		Michel L. Lapidus Professor of Mathematics University of California, Riverside.



		Office:		SURGE 231
		Email:		lapidus*
				*@math.ucr.edu
		Mailing address:		Department of Mathematics, University of California Riverside, CA 92521-0135, USA

The magnitude of the gradient of the 13^th eigenfunction of the Koch snowflake drum.

Research Interests

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.

The Fractal Research Group

Dr. 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.

Seminar on Mathematical Physics and Dynamical Systems

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.

Spring 2008 Schedule of Presentations.

Apr 3 Organizational Mtg.

Apr 10 John Huerta: The Octonions

*Apr 17 Dr. Henrique Bursztyn: Generalized Complex/Kahler Geometry

Apr 24 Robert G. Niemeyer: A Countable Class of Periodic Orbits of the Koch Snowflake

May 1 Michael Maroun: Entropy, Classical and Quantum Action.

May 8 Scot Childress: Fractal Geometry, Number Theory, and Physics II

May 15 Alex Hoffnung: Categorifying Hamiltonian Mechanics: Motivation and Theory

May 22 Chris Rogers: Categorifying Hamiltonian Mechanics: Applications to String Theory

May 29 Vicente Alvarez: Finite Element Method on the Sierpinski Gasket

Jun 5 John Huerta: Quantum Mechanics and Representation Theory

*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.

Images of Fractal Drums – Motivation and Realization

Seeing the sound of a fractal harp (or string) and hearing the pattern of its complex dimensions fall from the clouds.

Picture by Professor Emeritus John de Pillis.

Hearing the shape of a fractal drum.

Picture by Professor Emeritus John de Pillis.

Citation: (for the images that follow)

Lapidus, M. L., Neuberger, J. W., Renka, R. J. , Griffith, C. A.

“Snowflake Harmonics and Computer Graphics: Numerical Computation of Spectra on Fractal Drums.”

Int. J. Bifur. Chaos., Vol. 6, No. 7. (1996) 1185-1210.

The second snowflake harmonic: the second eigenfunction of the Dirichlet-Laplacian

on the Koch snowflake domain.

The Thirteenth eigenfunction.

Nodal lines of the 13^th eigenfunction (bottom view).

Teaching

In the Winter of 2008, Dr. Lapidus will be teaching Math 141, Fractal Geometry and Applications. 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.

In the Spring of 2008, Dr. Lapidus will be on sabbatical.

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.

Concise Curriculum Vitae

(Click here for a PDF version of Curriculum Vitae.)

A. CURRICULUM VITAE

Citizenship: U.S.A. (naturalized American since 1988)

Education:

Ph.D. & Doctorat d'Etat ès Sciences (Mathematics), ‘Summa Cum Laude’, Université Pierre et Marie Curie (Paris VI), France, 1980 & 1986. Habilitation to Direct Research (Paris VI), 1987.

Experience:

Professor, University of California, Riverside, 1990-

· Member, Mathematical Sciences Research Institute (MSRI), Berkeley, Spring 1999 and Spring 2001 & Institut Henri Poincaré (IHP), Paris, Spring 2003.

· Member of the Newton Institute for Mathematical Sciences, Cambridge University, UK, (part of) Spring 1999

· Member, Institut des Hautes Etudes Scientifiques (IHES), Bures-sur-Yvette, France, 1994-95 and for several parts of 1995-98

· Visiting Professor, Yale University, New Haven, 1990-91

· Associate Professor, University of Georgia, Athens, 1986-90

· Visiting Assistant Professor, University of Iowa, Iowa City, 1985-86

· Member, Mathematical Sciences Research Institute (MSRI), Berkeley, 1984-85

· Assistant Professor, University of Southern California, Los Angeles, 1980-85

· Research Associate, University of California, Berkeley, 1979-80

· Research Associate, Université Paris VI, France, 1978-80

Recent Visiting Positions (over the last five years): Visiting Professor, University of Rome, Italy and University of Copenhagen, Denmark (four times each), as well as Oxford University, England, UK (7/01), Univ. of Paris (VI & VII) and Hebrew Univ. of Jerusalem, Israel.

Recent membership in Research Institutes (Mathematics/Physics/Pluridisc.): Isaac Newton Institute (Cambridge), Max Planck Institute (Dresden & Bonn) , IHES (Bures-Sur-Yvette/Paris), In(Paris), MSRI (Berkeley), Fields Institute (Canada), Erwin Schroedinger Institute (Vienna), Feza Gurse National Inst. (Istanbul).

