Professor Michel L. Lapidus

Professor of Mathematics

Cooperating Faculty Member in the Departments of Physics and Astronomy, and of Computer Science and Engineering

Department of Mathematics
University of California, Riverside
900 Big Springs Road, Surge 231
Riverside, CA 92521-0135 USA
Webpage: http://www.math.ucr.edu/~lapidus
Email: lapidus*
*math.ucr.edu

 

Michel L. Lapidus


The research performed by Professor Lapidus and the construction of this Web site have been funded in part by the National Science Foundation (NSF).(*) (**)

 

 

 

 



Research Interests (Top)

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:


The Fractal Analysis and Mathematical Physics Research Group

Professor Michel L. Lapidus coordinates a weekly meeting between him and his PhD students and mentees. It is during The Fractal and Mathematical Physics 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!

Postdocs and Visiting Assistant Professors

Aviv Censor, Dana Clahane, Hafedh Herichi, Cristina Ivan [Antonescu], Michael Maroun, Stephen Muir, Michael Pang, Alexander Teplyaev, Steffen Winter, Maxim Zyskin

Recent and Current Graduate Students

Vicente Alvarez, Scot Childress, Tim Cobler*, Britta Daudert, Cheryl Griffith, Christina He, Hafedh Herichi, Nishu Lal, Hung (Tim) Lu, Michael Maroun, Robert Niemeyer, Erin Pearse, John Quinn, Scott Roby*, John Rock, Benjamin Sanders*, Jonathan Sarhad, Dominick Scaletta*, Sean Watson*

Visiting Scholars

David Carfi, Gerald Johnson, Jun Kigami, Richard Niemeyer

Recent and Current Undergraduate Students

Joseph Adams, Mutaz Alsayegh, Jason Payne, Diane Pell, Emad Totari, Sean Townsend, Leo Vu

Fractal and Mathematical Physics Research Group Fall 2011

Past Seminar Schedules

The schedule for Spring 2012

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The Seminar on Mathematical Physics and Dynamical Systems (Top)

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.

Past Seminar Schedules

The schedule for Spring 2012

*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.


Teaching (Top)

In the Spring of 2012, Dr. Lapidus will be on sabbatical leave at the Institute Des Hautes Scientifiques (IHES) in Bures-sur-Yvette and the Insitut Henri Poincaré (IHP) in Paris, France.

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 FRG seminar is held on Thursday at 11:10-12:30, and anyone interested in Fractal Geometry and its applications are invited to attend, as this is a more informal setting designed to foster discussion.


Curriculum Vitae(s)

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Research Collaborators

The indicated institutions is the current one (when it is known). Otherwise, it is the one corresponding to the first collaboration, in which case a star (*) is added.

Antonio Bivar Weinholtz (University of Lisbon, Portugal; Mathematics/ Mathematical Physics)

Erik Christensen (University of Copenhagen, Denmark; Mathematics)

Britta Daudert [Laser Interferometer Gravitational-Wave Observatory (LIGO), Caltech, Pasadena, CA, USA; Western Regional Climate Center, Desert Research Center, Reno, NV, USA; Applied Mathematics/ Mathematical Physics]

Brian DeFacio (University of Missouri, Columbia, MO, USA; Physics)

Kate E. Ellis (California Polytechnic State University, Pomona, CA, USA; Mathematics)

Jacqueline Fleckinger (or Fleckinger--Pellé), (Université Paul Sabatier, Toulouse, France)

Cheryl A. Griffith (Engineering and Systems, Algorithms GPA/ SBAS Algorithms; Applied Mathematics/ Engineering)

Daniele Guido (University of Rome, Tor' Vergata, Italy; Mathematics/ Mathematical Physics)

Ben Hambly (Oxford University, Oxford, England, UK; Mathematics/ Statistical Probability)

Christina Q. He (University of California, Riverside, CA, USA; Mathematics/ Financial Mathematics)

