Department of Mathematics
University of California, Riverside
Riverside, CA 92507.
Research performed by Professor Lapidus and the construction
of this website have been funded by the
National
Science Foundation,
Grant # xxxxxx
Mathematical Physics, Functional and Harmonic Analysis, Geometric Analysis, Partial Differential Equations (PDEs), Dynamical Systems, Spectral Geometry, Fractal Geometry; Connections with Number Theory, Arithmetic Geometry and Noncommutative Geometry.
Current Research Projects:
- Mathematical Theory of Feynman Path Integrals;
- Feynman’s Operational Calculus for Noncommutative Operators.
- Vibrations of Fractal Drums;
- Analysis and PDEs On or Off Fractals;
- Waves and Diffusions in Fractal Media;
- Origins and Formation of Fractal Structures in Nature;
- Noncommutative Fractal Geometry;
- Analogues of Dirac Operators and Geodesics on Self-Similar Fractals and Trees;
- Fractal Strings and Membranes;
- Theory of Complex Fractal Dimensions;
- Analogues in the p-adic and Adelic Realms;
- Fractal Curvatures and Cohomology;
- Modular Flows on Moduli Spaces of Fractal Membranes;
- Noncommutative Flows of Zeros and Zeta Functions;
- Connections with the Riemann Zeta Function and the Riemann Hypothesis.
- Ihara Zeta Functions on Infinite Periodic or Self-Similar Graphs.
Professor Michel L. Lapidus coordinates a weekly meeting between him and his PhD students and mentees. It is during the Fractal Research Group meetings that he talks informally with his students and keeps informed of each student's progress. It is also here that his students have the opportunity to present their research and discuss more specialized topics not fitting with the spirit of the more formal Seminar on Mathematical Physics and Dynamical Systems. Visitors are welcome!
Post Doc's
Aviv Censor
Graduate Students
Scot Childress
Vicente Alvarez
Hafedh Herichi
Elie Atalla
Michael Maroun
Robert Niemeyer
Nishu Lal
Jayson Payne
Undergraduate Students
Diane Pell
The Seminar on Mathematical Physics and Dynamical Systems (MPDS, Math 260) is a seminar led by Dr. Lapidus that meets every quarter, once a week. It is here that students and guest speakers may present their research on topics ranging from the Feynman Integral to Fourier Analysis to Fractal Billiards. Such an environment allows for the dissemination of new ideas for interested and motivated students, visitors and researching faculty, while at the same time exposing students to more specialized topics and allowing for necessary professional development.
Fall 2008 Schedule of Presentations
October 2, 2008: Scot Childress
October 9, 2008: No Seminar - Holiday.
October 16, 2008: Mike Maroun
October 23, 2008: Professor Harold Widom
October 30, 2008: Jonathan Sarhad
November 6, 2008: Elie Atalla
November 13, 2008: John Huerta
November 20, 2008: Hafedh Herichi
November 4, 2008: Scot Childress
*This will be a colloquium, joint with the MPDS Seminar; hence, it will be lasting from 4:10pm to 5:00pm, with refreshments and tea/coffee starting at about 3:40pm.
In the Fall of 2008, Dr. Lapidus will be teaching Math 211, Differential Equations. For students and observers currently enrolled in the course, you may access the online content from http://www.ilearn.ucr.edu. Please, use your student email address and perm-pin to log in.
Every quarter, Dr. Lapidus holds a seminar in Mathematical Physics and Dynamical Systems. In the past, this seminar has been offered every Thursday at 3:40-4:30 PM. Also, Dr. Lapidus meets weekly with his students and mentees in the Fractal Research Group. The time of the FRG varies from quarter to quarter, but anyone interested in Fractal Geometry and its applications are invited to attend, as this is a more informal setting designed to foster discussion.
| 250 word CV | ||
| One page CV | ||
| Four page CV |
- For a complete list of publications, including conference proceedings, book chapters, miscellaneous articles, papers in preparation or submitted, as well as a detailed table of contents for each book, click here.
- For a complete list of publications by theme and category (without table of contents), click here.
[JA1] “Formules de Moyenne et de Produit pour les Résolvantes Imaginaires d'Opérateurs Auto-Adjoints”, [Mean and Product Formulas for Imaginary Resolvents of Self-Adjoint Operators], Comptes Rendus de l’Académie des Sciences Paris Sér. A 291 (1980), pp. 451-454; MR 81j:47016; Zbl 446:47010.
