a very brief description of the lectures.

This page (together with its predecessors at Berkeley and MIT) has been accessed at least times since April 18, 1997.