Robert Brooks

One of my current interests is in the application of probabilistic techniques to geometry. Here's a thought:

"I believe that basically you are a fine fellow, and one day some form of analysis will bear me out."

---Woody Allen, "The Gossage-Vardabedian Papers"

For those of you who think I'm just goofing off while on sabbatical, click here for the announcement of the Special Trimester in Heat Equations and Random Walks at the Institut Henri Poincare, Paris.

For those of you who think that "random walks" means goofing off while on sabbatical, or are otherwise unconvinced, probably nothing I say here will convince you.

Click here for the notes accompanying my lectures at the IHP, Spectral Geometry of Riemann Surfaces . I expect these notes to be updated with some regularity over the coming weeks.

Click here for the transparencies of my Edinburgh lecture, which can also give an overview of my IHP lectures.

Click here for a link to the Geometry-Topology Group web page.

Click here for the schedule for the Seminar in Geometric Analysis.

Click here for a copy of the M. Sc. thesis of my student, Dan Mangoubi, "Riemann Surfaces and 3-Regular Graphs," and here for a copy of his paper "Conformal Extension of Metrics of Negative Curvature."

Click here for the course notes for the course "Riemann Surfaces."

Click here for information (homework, course notes) for the course Differential Geometry.

Here are some of my recent and not-so-recent mathematical papers. If you know what you are doing, you may download the .ps files for these papers. If you encounter difficulties with this, please let me know.

Here are some of my more recent papers.

• On the Asymptotic Isoperimetric Constants for Riemann Surfaces and Graphs(with A. Zuk), to appear.
• Book review of I. Chavel, "Isoperimetric Inequalities: Differential Geometric And Analytic Perspectives," to appear BAMS.
• Isoscattering on Surfaces (with Orit Davidovich), to appear J. Geom. Anal.
• Random Construction of Riemann Surfaces (with Eran Makover), to appear.

• On the Asymptotic Isoperimetric Constants for Riemann Surfaces and Graphs(with A. Zuk), to appear.
• On Cheeger's Inequality (with Peter Perry and Peter Petersen V), Comm. Math. Helv. 68 (1993), pp. 599-621.
• Compactness and Finiteness Theorems for Isospectral Manifolds (with Peter Perry and Peter Petersen V), Jour. Reine Ang. Math. 426 (1992), pp. 67-89.
• Constructing Isospectral Manifolds , Amer. Math. Monthly 95 (1988), pp. 823-839. (.ps file in bad shape)
• The Continued Fraction Parameter in the Deformation Theory of Classical Schottky Groups, Contemp. Math. 136 (1992), pp. 41-54.
• Correction to ``Spectral Geometry in Dimension 3'' (with Peter Perry and Peter Petersen V), to appear.
• Factorization of N^2 +1 and the Modular Group (with Hershel Farkas), to appear.
• The First Eigenvalue of a Riemann Surface (with Eran Makover), ERA-AMS 5 (1999), pp. 76-81.
• Inverse Spectral Geometry, Andersson and Lapidus (eds), Progress in Inverse Spectral Geometry, Birkhauser Trends in Mathematics, pp. 115-132.
• Isophasal Scattering Manifolds in Two Dimensions(with Peter Perry), Comm. Math. Phys. 223 (2001), pp. 465-474.
• Isoscattering Schottky Manifolds(with Ruth Gornet and Peter Perry), GAFA 10 (2000), pp. 307-326. (.pdf file)
• Isoscattering on Surfaces (with Orit Davidovich), to appear J. Geo. Anal.
• Isospectral Graphs and Isospectral Surfaces (in memory of Hubert Pesce), Sem. Th. Spec. Geom. 1996-1997, pp. 105-113.
• Isospectral Problems in Conformal Geometry, in Boyom, Morvan, and Verstraelen (eds), Geometry and Topology of Submanifolds II, World Scientific (1990), pp. 50-64.
• An L^p Spectral Bootstrap Theorem (with Paul Glezen), Contemp. Math. 173 (1994), pp. 89-97.
• Mutually Isospectral Riemann Surfaces (with Ruth Gornet and William Gustafson), Adv. Math. 138 (1998), pp. 306-322.
• Non-Sunada Graphs, Ann. Inst. Fourier 49 (1999), pp. 707-725.
• Number Theory, Theta Functions, and Modular Curves(with Hershel Farkas and Irwin Kra), Contemp. Math. 201 (1997), pp. 125-154.
• Platonic Surfaces, Comm. Math. Helv. 74 (1999), pp. 156-170. .pdf version
• Random Construction of Riemann Surfaces (with Eran Makover), to appear.
• Reflections on the First Eigenvalue, , Texas Tech Distinguished Lecture Series.
• Riemann Surfaces with Large First Eigenvalue (with Eran Makover), J. d'Anal. 83 (2001), pp. 243-258.
• Some Examples in L^p Spectral Geometry (with Peter Perry and Peter Petersen V), J. Geo. Anal. 3 (1993), pp. 293-313.
• Some Geometric Aspects of the Work of Lars Ahlfors , in Brooks and Sodin (eds.), Lectures in Memory of Lars Ahlfors, IMCP 14 (2000), pp. 31-39.
• Some Relations Between Graph Theory and Riemann Surfaces , Isr. Math. Conf. Proc. 11 (1997), pp. 61-73. (For an older version of the paper, but with pictures, click here.)
• Some Relations Between Spectral Geometry and Number Theory,in Apanosov et.al., Topology '90, Walter de Gruyter, 1992, pp. 61-75. (good text but bad pictures) same paper with crummy text but good pictures.
• The Spectral Geometry of Belyi Surfaces (with Eran Makover), Entire Functions in Modern Analysis: Boris Levin Memorial Conference , IMCP 15, pp. 37-46.
• Spectral Geometry and the Cheeger Constant , in: J Friedman (ed), Expanding Graphs, Proc. DIMACS Workshop, Amer. Math. Soc. (1993), pp. 5-19.
• The Spectral Geometry of Flat Disks (with C. McMullen and Y. Eliashberg), Duke Math. J. 61 (1990), pp. 119-131.
• Spectral Geometry in Dimension 3 (with Peter Perry and Peter Petersen V), Acta Math. 173 (1994), pp. 283-305.
• The Spectral Geometry of k-Regular Graphs, J. d'Analyse 57 (1991), pp. 120-151.
• The Sunada Method , Contemp. Math. 231 (1999), pp. 25-35.
• Trace Formula Methods in Spectral Geometry , J. Fourier Anal. App. Kahane Special Issue, 1995, pp. 87-95.
• Twist Surfaces , in Picardello and Woess (eds.) Random Walks and Discrete Potential Theory, Camb. Univ. Press (1999), pp. 85-103.
• Typical Surfaces and Random Graphs, Sem. Th. Spec. Geom. 17 (1999), pp.99-103.
• Uniformization of Some Quotients of Modular Curves (with Yaacov Kopeliovich), Contemp. Math. 201 (1997), pp. 155-164.

Click here for a copy of Figurate Numbers and the Algebra of Recursion Sequences, the Avodat Gemer (high school project) of my son, Shimon Brooks.

You can send me mail here.

Dr. Robert Brooks <rbrooks@tx.technion.ac.il>
Department of Mathematics 
Technion - Israel Institute of Technology
Haifa 32000, ISRAEL

+972-4-829-4218
+972-4-832-4654(FAX)     http://www.math.technion.ac.il/~rbrooks      

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Over served since 23-Jan-96.