I am a Professor in the Department of Mathematics of Stanford University.

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 dot stanford dot edu.

This year I am teaching (where for now only some links are working, and some are for past versions of the course):- Math 172, Lebesgue Integration and Fourier Analysis, in winter 2014. This is similar to 205A, but designed for undergraduate students, and for graduate students in other departments. It also includes basic Fourier analysis.
- Math 256B, Partial differential equations, in winter 2014. This is an advanced graduate PDE class, focusing on Melrose's so-called b-(pseudo)differential operators, but no PDE background is required. (Thus, 256A is not a prerequisite.) However, a thorough knowledge of functional analysis and Fourier analysis (as presented in the Math 205 sequence) is a must. In the Riemannian world b-analysis includes manifolds with cylindrical ends, and in the Lorentzian world such diverse spaces as Minkowski space, a neighborhood of the static patch of de Sitter space, and Kerr-de Sitter space, as well as various spaces which asymptotically have a similar (but not necessarily the same!) structure. In addition, b-analysis helps analyze the standard boundary value problems in wave propagation; time permitting this will be discussed as well.
- Math 256A, Partial differential equations, in spring 2014. This is an advanced graduate PDE class, but with no PDE background required. However, a thorough knowledge of functional analysis as in 205A-B is a must. The course relies on Leon Simon's lecture notes. This course should be appropriate for first year graduate students.

Some recent quals are posted here for fast dissemination:

- Fall 2013,
- Spring 2013,
- Fall 2012,
- Spring 2012,
- Fall 2011, with a clarification (see brackets),
- Spring 2011, with a correction (see brackets),
- Fall 2010.

Last year (2012-2013) I taught:

- Math 131P, Partial differential equations I, in autumn 2012. This is an undergraduate PDE class geared towards students interested in sciences/engineering.
- Math 220, Partial Differential Equations of Applied Mathematics, in autumn 2012.
- Math 205B, Real Analysis, in winter 2013.

- Math 256B, Partial differential equations, in winter 2012. This is an advanced graduate PDE class, focusing on scattering theory, but no PDE background is required. (Thus, 256A is not a prerequisite.) However, a thorough knowledge of functional analysis and Fourier analysis (as presented in the Math 205 sequence) is a must.
- Math 394, Classics in Analysis, in winter 2012.
- Math 171, Fundamental Concepts of Analysis, in spring 2012.
- Math 256A, Partial differential equations, in spring 2012. This is an advanced graduate PDE class, but with no PDE background required. However, a thorough knowledge of functional analysis as in 205A-B is a must. The course relies on Leon Simon's lecture notes. This course should be appropriate for first year graduate students.

In 2010-2011 I taught:

- Math 205B, Real Analysis, in winter 2011.
- Math 256B, Partial differential equations, in winter 2011. This is an advanced graduate PDE class, focusing on microlocal analysis, but no PDE background is required. (Thus, 256A is not a prerequisite.) However, a thorough knowledge of functional analysis and Fourier analysis (as presented in the Math 205 sequence) is a must. The course is based on Richard Melrose's lecture notes, volume 2 of Michael Taylor's PDE book, and additional material supplied by the instructor. We cover pseudodifferential operators, their use in elliptic and hyperbolic PDE, and hopefully scattering theory.
- Math 171, Fundamental Concepts of Analysis, in spring 2011.

In 2009-2010 I taught:

- Math 220, Partial Differential Equations of Applied Mathematics, in autumn 2009.
- Math 205B, Real Analysis, in winter 2010.

In 2008-2009 I taught:

- Math 205B, Real Analysis, in winter 2009.
- Math 171, Fundamental Concepts of Analysis, in spring 2009.
- Math 256A, Partial differential equations, in spring 2009. This is an advanced graduate PDE class, but with no PDE background required. However, a thorough knowledge of functional analysis as in 205A-B is a must. The syllabus is somewhat different from the previous year's version (see below) relying instead on Leon Simon's lecture notes. This course should be appropriate for first year graduate students.