Fellowships, Awards and Honors:

Research Fellowship D.G.R.S.T., Université Paris VI, 1978-80; George Lurcy Fellowship, University of California, Berkeley, 1979-80; My research on “Spectral and Fractal Geometry" was one of the two mathematical works presented by NSF to the “Office of the President" and to the U.S. Congress in the “National Science Foundation 1990 Fiscal Year Budget Proposal to Congress". Award of a university-wide “Creative Research Medal", Univ. of Georgia, Athens, 1989; Recipient of the M. G. Michael Award for Excellence in Research, Univ. of Georgia, Athens, 1989; Honorary Member, Research Board of Advisors, American Biography Institute (ABI), 1997-; Twentieth Century Award for Achievement, International Biographical Institute (IBC), Cambridge, UK, 1998; Outstanding Man of the 20th Century, ABI, 1999; nomin. “100 Geniuses of the 21st Century” (IBC, 2006). Elected Fellow of the American Association for the Advancement of Science (AAAS), Sept. 2000 [for “Distinguished Research Contributions to Mathematical Physics and Fractal Geometry"]; Member, AMS Council, and Associate Secretary of the American Mathematical Society (AMS), Western Section, Feb. 2002-. (Renominated and reconfirmed three times, now through Jan. 2010.) Nominated for membership in the American Academy of Arts and Sciences, 2003-04 &.2006-07.

Discussion in the general scientific press of my (joint) work on the “vibrations of fractal drums" and “the Weyl-Berry conjecture”. Please see the articles by Ian Stewart in Nature (vol. 333, 5/19/88), Jean-Pierre Fabre in La Recherche (vol. 202, 9/88), Barry Cipra in Science (vol. 259, 2/12/93), as well as the cover article by Ivars Peterson in Science News (vol. 146, No. 12, 9/17/94) and the book by the same author, entitled “The Jungles of Randomness: A Mathematical Safari" (Wiley, 1998). Also see the article by Ian Stewart in New Scientist (vol. 156, 12/20/97) and the recent article in La Recherche (vol. 383, 2/05).

Research Areas:

Current Research Projects: Mathematical Theory of Feynman Path Integrals; 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.

Research Grants (over the last twenty years): National Science Foundation Research Grants, DMS-8703138 (6/87- 6/89), DMS-8904389 (6/89-6/92), DMS-9207098 (7/92-7/96), DMS-9623002 (9/96-7/00), DMS-0070497 (7/00-7/05), and DMS-0707524 (7/07-6/10).

Selected Invited Talks (over the last five years):

(a) Plenary Speaker: Workshop on “Fractals and Dynamical Systems" (IMS, Hong Kong, China,12/00). Conf. “Fractals in Graz 2001" (Graz, Austria, 6/01). Conf. on “Analysis and Probability on Fractals" (Cornell Univ., Ithaca, 6/02). Internat. Sympos. on “Analysis, Geometry and Probability: Feynman's Legacy" (Lisbon, 6/02). Workshop (and Program) on “Topology in Condensed Matter Physics" (Max Planck Inst., Dresden, 7/02). Workshop on “Feynman Integrals and Applications" (MSRI, Berkeley, 12/02). Sympos. on “New Geometric Methods in Modern Science" (Paris, France, 03/02). Internat. Conf. on “Differential Equations and Asymptotic Theory” (Wuhan, China, 10/03) *. Workshop on “Mathematics of Schrödinger Operators, with Applications to Path Integrals” (Goa, India, 12/03)*. CMLA Annual Analysis Conf. (ENS Cachan, Paris, 5/04). 5th European Conf. on “Elliptic and Parabolic Problems: A Special Tribute to the Work of Haim Brezis” (Gaeta, Italy, 6/04). Internat. Conf. on “Physics of Irregular Systems” (Brazil, Fortaleza, 8/04) *. 2nd Conf. on “Analysis and Probability on Fractals” (Cornell Univ., Ithaca, 6/05). Internat. Workshop on “Fractal Analysis” (Eisenach, Germany, 9/05). Workshop on “Traces in Geometry, Number Theory and Quantum Fields” (Max-Planck Inst., Bonn, Germany,10/05). Conf. on “Feynman Integrals in Mathematics and Physics:" (Lincoln, 5/06). EMS School on “Arithmetic and Geometry Around Quantization” (Istanbul, Turkey, 6/06). Sympos. on “Contemporary Problems of Mathematical Analysis and Mathematical Physics” (Taormina, Italy, 6/06). Internat. Conf. on “Analysis on Fractals” (Kyoto, Japan, 9/06) *. Internat. Conf. on “Zeta Functions” (Moscow, Russia, 9/06). Interdisc. Conf on “Heat Kernels in Mathematics and Physics” (Ulm, Germany, 11/06)*. To take place: Workshop on “Geometric Measure Theoretic Approaches to Potential Theory on Fractals and Manifolds” (Oberwolfach, Germany, 03/07). Conf. on “Variational and Topological Methods, Applications and Simulations” (Flagstaff, 05/07). Workshop on “Analysis on Graphs and Fractals” (Newton Inst., Cambridge and Cardiff, UK, 5-6/07)