Hafedh Herichi (University of California, Riverside, CA, USA; Mathematics)

Titus Hilberdink (University of Reading, England, UK; Mathematics)

Tommaso Isola (University of Rome, Tor' Vergata, Italy; Mathematics/ Mathematical Physics)

Cristina Ivan (formerly Antonescu) (University of Texas MD Anderson, Houston, TX; Mathematics/ Mathematical Biology)

Gerald W. Johnson (University of Nebraska, Lincoln, NE, USA; Mathematics/ Statistics/ Probability)

Jun Kigami (Kyoto University, Kyoto, Japan; Mathematics/ Applied Analysis & Complex Dynamical Systems Group)

Jacques Levy--Vehel [Institut National de la Recherche en Informatique et Automatique (INRIA), Nantes & Paris, France; Engineering/ Applied Mathematics/ Computer Science]

Michael C. MacKenzie (University of Connecticut, Storrs, CT, USA; Mathematics)

Ryszard Nest (University of Copenhagen, Copenhagen, Denmark; Mathematics)

John W. Neuberger (North Texas State University, Denton, TX, USA; Mathematics/ Numerical Analysis/ Scientific Computing)

Robert N. Niemeyer (University of New Mexico, Albuquerque, NM, USA; Mathematics)

Michael M. M. Pang (University of Missouri, Columbia, MO, USA; Mathematics)

Erin P. J. Pearse (California Polytechnic State University, San Luis Obispo, CA, USA; Mathematics)

Goran Radunović (University of Zagreb, Zagreb, Croatia; Applied Mathematics/ Electrical Engineering and Computing)

Robert J. Renka (North Texas State University, Denton, TX, USA; Computer Science)

Scott S. Roby (University of California, Riverside, CA, USA; Mathematics)

John A. Rock (California State Polytechnic University, Pomona, CA, USA; Mathematics)

Rolando de Santiago (California State Polytechnic University, Pomona, CA, USA; University of Iowa, Iowa City, IA, USA; Mathematics)

Jonathan J. Sarhad (University of California, Riverside, CA, USA; Mathematics/ Mathematical Biology/ Biology)

Machiel van Frankenhuijsen (formerly van Frankenhuysen) (Utah Valley University, UT, USA; Mathematics)

Steffen Winter (Karlsruhe Institute of Technology, Karlsruhe, Germany; Mathematics)

Darko Žubrinić (University of Zagreb, Zagreb, Croatia; Applied Mathematics/ Electrical Engineering and Computing)

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Recent Preprints & Papers (Top)

“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.] ([BC8])

“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.] ([JA41])

“Beurling Zeta Functions, Generalized Primes, and Fractal Membranes”, Acta Applicandae Mathematicae No.1, 94 (2006), pp. 21-48, (with T. Hilberdink). [E-print: arXiv:math.NT/0410270, 2004.] ([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, (with E. P. J. Pearse). [E-print: arXiv:math-ph/0412029, 2005.] ([JA43])

“Localization on Snowflake Domains”, Fractals No. 3, 15 (2007), pp. 255-272, (with B. Daudert). [E-print: arXiv:math.NA/0609798, 2006.] ([JA46])

“Nonarchimedean Cantor Set and String”, Journal of Fixed Point Theory and Applications 3 (2008), pp. 181-190, (with H. Lu). (Special issue dedicated to Vladimir Arnold on the occasion of his Jubilee. Vol. I.) [E-print: IHES/M/08/29, 2008. re.pdf.] ([JA47])

“A Trace on Fractal Graphs and the Ihara Zeta Function”, Transactions of the American Mathematical Society No. 6, 361 (2009), pp. 3041-3070, (with D. Guido and T. Isola). [E-print: arXiv:math.OA/0608060v3, 2008. IHES/M/08/36, 2008.] ([JA48])