Winter quarter 2008 I taught

- Math 205B, Real Analysis. The second quarter of the graduate real analysis sequence covers functional analysis. We use Reed and Simon's Functional Analysis (volume 1 of `Methods of Mathematical Physics'), quickly covering Chapter 1 as background (except the measure theory part, which was covered in 205A), and start with Chapter 2 (Hilbert spaces). We cover Banach spaces, topological spaces, locally convex vector spaces, bounded operators, the spectral theorem, and hopefully unbounded operators. There will be an in-class midterm, a take-home midterm, and regular homework assignments (but no final).

Autumn 2007 I taught:

- Math 113. This is a `linear algebra done right' course (as is the title of the primary text). It does not assume any linear algebra background, or a background in writing proofs. However, one of the goals of the course is to make you proficient in proof-writing, which is a crucial skill for more advanced mathematics courses, as well as a mechanism by which you can test your understanding of the material. For an application-oriented linear algebra class, see Math 103. You may want to try out both courses at the beginning of the quarter to see which suits your taste better.
- Math 256A. Partial differential equations. This was an advanced graduate PDE class, but no PDE background was required. However, a thorough knowledge of functional analysis and Fourier analysis (as presented in the Math 205 sequence) was a must. The course was based on Michael Taylor's PDE book and Richard Melrose's lecture notes.

- Math 174A, Topics in Differential Equations with Applications. We use Michael Taylor's Partial Differential Equations: I (Basic Theory). This book covers both ODEs and PDEs, and has an approach that naturally leads to extensions in modern PDE theory that we cannot cover this quarter. The book is fairly advanced, but if read carefully, with attention paid in lectures too, it should be a great reference. We cover only small parts of the book: the first half of Chapter 1 (ODEs), and Chapter 3 (Fourier series and Fourier transform), with perhaps a little bit of Chapter 2. There will be a midterm, a final, and regular homework assignments.

In spring 2004, back at MIT, I taught 18.157, Introduction to Microlocal Analysis: here is the web page. The previous fall I taught 18.152, Introduction to Partial Differential Equations: here is the web page. In March 2004, I gave a lecture at the Clay Mathematics Institute on distribution theory; if you are an undergraduate interested in the flavor of modern analysis, please have a look at the overheads in postscript or pdf format.

I am co-organizing a meeting in honor of Gunther Uhlmann, and a meeting at Northwestern University on Microlocal Methods in Spectral and Scattering Theory.

Daniel Grieser, Stefan Teufel and I are organizing a meeting on Microlocal Methods in Mathematical Physics and Global Analysis in Tübingen on June 14-18, 2011.

On October 25-26, 2008, I co-organized a conference in honor of Richard Melrose's upcoming 60th birthday.

I am organizing the Analysis and PDE seminar. We are not meeting in Fall 2008 due to the special semester at MSRI on Analysis on Singular Spaces.

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.

Jared Wunsch and I organized a meeting on Scattering theory and singular spaces at Northwestern University on May 27-30, 2005.

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

My 2014 ICM slides.

My lecture notes from an AMS session talk at Penn State in October 2009 on asymptotically anti de Sitter spaces, and the updated version from Banff in March 2010.

My lecture notes from my talk at MSRI in October 2008 on de Sitter-Schwarzschild space.

My lecture notes (slightly preliminary version) for the 2007 joint meeting in New Orleans are here.

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.

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.

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, Dean Baskin, Hans Christianson, Kiril Datchev, Nick Haber, Andrew Hassell, Peter Hintz, Maarten de Hoop, Lizhen Ji, Robert Lauter, Rafe Mazzeo, Richard Melrose, Marius Mitrea, Werner Müller, Victor Nistor, Antonio Sa Barreto, Emmanuel Schenk, Plamen Stefanov, Michael Taylor, Gunther Uhlmann, Xue Ping Wang, Herwig Wendt, Jared Wunsch or Maciej Zworski, in postscript or pdf 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). Postscript version.#### 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. Postscript version.#### The spectral projections and the resolvent for scattering metrics, joint work with Andrew Hassell.

