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Dr Matteo Sommacal

Senior Lecturer

Department: Mathematics, Physics and Electrical Engineering

Dr Matteo Sommacal is Senior Lecturer in Applied Mathematics in the Department of Mathematics, Physics and Electrical Engineering at Northumbria University where he arrived in January 2012.

EE Matteo Sommacal Staffprofile 255Matteo Sommacal was born in Roma, Italy, on November 18, 1977. Currently, he is Senior Lecturer in Applied Mathematics in the Department of Mathematics, Physics and Electrical Engineering at Northumbria University, Newcastle upon Tyne, UK, where he arrived in January 2012. He earned a M.Sc. in Physics at the Università degli studi di Roma "La Sapienza", Italy (2002). Subsequently, he received a Ph.D. in Mathematical Physics at the International School for Advanced Studies in Trieste, Italy, with Professor Francesco Calogero as advisor (2005), discussing the thesis "The Transition from Regular to Irregular Motions, Explained as Travel on Riemann Surfaces".

After the Ph.D. studies, he got several academic positions as post-doctoral fellow and research scholar. He was "Chercheur post-doctorant" at the Laboratoire J.-L. Lions, Université Pierre et Marie Curie-Paris VI, France (November 2005 – October 2006). He then received a three-year post-doctoral fellowship ("assegno di ricerca") from the Department of Mathematics and Computer Science, Università degli Studi di Perugia, Italy (November 2006 – October 2009). After one year as post-doctoral fellow at the Department of Physics, Università degli Studi di Roma "La Sapienza", Italy (November 2009 – October 2010), he got an invitation as "Visiteur de longue durée" at the Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France (November 2010 – April 2011). He then was appointed a post-doctoral research scholarship at the Department of Mathematics, North Carolina State University, Raleigh, NC, USA (May 2011 – December 2011).

0191 227 4626

Qualifications

PhD

Research Themes and Scholarly Interests

My main research interests and relevant specialist knowledge are in Applied Mathematics, with a multidisciplinary approach, namely crossing mathematical competencies coming from Mathematical-Physics, Analysis and Numerics, and aiming at rigorously solving problems and assisting experimental observations in Physics, Biology and Chemistry (but also in Archaeology and Social Sciences). I have obtained several results belonging to the field of Integrable Systems and Dynamical Systems, in particular concerning the onset of irregular/chaotic motions for dynamical systems explained in terms of "travels" on Riemann surfaces. I am currently involved in a research program aimed at studying the magnetization dynamics in ferromagnetic materials at the nano-scale. I also obtained results in the study of the Kirchhoff elastic rod as a model for polymeric chains, as well as nonlinear PDEs in multidimension, boundary value problems and tame transformations.

Towards a theory of chaos explained as travel on Riemann surfaces
Recently, a mechanism to explain the onset of irregular (chaotic) motions in a dynamical system, in terms of the singularity structure of its solutions, was introduced. In particular, an example was provided of how the complex dynamics of a certain model, interpretable as a 3-body problem in the (complex) plane, can be understood via a detailed analysis of its associated Riemann surface.

Thanks to this geometric description an explicit formula for the period of the orbits can be derived, which is shown to depend on the initial data and the continued fraction expansion of a simple ratio of the coupling constants of the problem. For rational values of this ratio and generic values of the initial data, all orbits are periodic and the system is isochronous. For irrational values of the ratio, there exist periodic and quasi-periodic orbits for different initial data. Moreover, the dependence of the period on the initial data shows a rich behavior and initial data can always be found such that the period is arbitrarily high. Previous numerical works and recent more analytical publications along the same line suggest the generality of such an approach and include problems whose solution is obtained by inverting hyperelliptic integrals.

This theory promises to provide a new mean to explain isochronicity and the onset of disordered behaviours in dynamical systems, and it has been recognized as an important tool for understanding Chaos and referenced on the Wolfram Mathworld Encyclopedia (http://mathworld.wolfram.com/Chaos.html).

Collaborations: Jean-Pierre Françoise (Université "Pierre et Marie Curie"-Paris VI, Laboratoire J. L. Lions, Paris, France); Francesco Calogero and Paolo Maria Santini (Università degli Studi di Roma "La Sapienza", Roma, Italy); David Gomez-Ullate (Universidad Complutense de Madrid, Spain); François Leyvraz (Universidad de los Andes, Bogotà, Columbia); Piotr Grinevich (Landau Institut, Moscow, Russia).

The Kirchhoff elastic rod as a model for polymeric chains
The theory of elastic deformations in thin rods is very old, dating up to Euler and Kirchhoff, and has seen renewed interest since it supplies one of the most convenient frameworks to study the dynamics and the folding of polymeric chains (more in general, all phenomena taking place on time and space scales that cannot be covered by numerical simulations at the electronic and atomic level). Several new analytical results were obtained for Kirchhoff thin elastic rods, generalising previous results to rods with non-circular cross section and arc-length dependent elastic properties. It was shown that circular helical structures at equilibrium can be obtained only for (anisotropic) elastic rods possessing specific characteristics and that a connection can be established between microscopic characteristics (force constants, moments of inertia of the repeated units) and macroscopic characteristics (the whole structure).

Moreover, the stability of helical configurations was studied, showing explicitly, in several cases of applicative interest, that local minima of the deformation energy exist. Other results were obtained for the dynamical Kirchhoff equations, including the existence of travelling solutions with arbitrary curvature and torsion that can be interpreted as conformational solitons.

Collaborations: Mark J. Ablowitz (University of Colorado at Boulder, USA); Vincenzo Barone (Scuola Normale Superiore, Pisa, Italy); Silvana De Lillo (Università degli Studi di Perugia, Italy).

Key Publications

F. Calogero and M. Sommacal, "Periodic solutions of a system of complex ODEs. II. Higher periods", J. Nonlinear Math. Phys. 9, 1-33 (2002).

Calogero, J.-P. Françoise and M. Sommacal, "Periodic solutions of a many-rotator problem in the plane. II. Analysis of various motions", J. Nonlinear Math. Phys. 10, 157-214 (2003).

D. Gomez-Ullate, A. N. W. Hone and M. Sommacal, "New Many-Body Problems in the Plane with Periodic Solutions", New J. of Phys. 6, 24 (2004).

D. Gomez-Ullate and M. Sommacal, "Periods of the Goldfish Many-Body Problem", J. Nonlinear Math. Phys., Volume 12, Supplement 1, 351-362 (2005).

F. Calogero, D. Gomez-Ullate, P. M. Santini and M. Sommacal, "The Transition from Regular to Irregular Motions, Explained as Travel on Riemann Surfaces", J. Physics A 38, 8873-8896 (2005).

F. Calogero, M. Sommacal, "Solvable nonlinear evolution PDEs in multidimensional space", SIGMA 2 (2006), 088, 17 pages; nlin.SI/0612019.

To view my Northumbria Research Link page click here

My Google Scholar page can be found here


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