Curriculum Vitae of Mirko Tarulli Di Giallonardo
Name: Tarulli Di Giallonardo
First name: Mirko
Date of birth: March 02, 1977
Place of birth: Pescara - Italy
Nationality: Italian
Marital Status: Single
Current position: Ph.D. student at University of Pisa (Italy)
Languages: Italian, English.
Business address: Dipartimento di Matematica ”L. Tonelli” - Via F. Buonarroti, 2 - 56100
Pisa (Italy) e-mail: tarulli@mail.dm.unipi.it
Home address: Via Pola, 51/b - 67039 Sulmona (L’Aquila) (Italy)
Scientific Career
Academic Year 1995-96:
- June 1996
High school Diploma (Maturità di Liceo Scientifico) with full marks (60/60).
Academic Year 2001-02:
- September 2001 - December 2002
Three months of study in Sofia at the Sofia University and Technical University of
Sofia .
- July 2002
Taking the degree certificate (Laurea) in Mathematics at the University of L’Aquila
with full marks (110/110 cum laude).
Title of the thesis:
Stime a priori su variet`a di Riemann a curvatura costante negativa.
Adviser: Prof. V. Georgiev.
Accademic Year 2002-03:
- November 2003
Winning a competition for a Ph.D. position (Dottorato di Ricerca) in Mathematics at
the University of Pisa; research adviser: Prof. Vladimir Georgiev.
- 7- 14 June 2003
Talk entitled ”Some A Priori Estimates On Riemannian Manifold With Constant Negative
Curvature” in 29-th International Summer School ”Application of Mathematics
in Engineering and Economics” , Sozopol, Bulgaria
- 19 November 2003
Organizing Committee of INCONTRO DI FISICA MATEMATICA Equazioni dispersive
della fisica matematica, aspetti teorici e numerici Dipartimento di Matematica
Universit´a di Pisa, Italy.
Academic Year 2003-04:
- 20 January 2004
Talk entitled ”Resolvent Estimates For Compact Perturbations and Applications” nell’
ambito della Giornata di Incontro - Workshop Progetto GNAMPA 2003 ”Problemi
iperbolici in geometria e fisica”, Universit´a ”La Sapienza” di Roma, Italy.
- February 15 - May 15, 2004
Participation to ”Phase Space Analysis” Scuola Normale Superiore, Centro De Giorgi
Pisa, Italy.
- 7-14 July 2004
Talk entitled ”Resolvent estimates for scalar fields with electromagnetic perturbation”
in the NLW-HYKE : Summer School and Workshop on ”NonLinear Wave Equations”
Training Event of the HYKE network 7.- 14. July 2004, Vienna, Austria .
- 20-22 October 2004
Organizing Committee of the Meeting ”IPERPISA 2004 XI Incontro Nationale sulle
Equazioni Iperboliche” Dipartimento di Matematica Universit´a di Pisa, Italy.
- 14 December 2004
Talk entitled ”Smoothing and Strichartz Estimates for the Schr¨odinger Equation with
small Magnetic Potential” Universit´a de L’ Aquila, Italy.
Academic Year 2004-05:
- 1-2 March 2005
Talk entitled ”Scale invariant smoothing estimates for the Schr¨odinger Equation with
small Magnetic Potential” in the ”Equazioni Dispersive della Fisica-Matematica,
Aspetti Teorici e Numerici”, Universit´a ”La Sapienza” di Roma, Italy.
- 26 May-19 June 2004
Visit in the Institut f¨ur Angewandte Analysis Technische Universit¨at Bergakademie
Freiberg. Supervisor Prof. Dr. M. Reissig with two talk entitled ”Scale invariant
smoothing estimates for the Schr¨odinger Equation with small Magnetic Potential” and
”Strichartz Estimates for Wave and Schr¨odinger equations perturbed by Potential, An
Overview On General Results and Harmonic Analysis Techniques” .
Academic Year 2005-06:
- 8-13 November 2005
Participation to ”Phase Space Analysis Of Pde’s”, Pienza, Italy.
- 6 June 2006
Taking the Ph.D. degree (Dottorato di Riceraca) at the Department of Mathematics
”L. Tonelli”, University of Pisa.
Title of the thesis:
Smoothing-Strichartz estimates for dispersive equations perturbed by a first order differential
operator.
Adviser: Prof. V. Georgiev.
Scientific Research
My scientific research has been mainly devoted to the following fields,
- A Priori Sobolev Estimates On Riemannian Manifolds With Constant Negative Curvature
- Perturbative Theory for semilinear Wave Equation
- Strichartz Estimates for the Wave and Schr¨odinger Equation On Riemannian Manifolds
- A Priori Estimates On Riemannian Manifolds With Schwarzchild Metrics
- Smoothing And Strichartz Estimates for the Wave and Schr¨odinger Equation Perturbed
by a Magnetic Potential (Small and Large with respect to suitable norms)
- Wave Equation with Time depending perturbation (Resolvent and Microlocal analysis)
- Microlocal Analysis, Maslov Index and Initial Approach to Morse Theory
- All Aspects of Harmonic Analysis
Teaching Activities
- Academic Year 2004-05: Teaching activity (Assistant) in the course of Analysis II for
Physicists, teacher Prof. Vladimir Georgiev.
- Academic Year 2005-06: Teaching activity (Assistant) in the course of Analysis III for
Physicists, teacher Prof. Vladimir Georgiev.
Pubblications
-[1] Tarulli, M. Resolvent estimates for compact perturbation of the Laplace operator and
application. C. R. Acad. Bulgare Sci. 57 (2004), no. 4, 5–10.