Published in Journal d'Analyse Mathématique, 79:241-298 (1999). Also in postscript.#### 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. Postscript version.#### Semiclassical estimates in asymptotically Euclidean scattering, joint work with Maciej Zworski.

Published in Commun. Math. Phys., 212:205-217 (2000). Also in postscript.#### 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). Also in postscript.#### Resolvents and Martin boundaries of product spaces, joint work with Rafe Mazzeo.

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

Published in Proc. Lond. Math. Soc. (3) 94:545-593 (2007). 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). Also in postscript.#### 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). Also in postscript. 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.

Published in Annals of Mathematics, 168:749-812 (2008). There is also a correction to the proof of Proposition 7.3. 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.

Published in Analysis and PDEs, 1:127-196 (2008). The original version is available in pdf and postscript formats.#### Propagation of singularities for the wave equation on edge manifolds, joint work with Richard Melrose and Jared Wunsch.

Published in Duke Mathematical Journal, 144:109-193 (2008). The original version is here, and the November, 2007 version is also available.#### The wave equation on asymptotically de Sitter-like spaces.

Published in Advances in Mathematics, 223:49-97 (2010). Original version from 2007 -- the new version is more reader friendly. (A reference to Dafermos-Rodnianski had been added to the 2007 version posted here as compared to arxiv version.)#### Semiclassical second microlocal propagation of regularity and integrable systems, joint work with Jared Wunsch.

Published in Journal d'Analyse Mathematique, 108:119-157 (2009). Erratum from March, 2011. A link to the January, 2008 version.#### Diffraction by edges.

Published in Modern Physics Letters B 22:2287-2328 (2008).#### Asymptotics of solutions of the wave equation on de Sitter-Schwarzschild space, joint work with Richard Melrose and Antonio Sa Barreto.

Published in Commun. in PDEs, 39:512-529 (2014). The original version was from 2008, updated January 5, 2012.#### Diffraction of singularities for the wave equation on manifolds with corners, joint work with Richard Melrose and Jared Wunsch.

Published in Astérisque, 351 (2013), 136pp. Preprint, 2009. The originally posted version is here.#### Diffraction at corners for the wave equation on differential forms.

Published in Communications in PDE, 35:1236-1275 (2010). The original submitted version is here. The first posted version is here. Apart from minor changes, a new section was added to this first version regarding other boundary conditions covered by the method of the paper.#### Positive commutators at the bottom of the spectrum, joint work with Jared Wunsch.

Journal of Functional Analysis 259:503-523 (2010). This version fixes an equation numbering problem from the previously posted version. The original submitted version is here.#### The wave equation on asymptotically Anti-de Sitter spaces.

Published in Analysis and PDE 5:81-144 (2012). The original version from 2009 is here.#### Gluing semiclassical resolvent estimates, or the importance of being microlocal, joint work with Kiril Datchev.

Published in Oberwolfach Reports 7(2):1648-1651 (2010). The complete manuscript (rather than just this report) is available below.#### Gluing semiclassical resolvent estimates via propagation of singularities, joint work with Kiril Datchev.

Published in Int. Math. Res. Notices (IMRN), 2012 (23):5409-5443 (2012).#### Propagation through trapped sets and semiclassical resolvent estimates, joint work with Kiril Datchev.

Published in Annales de l'Institut Fourier, 62(6): 2347--2377 (2012).#### Morawetz estimates for the wave equation at low frequency, joint work with Jared Wunsch.

Published in Math. Annalen, 355(4):1221--1254 (2013). The originally posted version is here.#### Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces, with an appendix by Semyon Dyatlov.

