Dipartimento di Matematica Università degli Studi di Padova Via Trieste, 63
35121 - Padova (PD) Italy
Phone: (+39) 49 827 1460 Fax: (+39) 49 827 1428 Office: 530 Torre Archimede Email: [email protected]
Homepage: http://www.math.unipd.it/
~feleqi
Personal info
Born in Vlorë, Albania on October 23, 1983.
Resident in Italy since August, 2002.
Italian and Albanian citizen.
Current positions
Lecturer and researcher (tenured position)1 June 1, 2015 – Currently Head of the Department of Mathematics June 18, 2015 – Currently Institution: Universiteti i Vlorës “Ismail Qemali”, Vlorë, Albania.
Past academic positions
Postdoctoral researcher2 June 1, 2013 – May 30, 2015
Institution: Università degli Studi di Padova, Dipartimento di Matematica, Padua, Italy;
Postdoctoral mentor: Prof. Martino Bardi;
Research project title: Nonlinear Partial Differential Equations and Mean Field Games.
Postdoctoral researcher3 January 1, 2012 – May 30, 2013 Institution: Università degli Studi di Padova, Dipartimento di Matematica, Padua, Italy;
Postdoctoral mentor: Prof. Martino Bardi;
Research project title: Asymptotic problems is Partial Differential Equations arising in Control Theory and Differential Games.
1Albanian, “Punonjës mësimor-kërkimor”.
2Italian, “titolare di assegno per collaborazione ad attività di ricerca”, or briefly, “assegnista di ricerca”.
3see the first footnote.
Part-time lecturer4 January 1, 2014 - March 30, 2014 Taught the course “Mathematical Analysis II” to part-time (or working) students of Engineering disciplines at the Università degli Studi di Padvoa, Italy.
Postdoctoral researcher5 April 1, 2010 – December 31, 2011 Institution: Università degli Studi di Padova, Dipartimento di Matematica, Padua, Italy;
Postdoctoral mentor: Victor I. Burenkov;
Research project title: Some applications of the theory of functional spaces to pertur- bation problems for the eigenfunctions of elliptic differential operators.
Education
• Ph.D. in Mathematics January 2007 – December 2009
Institution: Università degli Studi di Padova, Italy. Defense date: April 14, 2010.
Title of the thesis: “Spectral stability estimates for eigenfunctions of second order elliptic operators”, Advisor: Prof. Victor I. Burenkov.
Evaluation: Excellent. Members of the evaluation commission: prof. Massimo Lanza de Cristoforis, prof. Gerassimos Barbatis, prof. José Maria Arrieta.
• M.Sc. in Mathematics October 2005 – June 2006
Institution: University of Bari, Italy. Mark: 110/110 summa cum laude. Defence date October 5, 2006.
Title of the thesis: “On the abstract approximation of functions” (Italian). Adviser:
Prof. Francesco Altomare
• B. Sc. in Mathematics October 2002 – June 2005
Institution: Universitá degli Studi di Bari, Italy. Mark: 110/110 summa cum laude.
Defence date: July 14, 2005.
Title of the thesis: “On some trigonometric series and wavelets” (in Italian). Advisors:
Prof. Enrico Jannelli, Dr. Sandra Lucente.
Publications and preprints
[1] M. Bardi, E. Feleqi, P. Soravia. Regularity of the minimum time and of solutions of eikonal equations via generalized Lie brackets (in preparation).
4Italian, “Professore a contratto”.
5see the first footnote.
[2] E. Feleqi, Joint time-state generalized semiconcavity of the value function of a jump diffusion optimal control problem, (submitted).
[3] E. Feleqi, F. Rampazzo, A nonsmooth Chow-Rashevski’s Theorem (in preparation).
[4] E. Feleqi, F. Rampazzo, Integral representations for bracket-generating multi-flows, accepted by Discrete Contin. Dyn. Syst. Ser. A., 35, No. 9, 4345–4366 (2015).
[5] E. Feleqi, Generalized semiconcavity of the value function of a jump diffusion optimal control problem, NoDEA - Nonlinear Differential Equations Appl., 22, No. 4, 777–809 (2015).
[6] E. Feleqi, The derivation of ergodic mean field game equations for several populations of players, Dyn. Games Appl., 4, 523–536 (2013).
[7] M. Bardi, E. Feleqi, Nonlinear elliptic systems and mean field games, (submitted).
[8] E. Feleqi, Estimates for the deviation of solutions and eigenfunctions of second-order elliptic Dirichlet boundary value problems under domain perturbation, accepted by J.
Differential Equation.
[9] V. I. Burenkov, E. Feleqi, Spectral stability estimates for the eigenfunctions of second order elliptic operators, Math. Nachr., 285, No. 11–12, 1357–1369 (2012).