[JA2] “Généralisation de la Formule de Trotter-Lie”, [Generalization of the Trotter-Lie Formula], Comptes Rendus de l’Académie des Sciences Paris Sér. A 291 (1980), pp. 479-500; MR 81k:47092; Zbl 447:47023.
[JA3] “Perturbation d'un Semi-groupe par un Groupe Unitaire”, [Perturbation of a Semigroup by a Unitary Group], Comptes Rendus de l’Académie des Sciences Paris Sér. A 291 (1980), pp. 535-538; MR 82d:47046; Zbl 447:47022.
[JA4] “Generalization of the Trotter-Lie Formula”, Integral Equations and Operator Theory 4 (1981), pp. 366-415; MR 83e:47057; Zbl 463:47824.
[JA5] “Modification de l'Intégrale de Feynman pour un Potentiel Positif Singulier: Approche Séquentielle”, [Modification of the Feynman Integral for a Nonnegative Singular Potential: Sequential Approach], Comptes Rendus de l’Académie des Sciences Paris Sér. I Math. 295 (1982), pp. 1-3; MR 83b:81028; Zbl 493:35038.
[JA6] “Intégrale de Feynman Modifiée et Formule du Produit pour un Potentiel Singulier Négatif”, [Modified Feynman Integral and Product Formula for a Negative Singular Potential], Comptes Rendus de l’Académie des Sciences Paris Sér. I Math. 295 (1982), pp. 719-722; MR 85h:35065; Zbl 508:35027.
[JA7] “Valeurs Propres du Laplacien avec un Poids qui Change de Signe”, [Eigenvalues of the Laplacian with an Indefinite Weight Function], Comptes Rendus de l’Académie des Sciences Paris Sér. I Math. 298 (1984), pp. 265-268; MR 85j:35139.
[JA8] “Eigenvalues of Elliptic Boundary Value Problems with an Indefinite Weight Function”, Transactions of the American Mathematical Society 295 (1986), pp. 305-324; MR 87j:35282, (with J. Fleckinger).
[JA9] “Product Formula for Imaginary Resolvents with Application to a Modified Feynman Integral”, Journal of Functional Analysis 63 (1985), pp. 261-275; MR 87c:47059.
[JA10] “Perturbation Theory and a Dominated Convergence Theorem for Feynman Integrals”, Integral Equations and Operator Theory 8 (1985), pp. 36-62; MR 86g:81036.
[JA11] “The Differential Equation for the Feynman-Kac Formula with a Lebesgue‑Stieltjes Measure”, Letters in Mathematical Physics 11 (1986), pp. 1-31; MR 87d:58033.
[JA12] “Generalized Dyson Series, Generalized Feynman Diagrams, the Feynman Integral and Feynman's Operational Calculus”, Memoirs of the American Mathematical Society No. 351, 62 (1986), pp. 1‑78, (with G.W. Johnson); MR 88f:81034.
[JA13] “The Feynman-Kac Formula with a Lebesgue-Stieltjes Measure and Feynman's Operational Calculus”, Studies in Applied Mathematics 76 (1987), pp. 93-132.
[JA14] “Remainder Estimates for the Asymptotics of Elliptic Eigenvalue Problems with Indefinite Weights”, Archives for Rational Mechanics and Analysis 98 (1987), pp. 329-356, (with J. Fleckinger); MR 88b:35149.
[JA15] “The Feynman-Kac Formula with a Lebesgue-Stieltjes Measure: An Integral Equation in the General Case”, Integral Equations and Operator Theory 12 (1989), pp. 163-210.
[JA16] “Une Multiplication Non Commutative des Fonctionnelles de Wiener et le Calcul Opérationnel de Feynman”, [A Noncommutative Multiplication of Wiener Functionals and Feynman's Operational Calculus], Comptes Rendus de l’Académie des Sciences Paris Sér. I Math. 304 (1987), pp. 523-526, (with G.W. Johnson).
[JA17] “Strong Product Integration of Measures and the Feynman-Kac Formula with a Lebesgue-Stieltjes Measure”, Supplemento ai Rendiconti del Circolo Matematico di Palerm, Ser. II, 17 (1987), pp. 271-312.
[JA18] “Tambour Fractal: Vers une Résolution de la Conjecture de Weyl-Berry pour les Valeurs Propres du Laplacien”, [Fractal Drum: Towards a Resolution of the Weyl‑Berry Conjecture for the Eigenvalues of the Laplacian], Comptes Rendus de l’Académie des Sciences Paris Sér. I Math. 306 (1988), pp. 171-175, (with the collaboration of J. Fleckinger).