Inventiones Math 194:381-513 (2013). Original preprint from 2010; revised 2011, again in 2012. Posted version added detail and explanation to the previous version, as well as the argument on pp.89-90 checking a hypothesis of the Wunsch-Zworski setup that was not checked in full generality in the earlier version, and correcting several imprecise statements in Subsection 3.3. It also rearranged the previous version, moving the large imaginary sigma part to the final section of the paper to make the rest more readable. This version added detail and explanation to the original version, resulting in new Subsections 3.3 and 2.7, as well as the expansion of Subsection 3.2, apart from minor changes. See the Acknowledgments section at the end of the preprint for additional information. The previous posted version fixed a few typos and minor issues compared to the originally posted version, and adds some references. The earlier posted version fixed a few typos compared to the arxiv version.#### Analytic continuation and semiclassical resolvent estimates on asymptotically hyperbolic spaces, joint work with Richard Melrose and Antonio Sa Barreto.

Published in Commun. in PDEs 39:452-511 (2014). The original version was from 2011, updated January 5, 2012. Original posted version from March 17, 2011.#### Microlocal analysis of asymptotically hyperbolic spaces and high energy resolvent estimates.

Published in `Inverse problems and applications. Inside Out II', edited by Gunther Uhlmann, Cambridge University Press, MSRI Publications, no. 60 (2012). Minor revision (2012) of version from May 30, 2011; most important it corrects some incorrect numerology in intermediate steps. The originally posted version from April 7, 2011, is available.-
#### Propagation of singularities around a Lagrangian submanifold of radial points, joint work with Nick Haber.

To appear in the Bulletin de la Société Mathématique de France. Preprint, 2011. -
#### Diffraction from conormal singularities, joint work with Maarten de Hoop and Gunther Uhlmann.

Preprint, 2012. -
#### Semiclassical resolvent estimates at trapped sets, joint work with Kiril Datchev.

Published in Annales de l'Institut Fourier, 62(6):2379--2384 (2012). -
#### From resolvent estimates to damped waves, joint work with Hans Christianson, Emmanuel Schenk and Jared Wunsch.

Published in Journal d'Analyse Mathematique 122:143-162 (2014). The original version is here. -
#### Spectral theory for the Weil-Petersson Laplacian on the Riemann moduli space, joint work with Lizhen Ji, Rafe Mazzeo and Werner Müller.

To appear in Commentarii Mathematici Helvetici. Preprint, 2012. -
#### Analytic continuation and high energy estimates for the resolvent of the Laplacian on forms on asymptotically hyperbolic spaces.

Preprint, 2012. This version fixes a couple of typos relative to the originally posted version. -
#### Multi-scale discrete approximations of Fourier integral operators associated with canonical transformations and caustics, joint work with Maarten de Hoop, Gunther Uhlmann and Herwig Wendt.

Published in Multiscale Model. Simul. 11(2):566--585 (2013). -
#### The inverse problem for the local geodesic ray transform, joint work with Gunther Uhlmann.

Preprint, 2012. -
#### Asymptotics of radiation fields in asymptotically Minkowski space, joint work with Dean Baskin and Jared Wunsch.

To appear in Amer. J. Math. Original preprint, 2012. -
#### Resolvents, Poisson operators and scattering matrices on asymptotically hyperbolic and de Sitter spaces.

Preprint, 2013. Slightly revised version of the originally posted version. -
#### Boundary rigidity with partial data, joint work with Plamen Stefanov and Gunther Uhlmann.

Preprint, 2013. This corrects the formulation of the global result, Theorem 1.2, in the originally posted version, and adds additional detail. -
#### Semilinear wave equations on asymptotically de Sitter, Kerr-de Sitter and Minkowski spacetimes, joint work with Peter Hintz.

Preprint, 2013. Revised version, with appendix removed into a separate paper; original was this. -
#### Non-trapping estimates near normally hyperbolic trapping, joint work with Peter Hintz.

Preprint, 2013. Original included in the appendix of this paper above; this is an expanded version as a separate paper. -
#### Global analysis of quasilinear wave equations on asymptotically Kerr-de Sitter space, joint work with Peter Hintz.

Preprint, 2014.

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.