-[2] Tarulli, M. Resolvent estimates for scalar fields with electromagnetic perturbation.
Electron. J. Differential Equations 2004, No. 146, 14 pp. (electronic).
-[3] Tarulli, M. Some A Priori Estimates On Riemannian Manifold With Constant Negative
Curvature.
Applications Of Mathematics in Engineering and Economics.
30th Jubilee International Conference, June 7- 11, 2004, Sozopol, Technical University
of Sofia (Bulgaria), 2005, 153-157.
-[4] Tarulli, M. Resolvent estimates for scalar fields with electromagnetic perturbation.
Communication (Poster) in the XI Incontro Nazionale sulle Equazioni Iperboliche
IPERPISA 2004 (october 2004).
-[6] Georgiev, V. Stefanov, A and Tarulli M. Strichartz Estimates For The Schr¨odinger
Equation With Small Magnetic Potential.
Journ´ees ´Equations aux d´eriv´ees partielles.
Forges-les-Eaux, 6 juin-10 juin 2005
GDR 2434 (CNRS).
-[5] Georgiev, V. and Tarulli M. Scale Invariant Energy Smoothing Estimates For The
Schr¨odinger Equation With Small Magnetic Potential.
(ArXiv: http://arxiv.org/abs/math.AP/0509015).
Asymptotic Analysis 47(1,2) (2006), IOS Press.
-[6] Georgiev, V. Stefanov, A and Tarulli M. Smoothing - Strichartz Estimates For The
Schr¨odinger Equation With Small Magnetic Potential.
(ArXiv: http://arxiv.org/abs/math.AT/0509416).
Submitted to Communications on Pure and Applied Analysis (CPAA).
-[7] Tarulli M. Energy - Strichartz Estimates For The Wave Equation With Small Magnetic
Potential. In Preparation.
Scientific Activity
My research interests involve several aspects of the Harmonic Analysis. I started to work
with Professor Vladimir Georgiev during the early years of my University study. I learned
tools of Harmonic analysis on Manifolds and on Symmetric Spaces and of Differential Geometry
and this activity produces my thesis Stime a priori su variet`a di Riemann a curvatura
costante negativa, where i generalized some estimates valid on Euclidean space Rn to Manifolds
with constant negative sectional curvature (Hyperbolic Spaces). Part of this thesis was
published as ”Some A Priori Estimates On Riemannian Manifold With Constant Negative
Curvature” (ref [3]).
A second project, again with Professor Vladimir Georgiev, concerned the use of the resolvent
and the scattering techniques in order to obtain stability and dispersive effect for some
hyperbolic equations perturbed by potentials. More precisely I obtained a priori estimates
for the resolvent of ”free” Laplacean and ”perturbed by a small magnetic field” Laplacian in
R3 in suitable weighted Hilbert spaces, after I applied these estimates to Schr¨odinger, wave
and Dirac Equations obtaining a smoothing effect (for wave equation and Dirac Equation a
L2 integrability of the Local Energy) and the problem of the resonances was also considered.
This is contained in the works ”Resolvent estimates for compact perturbation of the
Laplace operator and application.” (ref [2]) and ”Resolvent estimates for scalar fields with
electromagnetic perturbation.” (ref [3]).
Successively in a joint work with V.Georgiev ”Scale Invariant Energy Smoothing Estimates
For The Schr¨odinger Equation With Small Magnetic Potential” (ref [5]) , we considered
some scale invariant generalizations of the smoothing estimates for the free Schr¨odinger equation
obtained by Kenig, Ponce and Vega. Applying these estimates and using appropriate
commutator estimates, we obtain similar scale invariant smoothing estimates for perturbed
Schr¨odnger equation with small magnetic potential. These techniques were refined in a joint
work with Vladimir Georgiev and Atanas Stefanov (University of Kansas), ”Smoothing -
Strichartz Estimates For The Schr¨odinger Equation With Small Magnetic Potential” (ref
[6]). The work treats smoothing and dispersive properties of solutions to the Schr¨odinger
equation with magnetic potential. Under suitable smallness assumption on the potential
involving scale invariant norms we prove smoothing - Strichartz estimate (Bilinear Estimate)
for the corresponding Cauchy problem. An application that guarantees absence of pure
point spectrum of the corresponding perturbed Laplace operator is discussed too. Finally, it
is in preparation the work Smoothing - Strichartz Estimates For The Wave Equation With
Small Magnetic Potential where under a suitable modification of an interpolation lemma due
to Markus Keel and Terence Tao, I obtain smoothing - Strichartz bilinear estimate for the
corresponding Cauchy problems respectively for a class of dispersive equations and I apply
such result to the Wave Equation. Applications to the problem of perturbation of these
equation by a magnetic potential will be discuss too. On the other hand, I’m interested on
the problem of Wave equation on Schwarzschild Metric, in order to obtain some smoothing
and dispersive property (resolvent estimates, microlocal analysis pseudodifferential approach...).
The problem of Maslov index is also considered.
Another direction of my research is to consider wave and Klein-Gordon Equation with time
depending coefficient and with Time-Periodic perturbation in order to find Lp − Lq estimates
and to construct counterexamples where such estimates are not valid. The techniques
involved in this problem are use of some tools of microlocal analysis and pseudodifferential
operators and the project came from a collaboration with Professor Michael Reissig (Institut
f¨ur Angewandte Analysis Technische Universit¨at Bergakademie Freiberg). In the month of
June, i’ll present my Ph. D thesis entitled ”Smoothing - Strichartz Estimates for Dispersive
Equations Perturbed by a First Order Differential Operator” in the University of Pisa.
This thesis is the result of my study and research in the research group of Prof. Vladimir
Georgiev. |