“Ihara’s Zeta Function for Periodic Graphs and Its Approximation in the Amenable Case, Journal of Functional Analysis No. 6, 255 (2008), pp. 1339-1361, (with D. Guido and T. Isola). [E-print: math.OA/0608229. IHES/M/08/37, 2008. re.pdf.] ([JA49])

“Tube Formulas and Complex Dimensions of Self-Similar Tilings”, Acta Applicandae Mathematicae No. 1, 112 (2010), pp. 91-137, (with E. P. J. Pearse). (Springer Open Access: DOI 10.1007/S10440-010-9562-x) [E-print: arXiv:math.DS/0605527v6, 2010. IHES/M/08/27, 2008.] ([JA54])

“Dirac Operators and Spectral Triples for some Fractal Sets Built on Curves”, Advances in Mathematics No. 1, 217 (2008), pp. 42-78, (with E. Christensen and C. Ivan). [E-print: arXiv:math.MG/0610222v2, 2008.] ([JA50])

“Fractal Strings and Multifractal Zeta Functions”, Letters in Mathematical Physics No. 1, 88 (2009), pp. 101-129, (with J. Levy Vehel and J. A. Rock). (Special issue dedicated to the memory of Moshe Flato.) (Springer Open Access: DOI 10.1007/s11005-009-0302-y.) [E-print: arXiv:math-ph/0610015v3, 2009.] ([JA51])

“Towards Zeta Functions and Complex Dimensions of Multifractals”, Journal of Complex Variables and Elliptic Equations No. 6, 54 (2009), pp. 545-559, (with J. A. Rock). (Special issue dedicated to Fractal Analysis.) [E-print: arXiv.math.ph/0810.0789, 2008. IHES/M/08/34, 2008.] ([JA52])

“Self-Similar p-Adic Fractal Strings and Their Complex Dimensions”, p-Adic Numbers, Ultrametric Analysis and Applications (Russian Academy of Sciences, Moscow, and Springer-Verlag), No. 2, 1 (2009), pp. 167-180, (with H. Lu). [E-print: IHES/M/08/42, 2008.] ([JA53])

“Tube Formulas for Self-Similar Fractals”, in: “Analysis on Graphs and its Applications”, P. Exner, et al. (Eds.), Proceedings of Symposia in Pure Mathematics, American Mathematical Society 77 (2008), pp. 211-230, (with E. P. J. Pearse). (Springer Open Access.) [E-print: arXiv:math.DS/0711.0173v1, 2007. IHES/M/08/28, 2008.] ([CP11])

“Bartholdi Zeta Functions for Periodic Simple Graphs”, in: “Analysis on Graphs and its Applications”, P. Exner, et al. (Eds.), Proceedings of Symposia in Pure Mathematics, American Mathematical Society 77 (2008), pp. 109-122, (with D. Guido and T. Isola). [E-print: IHES/M/08/38, 2008.] ([BC12])

“Ihara Zeta Functions for Periodic Simple Graphs”, in: “C*-Algebras and Elliptic Theory II ”, Proceedings of a Conference held at the Banach Center in Warsaw, Poland, D. Burghelea, R. Melrose, et al. (Eds.), Trends in Mathematics, Birkhäuser-Verlag, Basel, 2008, pp. 103-121, (with D. Guido and T. Isola).  [E-print: arXiv:math.OA/0605753, 2006. IHES/M/08/39, 2008.] ([BC11])

“Towards the Koch Snowflake Fractal Billiard: Computer Experiments and Mathematical Conjectures, in: "Gems in Experimental Mathematics", T. Amdeberham, L. A. Medina and V. H. Moll, (Eds.), Contemporary Mathematics, American Mathematical Society, 517 (2010), pp. 231-263, (joint with R. G. Niemeyer). [E-print: arXiv:math.DS:0912.3948.v1, 2009.] ([BC13])

“Pointwise Tube Formulas for Fractal Sprays and Self-Similar Tilings with Arbitrary Generators", Advances in Mathematics 227 (2011), pp. 1349-1398, (with E. P. J. Pearse and S. Winter). [E-print: arXiv:math.DG:1006.3807v3, 2011] ([JA55])


Publications

Published Research Journal Articles:

[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, No. 1, 4 (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 No. 4, 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 Applicandae Mathematicae 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). (re.pdf)

[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, (with E. 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, (with T. Hilberdink). [E-print: arXiv:math.NT/0410270, 2004.]