[10] V. I. Burenkov, E. Feleqi, Extension of the notion of a gap to differential operators defined on different open sets, Math. Nachr. 286, No. 5–6, 518–535 (2013).
[11] E. Feleqi, Spectral stability estimates for the eigenfunctions of second order elliptic operators, Ph. D. thesis, Università degli Studi di Padova, Padua, Italy (2010).
[12] E. Feleqi, On the abstract approximation of functions (Italian) M.Sc. Thesis, Univer- sità degli Studi di Bari, Bari, Italy (2006).
N.B. Publications may be accessed also through my website: http://www.math.unipd.it/
~feleqi/publications.html
Teaching
- “Numerical Methods” (lecturer), a MSc course for Mathematics and Computer Sciences students, University of Vlora “Ismail Qemali”, 45 hours, Spring 2015.
- “Mathematical Analysis II” (lecturer) an intensive course addressed to working students of engineering courses at the Università degli Studi di Padova, 52 hours, 2013/14.
- “Mathematical Analysis II” (Teaching Assistant) Department of Physics and Astron- omy at the Università degli Studi di Padova, 24 hours, 2013/14.
- “A theorerm of Chow and Rashevski and nonsmooth extensions” (lecturer), Ph.D.
School Course, Department of Mathematics at the Università degli Studi di Padova, 10 hours, 2012/13.
- “Semigroups of Lévy and jump-diffusion processes” (lecturer) Ph.D. School Course, Department of Mathematics at the Università degli Studi di Padova, 8 hours, 2011/12.
- “Some results in Mean Field Games theory” (lecturer) Ph.D. School Course, Depart- ment of Mathematics at the Università degli Studi di Padova, 6 hours, 2011/12.
- “Calculus II” (Peer tutor) Faculty of Statistical Sciences at the Università degli Studi di Padova, 100 hours, 2009-2010.
Fields of Study and Research Interest
- (Nonsmooth) geometric control theory; optimal control and differential games; partial (integro)-differential equations; stability estimates for elliptic boundary value problems subject to domain perturbations; semigroups theory and evolution equations; calculus of variations; geometric measure theory, nonsmooth analysis; stochastic analysis...
Research Activity
Extensions of the concept of iterated Lie bracket and applications to geometric control theory, sub-Riemannian distances, and eikonal equations
- (Integral formulas for bracket-generating multi-flows). An extension of the concept of iterated Lie bracket, called integrating iterated Lie bracket is proposed [4], and is used to derive exact integral formulas for bracket-generating multlflows, improving well-known asymptotic formulas. The construction is intrinsic and produces another vector field, which depends not only on vector fields but also on certain “time” parameters (which reduces to a usual iterated bracket when these parameters vanish).
- (A set-valued iterated Lie bracket). A set-valued iterated Lie bracket is defined for collections of vector fields which are as many times differentiable as it is necessary to compute the corresponding classical Lie bracket only almost everywhere, thus possess- ing highest order derivatives defined only almost everywhere.
- (A nonsmooth version of Chow-Rashevski’s theorem.) A classical theorem of Chow and Rashevski, asserting the sufficiency of the well-known LARC (Lie algebra rank condi- tion) to complete controllability of linearly controlled systems of ODEs, is extended to cases where the involved vector fields have a limited degree of smoothness [3]. This extension relies on a “generalized differentiation theory”, developed by H. Sussmann, which possesses good “chain rule” and “open mapping theorem” properties.
- The generalized iterated Lie brackets and the related asymptotic formulas are being uti- lized to establish results on the controllability and (Hölder) regularity of the minimum time function of rough control systems with rough targets. Similarly, the solvabil- ity and regularity of viscosity solutions of eikonal PDEs structured on rough vector fields on rough domains subject to suitable Dirichlet boundary conditions are being obtained. In particular, results on the well-posedness and regularity of the so-called Carnot-Caratheodory distances in subreiemannian geometry are being established.
Partial integro-differential equations arising from the optimal control of jump diffusions
- Semiconcavity estimates with a general semiconcavity modulus for solutions of a class of Hamilton-Jacobi-Bellman partial integro-differential equations arising from problems of stochastic optimal control of jump diffusions [5].
Nonlinear systems of PDEs and Mean Field Games
- Results about existence of solutions for a class of elliptic systems arising from the mean field games theory of J.-M. Lasry and P.-L. Lions, which consist of N Hamilton-Jacobi- Bellman (abbr. HJB) PDEs coupled with N Kolmogorov-Fokker-Planck (abbr. KFP) PDEs [7].
- Rigorous (or mathematical) derivation of the mean field game (abbr. MFG) equations for a class of ergodic games with agents belonging to several homogeneous populations.