(b) Invited Speaker, Mini-Symposia/Special Sessions: Joint Internat. Mtg. AMS/SMF, Special Session on “Fractal Geometry, Number Theory and Dynamical Systems" (ENS Lyon, France, 7/01). Joint Internat. Mtg. AMS/SMM, Sp. Session on “Dynamical Systems, with Emphasis on Geometric Aspects and Symbolic Dynamics" (Morelia, Mexico, 5/01), featured speaker. AMS Reg. Mtg., Sp. Session on “Spectral and Inverse Spectral Theories of Schrödinger Operators” (Bangalore, India, 12/03). Sp. Session on “Fractals" (Columbus, 9/01). AMS Annual Mtg., Sp. Session on “Fractal Geometry and Applications" (San Diego, 1/02). ICM Satellite Conf. on “Fractal Geometry and Applications" (Nanjing, China, 7-8/02). AMS Reg. Mtg., Sp. Session on “Analytic Number Theory" (Salt Lake City, 10/02). Joint Internat. Mtg. AMS/Indian Math. Soc., AMS Sp. Sessions on “Topics in Spectral Geometry” (Albuquerque, 9/04) and on “Iterated Functions Systems and Analysis on Fractals” (Northwestern Univ., Evanston, 9/04). First Joint Mtg. French/Nordic Math. Soc., Sp. Session on “Spectral Geometry” (Reykjavik, Iceland, 1/05) *. ISAAC Sp. Session on “Elliptic and Parabolic Nonlinear Problems” (Catania, Italy, 7/05). “Einstein Century” Internat. Sympos., Sp. Session on “Gravitation and Nature of Spacetime” (Paris, France, 7/05). AMS Sp. Session on “Fractal Geometry in Analysis, Mathematical Physics, Number Theory and Dynamical Systems” (San Francisco, 5/06). Conf. on “Topology”, Sp. Session on “Fractals and Tilings” (GA, 7/06) ^* AMS Sp. Session on “Analysis and Probability on Fractals” (Storrs, 9/06). To take place: German Math. Soc. Annual Meeting, Sp. Session on “Geometry of Random Fractals” (Berlin, 3/07)^*

(c) Invited Participant: Research Programs on “Spectral Invariants” (MSRI, Berkeley, 3-6/01) and on “Noncommutative Geometry and K-Theory" (Inst. Henri Poincaré (IHP), Paris, 3-7/04), and (to take place) on “Analysis on Graphs and Fractals” (Newton Inst., Cambridge & Cardiff Univ., UK, 5-6/07).

* Unable to attend due to other duties.

B. Assorted Publications:*

1. “The Feynman Integral and Feynman’s Operational Calculus”, Oxford Mathematical Monographs, Oxford Univ. Press, Oxford and New York, 2000, approx. 800 pp. (precisely, 771+(xviii) pp.), (with G. W. Johnson). (Paperback edition: 2002, OUP). [Refereed research monograh.]

2. “Fractal Geometry and Number Theory: Complex dimensions of fractal strings and zeros of zeta functions”, Birkhäuser, Boston, 2000, 268+(xii) pp., (with M. van Frankenhuysen). [Refereed research monograph.]

3. “A Koch Tube Formula for the Koch Snowflake Curve, with Applications to Complex Dimensions”, J. London Math. Soc. No.2, 74(2006), 397-414, (with E. P. J. Pearse).

4. “Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and spectra of fractal strings”, Springer Monographs in Mathematics,

Springer-Verlag, New York, 2006, approx. 500 pp. (precisely, 460+(xxiv) pp.), (with M. van Frankenhuijsen). [Refereed research monograph.]

5. “In Search of the Riemann Zeros: Strings, fractal membranes and noncommutative spacetimes”, Amer. Math. Soc., Providence, R I, 2008, 600 pages (precisely, 558+(xxix) pp.), February, 2008. ISBN-10: 0-8218-422-5. US Library of Congress Classification: QA333.L37 2007.

6. “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).

7. “Remainder Estimates for the Asymptotics of Elliptic Eigenvalue Problems with Indefinite Weights”, Archives for Rational Mechanics & Analysis 98(1987), pp. 329-356, (with J. Fleckinger).

8. “Counterexamples to the Modified Weyl-Berry Conjecture”, Mathematical Proceedings of the Cambridge Philosophical Society, 119(1996), pp. 167-178, (with C. Pomerance).