[JA45]   “Feynman's Operational Calculi: Auxiliary Operations and Related Disentangling Formulas”, Integration: Mathematical Theory and Applications No. 1, 1 (2008), pp. 29-48, (with G. W. Johnson).

[JA46]    “Localization on Snowflake Domains”, Fractals No. 3, 15 (2007), pp. 255-272, (with B. Daudert). [E-print: arXiv:math.NA/0609798, 2006.]

[JA47]    “Nonarchimedean Cantor Set and String”, Journal of Fixed Point Theory and Applications 3 (2008), pp. 181-190, (with H. Lu). (Special issue dedicated to Vladimir Arnold on the occasion of his Jubilee. Vol. I.) [E-print: IHES/M/08/29, 2008. re.pdf.]

[JA48]   “A Trace on Fractal Graphs and the Ihara Zeta Function”, Transactions of the American Mathematical Society No. 6, 361 (2009), pp. 3041-3070, (with D. Guido and T. Isola). [E-print: arXiv:math.OA/0608060v3, 2008. IHES Preprint: IHES/M/08/36, 2008.]

[JA49]   “Ihara’s Zeta Function for Periodic Graphs and Its Approximation in the Amenable Case”, Journal of Functional Analysis, No. 6, 255 (2008), pp. 1339-1361, (with D. Guido and T. Isola). [E-print: math.OA/0608229. IHES Preprint: IHES/M/08/37, 2008. re.pdf.]

[JA50]   “Dirac Operators and Spectral Triples for some Fractal Sets Built on Curves”, Advances in Mathematics No. 1, 217 (2008), pp. 42-78, (with E. Christensen and C. Ivan). [E-print: arXiv:math.MG/0610222v2, 2008.]

[JA51]   “Fractal Strings and Multifractal Zeta Functions”, Letters in Mathematical Physics No. 1, 88 (2009), pp. 101-129 (with J. Levy Vehel and J. A. Rock). (Special issue dedicated to the memory of Moshe Flato.) (Springer Open Access: DOI 10.1007/s11005-009-0302-y.) [E-print: arXiv:math-ph/0610015v3, 2009.]

[JA52]   “Towards Zeta Functions and Complex Dimensions of Multifractals”, Journal of Complex Variables and Elliptic Equations No. 6, 54 (2009), pp. 545-559, (with J. A. Rock). (Special issue dedicated to Fractal Analysis.) [E-print: arXiv.math.ph/0810.0789, 2008. IHES Preprint: IHES/M/08/34, 2008.]

[JA53]   “Self-Similar p-Adic Fractal Strings and Their Complex Dimensions”, p-Adic Numbers, Ultrametric Analysis and Applications (Russian Academy of Sciences, Moscow, and Springer-Verlag), No. 2, 1 (2009), pp. 167-180, (with H. Lu). [E-print: IHES/M/08/42, 2008.]

[JA54]   “Tube Formulas and Complex Dimensions of Self-Similar Tilings”, Acta Applicandae Mathematicae No. 1, 112 (2010), pp. 91-137 (with E. P. J. Pearse). (Springer Open Access: DOI 10.1007/0440-010-9562x)  [E-print: arXiv:math.DS/0605527v6, 2011. IHES Preprint: IHES/M/08/27, 2008.]

[JA55] “Pointwise Tube Formulas for Fractal Sprays and Self-Similar Tilings with Arbitrary Generators”, Advances in Mathematics No. 4, 227 (2011), pp. 1349-1398, (with Erin P. J. Pearse and Steffen Winter). [E-print: arXiv:math.DG:1006.3807v3, 2011.]