[JA19] “Noncommutative Operations on Wiener Functionals and Feynman's Operational Calculus”, Journal of Functional Analysis 81 (1988), pp. 74-99, (with G.W. Johnson).
[JA20] “Schrödinger Operators with Indefinite Weights: Asymptotics of Eigenvalues with Remainder Estimates”, Differential and Integral Equations 7 (1994), pp. 1389-1418, (with J. Fleckinger).
[JA21] “Fractal Drum, Inverse Spectral Problems for Elliptic Operators and a Partial Resolution of the Weyl-Berry Conjecture”, Transactions of the American Mathematical Society 325 (1991), pp. 465-529.
[JA22] “Quantification, Calcul de Feynman Axiomatique et Intégrale Fonctionnelle Généralisée”, [Quantization, Axiomatic Feynman's Operational Calculus and Generalized Functional Integral], Comptes Rendus de l’Académie des Sciences Paris Sér. I Math. 308 (1989), pp. 133-138.
[JA23] “Product Formula for Normal Operators and the Modified Feynman Integral”, Proceedings of the American Mathematical Society 110 (1990), pp. 449-460, (with A. Bivar Weinholtz).
[JA24] “La Fonction Zêta de Riemann et la Conjecture de Weyl-Berry pour les Tambours Fractals”, [The Riemann Zeta-Function and the Weyl-Berry Conjecture for Fractal Drums], Comptes Rendus de l'Académie des Sciences Paris Sér. I Math. 310 (1990), pp. 343‑348, (with C. Pomerance).
[JA25] “Hypothèse de Riemann, Cordes Fractales Vibrantes et Conjecture de Weyl‑Berry Modifiée”, [The Riemann Hypothesis, Vibrating Fractal Strings and the Modified Weyl-Berry Conjecture], Comptes Rendus de l'Académie des Sciences Paris Sér. I Math. 313 (1991), pp. 19-24, (with H. Maier).
[JA26] “The Riemann Zeta-Function and the One-Dimensional Weyl-Berry Conjecture for Fractal Drums”, Proceedings of the London Mathematical Society (3) 66, No. 1 (1993), pp. 41-69, (with C. Pomerance).
[JA27] “The Riemann Hypothesis and Inverse Spectral Problem for Fractal Strings”, Journal of the London Mathematical Society (2) 52, No. 1 (1995), pp. 15-35, (with H. Maier).
[JA28] “Weyl's Problem for the Spectral Distribution of Laplacians on P.C.F. Self-Similar Fractals”, Communications in Mathematical Physics 158 (1993), pp. 93-125, (with J. Kigami).
[JA29] “Indefinite Elliptic Boundary Value Problems on Irregular Domains”, Proceedings of the American Mathematical Society 125 (1995), pp. 513-526, (with J. Fleckinger).
[JA30] “Analysis on Fractals, Laplacians on Self-Similar Sets”, Noncommutative Geometry and Spectral Dimensions", Topological Methods in Nonlinear Analysis 4, No. 1 (1994), pp. 137-195.
[JA31] “Eigenfunctions of the Koch Snowflake Drum”, Communications in Mathematical Physics 172 (1995), pp. 359-376, (with M. Pang).
[JA32] “Fractals and Vibrations: Can You Hear the Shape of a Fractal Drum?”, Fractals 3, No. 3 (1995), pp. 725-736.
[JA33] “Counterexamples to the Modified Weyl-Berry Conjecture”, Mathematical Proceedings of the Cambridge Philosophical Society, 119 (1996), pp. 167-178, (with C. Pomerance).
[JA34] “Generalized Minkowski Content and the Vibrations of Fractal Drums and Strings”, Mathematical Research Letters 3 (1996), pp. 31-40, (with C.Q. He).
[JA35] “Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann Zeta-Function”, Memoirs of the American Mathematical Society No. 608, 127 (1997), pp. 1-97, (with C.Q. He).
[JA36] “Snowflake Harmonics and Computer Graphics: Numerical Computation of Spectra on Fractal Domains”, International Journal of Bifurcation and Chaos 6 (1996), pp. 1185-1210, (with J.W. Neuberger, R.J. Renka, and C.A. Griffith). (Includes 23 computer graphics color plates.)
[JA37] “Feynman's Operational Calculus and Evolution Equations”, Acta Mathematicae Applicandae 47 (1997), pp. 155-211, (with B. DeFacio and G. W. Johnson).
