Stefano Luzzatto
Research Scientist - Abdus Salam International Centre for Theoretical Physics (ICTP)
My mathematics path started with a BA in Mathematics at the University of Warwick in 1987, followed by an MSc in “Philosophy of Mathematics Education” at Marlboro College, Vermont, USA in 1990 and a 1995 PhD in Dynamical Systems at the International School of Advanced Studies (SISSA), Trieste, Italy and the Instituto de Matematica Pura e Aplicada, IMPA, Rio de Janeiro, Brazil.
I then went on to be a postdoc at the Laboratoire de Probabilité ́ in Paris VI and at the University of Warwick from 1995 to 1999, where I also became a lecturer.
I was subsequently a lecturer at the University of Manchester Institute of Science and Technology, UMIST, in 2000 and a tenure-track lecturer at Imperial College London, UK where I was later promoted to Reader.
I took up my current position of Research Scientist at the Abdus Salam International Centre for Theoretical Physics, ICTP, Trieste, Italy in 2009.
I have held short-term visiting positions at the University of Uppsala, Sweden (1998), IMPA, Rio de Janeiro, Brazil (2000 & 2005), Centre de Physique Theorique (CPT), Marseille, France (2001), Universidade Federal da Bahia (UFBA), Salvador, Brazil (2002), University of Kyoto, Japan (2004), University of Barcelona, Spain (2005), Penn State University, USA (2006 & 2008), Chinese Academy of Sciences, Beijing and Soochow University, China (2007).
My research includes collaborations with researchers from 18 institutions in 9 countries and 4 continents, some of which are: IMPA, UFAL, UFBA, UFF, UFRJ (Brazil), Soochow University, Academy of Sciences (China), CPT Marseille (France), Kyoto (Japan), Porto (Portugal), Uppsala (Sweden), Exeter, Surrey, Warwick (UK), Georgia Tech, Houston, Penn State (USA), Maracaibo (Venezuela).
I have taught over 120 lectures and seminars around the world, supervised 50 post-doc/PhD/MSc/BSc/diploma students, organized 30 schools and conferences and taught 26 long and short-term courses.
What all of these hefty numbers add up to is a profound passion for my field of work and the untiring motivation to share them with others.
How would you define your field of study? What is your vision about it? Which are the topics you're most passionate about?
My field is popularly known as “Chaos Theory” and more technically as “Dynamical Systems” or “Smooth Ergodic Theory”. It starts from the basic observation, already well known to great classical mathematicians such as Laplace and Poincaré but brought to the general attention by Lorenz in the 1960’s, that in the world of differential equations and dynamical systems, the fact that “identical initial conditions have identical outcomes” does not imply that “almost identical initial conditions have similar outcomes”. Indeed, we know understand that many systems exhibit very “sensitive dependence on initial conditions” meaning that extremely small changes in the initial conditions can lead very quickly to major changes in the outcomes. This has completely undermined a lot of strategies for understanding such systems which rely for example on numerical methods which are always approximations and thus susceptible to small errors in initial conditions and thus large errors in outcomes!
A revolutionary approach was proposed in the 1960’s and 1970’s by Sinai who introduced the methods of statistical mechanics and the language of probability theory and suggested a “statistical” study of such systems. In the course of the following decades this has led to a huge amount of research and very deep results showing that many such “chaotic” systems are nevertheless well behaved and very stable from a statistical point of view. As an example one can think of flipping a coin where the outcome of each flip is unpredictable but in the long run the average number of heads and tails tends to be the same, or even the weather which is unpredictable in the short term but where, for example, the average temperatures tend to be very stable.
Most of my own research falls within the scope of a broad conjecture of Palis which can be informally stated as saying that “most systems are statistically well behaved”. It is a fascinating subject, both for its philosophical meaning and for the wide range of mathematical techniques involved, ranging from probability to topology to analysis and many other areas. Of particular interest are recent approaches which involve “rigorous computational methods” to obtain explicit and concrete numerical results.
How do you expect your IMM experience to be?
I am honoured to be the Scientific Coordinator of the International Mathematics Master. This programme represents for me the embodiment of the philosophy and mission of Abdus Salam, visionary founder of the Abdus Salam International Centre for Theoretical Physics (ICTP), in Trieste, Italy, where I have the privilege of working since 2009. Notwithstanding the huge worldwide changes to the accessibility of science brought by the internet and cheaper international travel, many students in many countries are still academically and intellectually isolated and unable to fulfil their potentials. I hope that this programme can start to address this issue and in time benefit many students and thus, indirectly, their home countries.
Setting up the first IMM programme in Pakistan has had its fair share of challenges but I have been rewarded by an amazing enthusiasm from many actual and potential lecturers and many talented students who have embraced the programme and are giving it all they can. Certainly it would not have been possible without the indefatigable commitment and hard work of Sarfraz Ahmad, Head of the Mathematics department at CUI Lahore, where the programme is hosted.
I look forward to seeing the programme develop and the first students grow mathematically and move on to ambitious and prestigious PhD programmes and eventually return to their countries to become teachers and researchers and role models for other students.