[JA56] "Hyperfunctions and Spectral Zeta Functions of Laplacians on Self-Similar Fractals", Journal of Physics A: Mathematical and Theoretical, 45 (2012), 365205 14p., (with Nishu Lal). [E-print: arXiv:1202.4126v2 [math-ph], 2012. IHES Preprint: IHES/M/12/23, 2012.]

[JA57] "Riemann Zeros and Phase Transitions via the Spectral Operator on Fractal Strings", Journal of Physics A: Mathematical and Theoretical, 45 (2012), 374005 23p., (with Hafedh Herichi). [E-print: arXiv: 1203.4828v2 [math-ph], 2012. IHES Preprint: IHES/M/12/09, 2012.]

[JA58] "Sequences of Compatible Periodic Hybrid Orbits of Prefractal Koch Snowflake Billiards", Discrete and Continuous Dynamical Systems, Ser. A, No. 8 33 (2013), pp. 3719-3740, (with Robert G. Niemeyer). [E-print: arXiv: 1204.3133v2 [math.DS], 2012. IHES Preprint: IHES/M/12/16, 2012.]

[JA59] "Multifractal Analysis via Scaling Zeta Functions and Recursive Structure of Lattice Strings", Contemporary Mathematics, American Mathematical Society, 600 (2013), p. 205-238, 34p., (with R. de Santiago, S. A. Roby and J. A. Rock). [E-print: arXiv: 1207.6680v3 [math-ph], 2013. IHES Preprint: IHES/M/12/19, 2012.]

[JA60] "Minkowski Measurability Results for Self-Similar Tilings and Fractals with Monophase Generators", Contemporary Mathematics, American Mathematical Society, 600 (2013), p. 185-203, 19p. (with E. P. Pearse and S. Winter). [E-print: arXiv: 1104.1641v3 [math.MG], 2013. IHES Preprint: IHES/M/12/33, 2012.]

[JA61] "Minkowski Measurabilitiy and Exact Fractal Tube Formulas for p-Adic Self-Similar Strings", Contemporary Mathematics, American Mathematical Society, 600 (2013), p. 161-184, 24p. (with M. van Frankenhuijsen and L. Hung). [E-Print: arXiv:1209.6440v1 [math.MG], 2012. IHES Preprint: IHES/M/12/23, 2012.]

Submitted Research Journal Articles:

[JA62] "Dirac Operators and Geodesic Metric on the Harmonic Sierpinski Gasket and Other Fractals", (with Jonathan J. Sarhad), Nov. 2012, 28 pages. [E-print: arXiv: 1212.0878v1 [math.MG], 2012. IHES preprint: IHES/M/12/32, 2012.]

[JA63] "Truncated Infinitesimal Shifts, Spectral Operators, and Quantized Universality of the Riemann Zeta Function", (with Hafedh Herichi), Apr. 2013, 35pp. [E-print: arXiv: 1305.3933v1 [math.NT], 2013. IHES preprint: IHES/M/13/12, 2013.]

Research Journal Articles in Preparation:

[JA64] "Fractal Strings, the Spectral Operator and the Riemann Hypothesis: Zeta Values, Riemann Zeros, Phase Transitions and Quantized Universality", Research Memoir, 139 pages (as of Aug. 2013), (with Hafedh Herichi).

[JA65] "Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings, 23 pages (as of Aug. 2013), (with Machiel van Frankenhuijsen and Hung Lu).

[JA66] "Spectral Operator and Convergence of its Euler Product in the Critical Strip", work in progress, 14 pages (as of Aug. 2013), (with Hafedh Herichi).

[JA67] "Analytic Continuation of a Class of Multifractal Zeta Functions", work in progress, 10 pages (as of Aug. 2013), (with Driss Essouabri and John A. Rock).

[JA68] "Distance and Tube Zeta Functions on Fractals in Euclidean Spaces", 32 pages (as of Aug. 2013), (with Goran Radunović and Darko Žubrinić).