[JA38] “Feynman's Operational Calculus: A Heuristic and Mathematical Introduction”, Annales Mathématiques Blaise Pascal 3 (1996), pp. 89-102. (Special issue dedicated to the memory of Prof. Albert Badrikian.)
[JA39] “Self-Similarity of Volume Measures for Laplacians on P.C.F. Self-Similar Fractals”, Communications in Mathematical Physics 217 (2001), pp. 165-180, (with J. Kigami).
[JA40] “Complex Dimensions of Self-Similar Fractal Strings and Diophantine Approximation”, Journal of Experimental Mathematics No.1, 12 (2003), pp. 41-69, (with M. van Frankenhuysen).
[JA41] “Fractality, Self-Similarity and Complex Dimensions”, Proceedings of Symposia in Pure Mathematics, American Mathematical Society, 72 (2004), Part 1, pp. 349-372, (with M. van Frankenhuijsen). [E-print: arXiv:math.NT/0401156, 2004.]
[JA42] “Random Fractal Strings: Their Zeta Functions, Complex Dimensions and Spectral Asymptotics”, Transactions of the American Mathematical Society No.1, 358 (2006), pp. 285-314, (with B. Hambly).
[JA43] “A Tube Formula for the Koch Snowflake Curve, with Applications to Complex Dimensions”, Journal of the London Mathematical Society No. 2, 74 (2006), pp. 397-414. (Joint with Erin P. J. Pearse) . [E-print: arXiv:math-ph/0412029, 2005.]
[JA44] “Beurling Zeta Functions, Generalized Primes, and Fractal Membranes”, Acta Applicandae Mathematicae, No.1,94 (2006), pp. 21-48, (joint with Titus Hilberdink).[E-print: arXiv:math.NT/0410270, 2004.]
[JA45] “Feynman’s Operational Calculi: Auxiliary Operations and Related Disentangling Formulas”, to appear in the journal Modern Problems of Integration Theory, 29 pages, 2006, (joint with Gerald W. Johnson).
[JA46] “Localization on Snowflake Domains”, (joint with Britta Daudert) to appear in the journal Fractals, 38 pages, 2006. [E-print: arXiv:math.NA/0609798, 2006.]
[JA47] Nonarchimedean Cantor Set and String”, (joint with H. Lu) Journal of Fixed Point Theory and Applications 3 (2008), 181-190. (Special issue dedicated to Vladimir Arnold and the occurrence of his Jubilee.) [E-print: IHES/M/M-08-29] (2008).
[JA49] “Ihara’s Zeta Function for Periodic Graphs and Its Approximation in the Amenable Case, (joint with Daniele Guido and Tommaso Isola) Journal of Functional Analysis, No. 6, 255 (2008) 1339-1361.
Submitted Research Articles:
[JA48] “A Trace on Fractal Graphs and the Ihara Zeta Function”, 29 typed pages, 2006, (joint with Daniele Guido and Tommaso Isola). [E-print: arXiv:math.NA/0609798, 2006.]
[JA50] “Tube Formulas and Complex Dimensions of Self-Similar Tilings”, 49 typed pages, 2006, (joint with Erin P. J. Pearse). [E-print: arXiv:math.DS/0605527, 2006.] [E-print: IHES/M/M-08-27]
[JA51] “Dirac Operators and Spectral Triples for some Fractal Sets Built on Curves”, (joint with Erik Christensen and Christina Ivan) Advances in Mathematics, No. 1, 217 (2008), 42-78. [E-print: arXiv:math.MG/0610222.]
[JA52] “Fractal Strings and Multifractal Zeta Functions”, 40 typed pages, 2006, (joint with Jacques Levy Vehel and John Rock). [E-print: arXiv:math-ph/0610015.]
Research Books:
R [RB1] “The Feynman Integral and Feynman's Operational Calculus”, Oxford Mathematical Monographs, Oxford Science Publications, Oxford University Press, Oxford, London and New York, approx. 800 pages (precisely, 771 + (xviii) pages & 21 illustrations), March 2000. ISBN 0 19 853574 0 (Hbk). (With Gerald W. Johnson.) [Corrected Reprinting, Jan. 2001. First Paperback Edition, Jan. 2002. ISBN 0 19 851572 3 (Pbk). Second Reprinting: Jan. 2003. Electronic Edition: forthcoming.] US Library of Congress Classification: QA312.J54 2000.