9. “Towards a Noncommutative Fractal Geometry? Laplacians and Volume Measures on Fractals”, Contemporary Mathematics, American Mathematical Society 208(1997), pp. 211-252.

10. “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).

D. Four Examples of Synergistic Activities: (i) Associate Secretary, Amer. Math. Soc. (in charge of all the scientific meetings in the Western Section and of some of the national and international meetings of the AMS (including the forthcoming Joint National Meeting in San Diego, 1/08). (ii) Organizer of many research conferences in Mathematical Physics, Analysis, Dynamical Systems, Spectral Geometry or Fractal Geometry. (iii) Editor of several collective volumes at the interface of the above areas. (iv) Creation of a number of (Honors, undergraduate or graduate) courses and seminars in those areas as well as in Mathematical Biology.

E. Recent and Current Ph.D. Students and Postdocs: Elie Atallah*, Vicente Alvarez, Scot Childress, Britta Daudert, Cheryl Griffith, Christina He, Hafedh Herichi*, Nishu Lal*, Hung (Tim) Lu, Robert Niemeyer*, Erin Pearse, John Rock, Jonathan Sarhad*. NSF Postdocs: Alexander Teplyaev (8/00-12/02). Michael Anshelevich (9/02-8/04). Recent Postdoc: Steffen Winter (07). Recent VAPs: Mark Comerford (02-04), Cristina Ivan/Antonescu (04-05), Dana Clahane (05-08).

* This is an excerpt from a publication list of 94 entries, of which 83 have appeared or are in press.

Recent Preprints & Papers

· “Localization on Snowflake Domains”, Fractals, No. 3, 15 (2006),pp. 255-272 (joint with Britta Daudert). [E-print: arXiv:math.NA/0609798, 2006.]

· “A Prime Orbit Theorem for Self-Similar Flows and Diophantine Approximation”, Contemporary Mathematics, American Mathematical Society 290 (2001), pp. 113-138, (with M. van Frankenhuysen). [E-print: arXiv:math.SP/0111067, 2001.]

· “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.]

· “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.]

· “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.]

· “Fractal Strings and Multifractal Zeta Functions”, 40 typed pages, 2006, (joint with Jacques Levy Vehel and John Rock). [E-print: arXiv:math-ph/0610015.]

· “Dirac Operators and Spectral Triples for some Fractal Sets Built on Curves”, (joint with Erik Christensen and Christina Ivan) Advances in Mathematics, 217 (2008), 42-78 [E-print: arXiv:math.MG/0610222.]

· “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.]

· “A Trace on Fractal Graphs and the Ihara Zeta Function” (joint with Daniele Guido and Tommaso Isola), 29 typed pages, Transactions of the American Mathematical Society, 2008, in press. [E-print: arXiv:math.OA/0608060, 2007.]

· “Ihara’s Zeta Function for Periodic Graphs and Its Approximation in the Amenable Case, 21 typed pages, 2006, (joint with Daniele Guido and Tommaso Isola). [E-print: arXiv:math./0608229, 2006.]

· “Ihara Zeta Functions for Periodic Simple Graphs”, 17 typed pages, 2006, to appear in “C*-Algebras and Elliptic Theory II”, Proceedings of a Conference held at the Banach Center (Bedlewo, Poland, in January 2006) A. Mischenko & E. Toitsky, Eds. Trends in Mathematics, Birkhäuser-Verlag, Basel, 2007, pp. 103-121. (joint with Daniele Guido and Tommaso Isola). [E-print: arXiv:math.OA/0605753v3, 2007.]

· “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).

List of Publications

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 Palermo, 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] “A Prime Orbit Theorem for Self-Similar Flows and Diophantine Approximation”, Contemporary Mathematics, American Mathematical Society 290 (2001), pp. 113-138, (with M. van Frankenhuysen). [E-print: arXiv:math.SP/0111067, 2001.]

[JA41] “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).

[JA42] “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.]

[JA43] “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).

[JA44] “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.]

[JA45] “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.]

[JA46] “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).

[JA47] “Localization on Snowflake Domains”, (joint with Britta Daudert)to appear in the journal Fractals, 38 pages, 2006. [E-print: arXiv:math.NA/0609798, 2006.]

[JA48] Nonarchimedean Cantor Set and String.” Journal of Fixed Point Theory and Applications, 2008 (Special issue dedicated to Vladimir Arnold and the occurrence of his Jubilee, vol I) in press. (Joint with Hung Lu).

List of Books

For a complete list of books with a table of contents for each, click here.

For an excerpt of Dr. Lapidus’ latest book, “In Search of the Riemann Zeros”, click here. (Front cover, Back cover).

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.)