[JA69] "Meromorphic Extensions of Fractal Zeta Functions", 38 pages (as of Aug. 2013), (with Goran Radunović and Darko Žubrinić).

[JA70] "Box-Counting Fractal Strings and Zeta Functions", work in progress, 12 pages (as of Aug. 2013), (with John A. Rock and Darko Žubrinić).

[JA71] "Fractal Curvatures and Local Tube Formula", work in progress, (with Erin P.J. Pearse and Steffen Winter).

[JA72] "Minkowski Measurability of Fractal Sprays and Self-Similar Tilings", work in progress, (with Erin P.J. Pearse and Steffen Winter).

[JA73] "Families of Periodic Orbits of the Koch Snowflake Billiard", Technical Report, 58 pages, 2011. [E-print: arXiv: 1105.0737v1 [math.DS], 2011.]

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Books (Top)

Research Books:

[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.

[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 Frankenhuysen.)  ISBN  0-8176-4098-3. (Boston). ISBN 3-7643-4098-3 (Basel). US Library of Congress Classification:  QA614.86.L36   1999.

[RB3] “Fractal Geometry, Complex Dimensions and Zeta Functions”. (Subtitle: “Geometry and Spectra of Fractal Strings”.) Refereed Research Monograph, Springer Monographs in Mathematics, Springer, New York, approx. 490 pages (precisely, 460 + (xxiv) pages & 54 illustrations), August 2006. (With Machiel van Frankenhuijsen.) 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.

[RB4] “In Search of the Riemann Zeros” (Subtitle:  “Strings, Fractal Membranes, and Noncommutative Spacetimes”.)  Refereed Research Monograph, American Mathematical Society, Providence, RI, approx. 600 pages (precisely, 558 + (xxix) pages), January 2008. ISBN-10:0-8218-422-5. ISBN-13:978-0-8218-4222-5. US Library of Congress Classification: QA333.L37 2007.

[RB5] "Fractal Geometry, Complex Dimensions and Zeta Functions". (Subtitle: Geometry and Spectral of Fractal Strings.) Second Revised and Enlarged Edition (of the 2006 edition, [RB3]). Refereed Research Monograph, Springer Monographs in Mathematics, New York, approx. 600 pages (precisely, 594 = 568 + (xxvi) pages and 73 illustrations), January 2013. (With Machiel van Frankenhuijsen.) ISSN 1439-7382. ISBN 978-1-4614-2175-7. ISBN 978-1-4614-2176-4 (eBook). DOI 10.1007/978-1-4614-2176-4. US Library of Congress Control Number: 2012948283.

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. 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), Vol. 72, Parts 1 & 2, American Mathematical Society, Providence, RI, Dec. 2004. (Co-Editor:  Machiel van Frankenhuijsen.)  ISBN 0-8218-3292-1 (set). ISBN 0-8218-3637-4 (part 1). ISBN 0-8218-3638-2 (part 2).  US Library of Congress Classification: QA325.F73   2004.S

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.)

[EB6] "Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics I". (Subtitle: "Fractals in Pure Mthematics".) Contemporary Mathematics, Vol. 600, American Mathematical Society, Providence, RI. 2013. 399p. (Co-Editors: David Carfi, Erin P.J. Pearse and Machiel van Frankenhuijsen.) ISBN-10: 0-8218-9147-2. ISBN-13:978-0-8218-9147-6.

[EB7] "Fractal Geometry and Dynnamical Systems in Pure and Applied Mathematics II". (Subtitle: Fractals in Applied Mathematics".) Contemporary Mathematics, Vol. 601, American Mathematical Society, Providence, RI. 2013. 372p. (Co-Editors: David Carfi, Erin P.J. Pearse and Machiel van Frankenhuijsen.) ISBN-10: 0-8218-9148-0. ISBN-10: 978-0-8218-9148-3.