R[RB2] “Fractal Geometry and Number Theory”. (Subtitle: “Complex Dimensions of Fractal Strings and Zeros of Zeta Functions”.) Research Monograph, Birkhäuser-Verlag, Boston, approx. 300 pages (precisely, 268 + (xii) pages & 26 illustrations), January 2000. (With Machiel van Frankenhuijsen.) ISBN 0-8176-4098-3. (Boston). ISBN 3-7643-4098-3 (Basel). US Library of Congress Classification: QA614.86.L36 1999.
R[RB3] “Fractal Geometry, Complex Dimensions and Zeta Functions”, (Subtitle: “Geometry and Spectra of Fractal Strings”.) Refereed Research Monograph, Springer Monographs in Mathematics, Springer-Verlag, New York, approx. 490 pages (precisely, 460 + (xxiv) pages & 54 illustrations), August 2006. (With Machiel van Frankenhuysen.). ISBN-10: 0-387-33285-5. e-ISBN: 0-387-35208-2. ISBN-13: 978-0-387-33285-7. US Library of Congress Control Number: 2006929212.
R [RB4] “In Search of the Riemann Zeros” (Subtitle: “Strings, Fractal Membranes, and Noncommutative Spacetimes”.) Refereed Research Monograph, American Mathematical Society, Providence, RI, approx. 500 pages (precisely, 496 + (xix) pages), 2007, in press. (Expected to appear in the Summer of 2007.)
Edited Books:
[EB1] Editor: "Progress in Inverse Spectral Geometry", Proc. Summer School on "Progress in Inverse Spectral Geometry", held in Stockholm (Sweden) in June-July 1994, Trends in Mathematics Series, Vol. 1, Birkhäuser-Verlag, Basel, November 1997, 204 pages, (Co-Editor: Stig I. Andersson). ISBN 3-7643-5755-X (Basel). ISBN 0-8176-5755-X (Boston). US Library of Congress Classification: QA614.95.P78 1997.
[EB2] Coordinating Editor: “Harmonic Analysis and Nonlinear Differential Equations”, (Volume in honor of Prof. Victor L. Shapiro), Contemporary Mathematics, Vol. 208, American Mathematical Society, Providence, RI, August 1997, 350 + (xii) pages. (Includes a Dedication and a Preface.) (Co-Editors: Lawrence H. Harper and Adolfo J. Rumbos.) ISBN 0-8218-0565-7 (alk. paper). US Library of Congress Classification: QA403.H223 1997.
[EB3] Editor: “Dynamical, Spectral and Arithmetic Zeta Functions”, Proc. Special Session (Amer. Math. Soc. Annual Meeting, San Antonio, Texas, Jan. 1999), Contemporary Mathematics, Vol. 290, American Mathematical Society, Providence, RI, December 2001, 195 + (x) pages. (Co-Editor: Machiel van Frankenhuysen.) ISBN 0-8218-2079-6 (alk. paper). US Library of Congress Classification: QA351.A73 1999.
[EB4] & [EB5]: Managing Editor: “Fractal Geometry and Applications”. (Subtitle: “A Jubilee of Benoît Mandelbrot”.) [Two volumes (Parts 1 & 2), totaling approximately 1,100 pages (precisely, 1,080 + (xxvi) pages).] Proceedings of Symposia in Pure Mathematics (PSPUM), American Mathematical Society, Vol. 72, Parts 1 & 2, Providence, RI, Dec. 2004. (Co-Editor: Machiel van Frankenhuijsen.) ISBN 0-8218-3292-1 (set: acid free paper).ISBN 0-8218-3637-4 (part 1: acid free paper). ISBN 0-8218-3638-2 (part 2: acid free paper). US Library of Congress Classification: QA325.F73 2004.
Subtitle of Part 1: Analysis, Number Theory, and Dynamical Systems. (Approx. 520 pages; precisely, 508 + (xiii) pages.)
Subtitle of Part 2: Multifractals, Probability and Statistical Mechanics, Applications. (Approx. 560 pages; precisely, 546 + (xiii) pages.)
Books in Preparation:
[RB5] Research Monograph: “The Feynman Integral and Feynman’s Operational Calculus for Noncommuting Operators” (tentative title). (With Gerald W. Johnson and Lance Nielsen.)
[TB] Textbook: “An Invitation to Fractal Geometry and Its Applications”. (With Dana Clahane, Erin Pearse and Robert Niemeyer.)