Letting the number of the individuals/players of each population go to infinity, under suitable assumptions on each population of players (similar to those in the case of a single homogeneous population), the MFG system of PDEs, consisting of as many pairs of a HJB equation and a KFP equation as the number of populations of agents, has been derived. (The case of finite horizon games is still open, with only some partial results claimed by Lasry and Lions.)
- Study of mean field game models arising from stochastic games whose dynamics are de- scribed by stochastic processes with jumps which are solutions to quite general stochas- tic differential equations. Hence, study of their infinitesimal generators, which turn out to be integro-differential operators, their invariant measures, regularity properties of their solutions etc.
Stability estimates for solutions and eigenfunctions of second-order elliptic boundary value problems subject to domain perturbation
- Extension of the notion of “gap” or “opening” or “aperture” between (possibly un- bounded) linear operators acing on normed spaces (a notion introduced in the 40s by M G. Krein and coworkers, which should be thought of as a kind of a sine of an angle between the operators graphs, and which is meant to serve as replacement of the norm
in the case of unbounded operators) to linear operators arising from second-order el- liptic boundary value problems in domains of the n-dimensional Euclidean space. This extension is not obvious; one has to overcome the difficulty that operators are defined and take values on different normed spaces depending on the domain, cf. [10,11].
- Application of the above-mentioned notion of “gap” to spectral stability estimates for second-order elliptic Dirichlet boundary value problems, cf. [9, 11].
- Much more general estimates, in Lp-norms (for p-s taking values in a subinterval of [1, ∞] having 2 as an inner point) for the variation of solutions, eigenfunctions of second-order elliptic boundary value problems, and their gradients, as a result of do- main perturbation, in terms of suitable distances (so-called “atlas”) distances between domains. As a corollary, estimates in terms of more “classical” distance between do- mains (such as the Hausdorff distance, or the Lebesgue measure of the symmetric difference of domains) have also been derived, cf. [8].
Conferences, Congresses, Workshops, Schools
Invited talks
- High order controllabilty results for nonsmooth vector fieldsŤ, University of Bath, UK, January 21, 2015.
- A set-valued iterated Lie bracket, Analysis and Geometry in Control Theory and its Applications (with a special tribute to Hélène Frankowska and Hector J. Sussmann), Istituto Nazionale di Alta Matematica (INDAM), Università di Roma “La Sapienza”, June 9-13, 2014.
- Ergodic MFG equations for several populations of agents as a ”continuum limit” of games with a finite but large number of agentsř, in Wokrshop on Mean field Games, Università degli Studi di Roma Tor Vergata, April 14 Ű 15, 2014.
- Generalized semiconcavity of the value function in (stochastic) optimal control, Dif- ferential Equations and Applications, Università degli Studi di Padova, February 7, 2014.
- Stability estimates for the deviation of solutions and eigenfunctions of second-order el- liptic Dirichlet BVPs under domain perturbation, Minicourses in Mathematical Anal- ysis 2013, Universitá degli Studi di Padova, June 10-14, 2013.
- Domain perturbation problems and estimates for the eigenfunctions of Dirichlet second- order elliptic BVPs, Karl-Franzenes-Universität, Graz, Austria, February 13, 2013.
- A brief introduction to the theory of mean field games, Karl-Franzenes-Universität, Graz, Austria, February 13, 2013.
- Some aspects of mean field games, RWTH-Aachen, Germany, July 25, 2012.
- Extension of the notion of a gap to linear differential operators defined on differ- ent domains and spectral stability estimates, in the 8th International Congress of the ISAAC (International Society for Analysis, its Applications, and Computation), Peo- ple’s Friendship University of Russia, Moscow Russia, August 22-27, 2011.
- Stability bounds for the spectral subspaces of the Dirichlet Laplacian under domain perturbation, in Minicourses in Mathematical Analysis 2011, Università degli Studi di Padova, Italy, June 13-17, 2011.
- Extension of the notion of a gap to linear differential operators defined on different domains and spectral stability estimates, in Operators in Morrey Type Space and Ap- plications - OMTSA 2011, Ahi Evran University, Turkey, May 20-27, 2011.
Participation
- “Control Day 2013: Control Theory at the University of Padova”, Università degli Studi di Padova, Padua, September,20, 2013
- “Mean Field Games and Related Topics - 2”, Università degli Studi di Padova, Padua, September 4-6, 2013
- SADCO Summer school and workshop, ”New Trends in Optimal Control”, 3-7 Septem- ber 2012, Ravello, Italy.
- “Differential Equations and Dynamical Systems, Serapo (Latina) Italy, June 11-15, 2012.
- “Geometric PDEs and applications”, Padova, Italy, April 19-20, 2012.
- “Analysis and Numerics of PDEs - In memory of Enrico Magenes”, Pavia, Italy, Novem- ber 2-4, 2011.
- “Eighth International ISDG (The International Society of Dynamic Games) Workshop”, Padua, Italy, July 21-23, 2011.