Book Chapters:

[BC1] "Towards the Koch Snowflake Fractal Billiard: Computer Experiments and Mathematical Conjectures", in Contemporary Mathematics, T. Amdeberhan et al (Eds). American Mathematical Society. Providence, RI. 517 (2010) p. 231-263. 33p. (with R. Niemeyer). [E-Print: arXiv:math.DS.0912.3948, 2009.]

[BC2] "The Geometry of p-Adic Fractal Strings: A Comparative Survey", in Contemporary Mathematics, J. Araujo et al (Eds). American Mathematical Society. Providence, RI. 551 (2011) p. 163-206. 44p. (with H. Lu). [E-print: arXiv: 1105.2966v1 [math.MG], 2011.]

[BC3] "Box-Counting Fractal Strings, Zeta Functions, and Equivalent Forms of Minkowski Dimension", in Contemporary Mathematics, D. Carfi et al (Eds). American Mathematical Society. Providence, RI. 600 (2012) p. 239-271. 33p. (with J. A. Rock and D. Žubrinić). [E-Print: arXiv:1207.6681v2 [math-ph], 2013. IHES Preprint: IHES/M/12/22, 2012.]

[BC4] "Fractal Complex Dimensions, Riemann Hypothesis and Invertibility of the Spectral Operator", in Contemporary Mathematics, D. Carfi et al (Eds). American Mathematical Society. Providence, RI. 600 (2012) p. 51-89. 39p. (with H. Herichi). [E-Print: arXiv: 1210.0882v3 [math.FA], 2013. IHES Preprint: IHES/M/12/25, 2012.]

Books in Preparation:        

[RB6] Research Monograph: “Feynman's Operational Calculus and Beyond”. (Subtitle: "Noncommutativity and Time-Ordering".) Oxford Mathematical Monograph, Oxford Science Publications, Oxford University Press, Oxford, London and New York. Expected date of publication: Dec. 2014. Approx. 328 pages. (With Gerald W. Johnson and Lance Nielsen.) [Contract signed with Oxford University Press in May 2013.]

[RB7] Research Monograph: "Fractal Zeta Functions" (Subtitle: Higher-Dimensional Theory of Complex Dimensions.) Approx. 240 pages, as of 08/10/13. (With Goran Radonović and Darko Žubrinić.)

[TB]    Textbook:  “An Invitation to Fractal Geometry”. (Subtitle: "Dimension Theory, Zeta Functions, and Applications".) (With Robert G. Niemeyer and John A. Rock.)


Useful Links

Books by Dr. Lapidus The American Mathematical Society Front for the Mathematics Arxives UC Riverside homepage Journal of Fractal Geometry (JFG)

JFG Cover
(Click to enlarge image)

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Images (Top)

 

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.

(Full-scale version)

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.

(Full-scale version)

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

on the Koch snowflake domain.

(Full-scale version)

Nodal curves of the second eigenfunction.

(Full-scale version)

The eigenfunctions of the Dirichlet Laplacian can be considered as a mathematical model for the study of the

(steady-states) "vibrations" of a drum.

(Full-scale version)

The 13th eigenfunction.

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Nodal curves of the 13th eigenfunction (bottom view).

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

(Full-scale version)

Another view of the magnitude of the gradient of the 13th eigenfunction.

(Full-scale version)

The first snowflake harmonic: the first eigenfunction of the Dirichlet Laplacian on the Koch snowflake domain.

(Full-scale version)

Level curves of several eigenfunctions.

(Full-scale version)

Nodal curves of several eigenfunctions.

(Full-scale version)

(*) Especially under the NSF Grants DMS-8703138, DMS-8904389, DMS-9207098, DMS-9623002, DMS-0070497, DMS-0707524, and DMS-1107750.

 

(**) The contents of this Web site have neither been reviewed nor approved by the National Science Foundation (NSF) and the views and opinions expressed in this Web site are strictly those of Michel L. Lapidus.