What is your teaching philosophy? What would you like to transmit to your students? How do you motivate them?
I have taught students at many different levels and I have found that in many cases the greatest motivation, for them as for me, is the sense of fascination with the subject.
Mathematics is a language without which it would be impossible to describe the mind-boggling landscape of different ideas which have been developed in mathematics over the centuries. Learning mathematics is really about gaining access to these amazing concepts.
I feel privileged, as a mathematician, to be able to glimpse even a very minuscule portion of this landscape and I would like the students to feel the same.
Do you have one of two favorite quotes you would like to share and/or a personal “motto”?
“There was nowhere to go but everywhere, so keep on rolling under the stars” - Jack Kerouac
Lectures
The Palis Conjecture, Resistencia Dinamica, PUC (Rio de Janeiro by webinar), 2020
Fractal Dimensions in Nature and Mathematics, Basic Notions Seminars, ICTP (Trieste), 2019
ODE & Dynamical Systems - playlist, Postgraduate Diploma Programme, ICTP (Trieste), 2019
Hyperbolicity and Physical Measures - lecture 1, Smooth and Homogeneous Dynamics, ICTS (Bangalore), 2019
Hyperbolicity and Physical Measures - lecture 2, Smooth and Homogeneous Dynamics, ICTS (Bangalore), 2019
Hyperbolicity and Physical Measures - lecture 3, Smooth and Homogeneous Dynamics, ICTS (Bangalore), 2019
Hyperbolicity and Physical Measures - lecture 4, Smooth and Homogeneous Dynamics, ICTS (Bangalore), 2019
Introduction to Ergodic Theory, Summer School in Dynamics, ICTP (Trieste), 2018
Young Towers and Sinai-Ruelle-Bowen measures for non-uniformly hyperbolic surface diffeomorphisms, School on Hyperbolic Dynamics, SNS (Pisa), 2017
Differential equations and dynamical systems - playlist, Diploma Programme in Mathematics, ICTP (Trieste), 2014
Ergodic theory - playlist, Diploma Programme in Mathematics, ICTP (Trieste), 2013
Geometry of Positive Lyapunov Exponents, UNAM (Cuernavaca), 2013
Selected Publications
S. Luzzatto, S. Tureli, K. War. Integrability of Continuous Bundles J. Reine Angew. Math. (Crelle's Journal) 752 (2019) 229-264 (ArXiv, Journal)
Alves, J.F.; Dias, C.L.; Luzzatto, S. Pinheiro, V. SRB measures for partially hyperbolic systems whose central direction is weakly expanding. J. Eur. Math. Soc. (JEMS), 19 (2017) no. 10, 2911-2946 (ArXiv, Journal)
S. Luzzatto, I. Melbourne. Statistical properties and decay of correlations for interval maps with critical points and singularities. Comm. Math. Phys. 320 (2013) 21-35 (pdf)
J.F. Alves, C. Dias, S. Luzzatto. Geometry of absolutely continuous expanding measures and the liftability problem Ann. Inst. H. Poincaré Anal. Non Linéaire. 30 (2013) 101-120. (pdf)
F Alves, J.M. Freitas, S. Luzzatto, S. Vaienti, From rates of mixing to recurrence times via large deviations. Adv. Math., 228 (2011), 1203-1236. (pdf)
Luzzatto, P. Pilarczyk. Finite resolution dynamics. Found. Comput. Math., Vol. 11, No. 2 (2011), 211-239 (pdf)
V. Araujo, S. Luzzatto, M. Viana. Invariant measures for interval maps with critical points and singularities. Adv. Math. pp. 1428-1444 (2009) (pdf)
M. Holland, S. Luzzatto. Stable manifolds under very weak hyperbolicity conditions. J. Differential Equations, 221, pp. 444-469 (2006) (pdf)
S. Luzzatto, L. Wang. Topological invariance of generic non-uniformly expanding one-dimensional maps. Math. Res. Lett., 13, pp. 343-357 (2006). (pdf)
J. F. Alves, S. Luzzatto, V. Pinheiro. Markov structures and decay of correlations for non-uniformly expanding maps. Ann. Inst. H. Poincaré Anal. Non Linéaire., 22 (2005), no. 6, pp. 817-839. (pdf)
S. Luzzatto, I. Melbourne, F. Paccaut. The Lorenz attractor is mixing. Comm. Math. Phys., 260 (2005), pp. 393-401. (pdf)
S. Luzzatto, M. Viana. Parameter exclusions in Hénon-like systems. Russian Math. Surveys. 58 (2003) pp. 1053-1092 (pdf)
H. Bruin, S. Luzzatto, S. van Strien. Rates of decay of correlations for one-dimensional dynamics, Ann. Sci. Ec. Norm. Super., 36 (2003), 621-646 (pdf)
S.Luzzatto, W. Tucker. Non-uniformly expanding dynamics in maps with criticalities and singularities. Publ. Math. IHES, 89 (1999), 179–226. (pdf)