Organizing activity
- Member of the organizing committee of the workshop and school “Mini-courses in Mathematical Analysis”, which is held each year, usually on June, at “Università degli Studi di Padova” in Padova, Italy.
- Member of the organizing committee of a cycle of seminars entitled “Differential equa- tions and applications” which is held regularly at the Department of Mathemamtics at
“Università degli Studi di Padova”, Padova, Italy (roughly each month, an internation- ally renown expert in differential equations and their applications is invited to deliver a talk).
- Engaged in representation activities on behalf of post docs at the Department of Math- ematics at “Università degli Studi di Padova”, Padova, Italy.
Editorial work
- Served as referee for the journals “Indiana University Journal of Mathematics” and
“Dynamic Games and Applications”.
Funded research projects [as participant]
Period Title and scientific responsible or coor- dinator of the research project
Institution and financier of the project
2012-2013 CaRiPaRo Excellency Project: “Non- linear Partial Differential Equations:
models, analysis, and control- theoretic problems”, Headed by Martino Bardi
Università degli Studi di Padova and Foundation CaRi- PaRo
2013 PRIN 2009 project: “Viscosity, metric, and control theoretic methods for non- linear partial differential equations”, coordinatore nazionale Italo Capuzzo Dolcetta
Università degli studi di Padova
2013 “Ricerca Scientifica fondi quota EX 60%” - 2013: “Equazioni differenziali nonlineari”, Coordinator: Pierpaolo So- ravia
Università degli Studi di Padova
2010-2012 PRIN 2008 project: “Proprietà e metodi geometrici nelle equazioni alle derivate parziali, disuguaglianze di Sobolev e convessità”, National coordi- nator: Andrea Cianchi
Università degli Studi di Padova
2012 “Ricerca Scientifica fondi quota EX 60%” - 2012: “Spazi funzionali, oper- atori differenziali e teoria del poten- ziale” Coordinator: Massimo Lanza de Cristoforis
Università degli Studi di Padova
2011 “Ricerca Scientifica fondi quota EX 60%” - 2011: “Spazi funzionali, oper- atori differenziali e teoria del poten- ziale” Coordinator: Massimo Lanza de Cristoforis
Università degli Studi di Padova
2010 “Ricerca Scientifica fondi quota EX 60´’
- 2010: “Spazi funzionali, operatori dif- ferenziali e teoria del potenziale” Coor- diantor: Massimo Lanza de Cristoforis
Università degli Studi di Padova
Miscellaneous
- Member of GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni, transl. from Italian: National Group for Mathematical Analysis, Probability and their Applications) 2008–Present.
- Graduated one year in advance with roughly 50 ECTS-credits more than required by the program.
- Attended the double of the courses required by the program of the Ph.D. school.
Continue to attend courses of the PHD school of Mathematics, Padova, Italy.
Honors & Awards
- Third Prize at Albanian Physics Olympiad6 - 2002.
- Participant and winner of an “Honorable Mention” at the International Physics Olympiad6- 2002, Bali, Indonesia.
Reference Contacts
Prof. Martino Bardi Prof. Victor I. Burenkov
Dipartimento di Matematica Cardiff University (Cardiff, UK) Università degli Studi di Padova L.N. Gumilyov Eurasian National Via Trieste 63 - 35131 University (Astana, Kazakhstan)
Padova (PD) Italy Vice-President of the International Society
Tel: +(39) 49 827 1468 [email protected]
Fax: +(39) 49 827 1428 e-mail: [email protected]
Prof. Massimo Lanza de Cristoforis Prof. Lawrence C. Evans Università degli Studi di Padova Department of Mathematics Via Trieste 63 - 35131 Berkeley, CA, 94720-3840
Padova (PD) Italy e.mail: [email protected]
Fax: +(39) 49 827 1428 e-mail: [email protected]
6An annual competition for secondary school students
Prof. Franco Rampazzo Prof. Pierre Cardaliaguet Università degli Studi di Padova Ceremade (UMR CNRS 7534) Via Trieste 63 - 35131 Université Paris-Dauphine
Padova (PD) Italy Place du Maréchal De Lattre De Tassigny
Tel: +(39) 49 827 1372 75775 PARIS CEDEX 16 - FRANCE
Fax: +(39) 49 827 1428 +(33) 1 44 05 46 01
e.mail: [email protected] [email protected] Prof. José Maria Arrieta Prof. Gerassimos Barbatis Departamento de Matemática Aplicada Department of Mathematics
Universidad Complutense de Madrid Univ. of Athens, Panepistimioupolis 28040 Madrid, España GR-157 84 - Athens, Greece
Tel: +(34) 91-3944232 Tel: +(30) 210-7276406
Fax: +(34) 91-3944102 Fax: +(30) 210-7276398
e-mail: [email protected] e-mail: [email protected]
