Doctor of Philosophy (Ph.D.)

Pure Mathematics
University of Maryland at College Park
2000

Master of Arts (M.A.)

Pure Mathematics
University of Maryland at College Park
1998

Laurea

Mathematics
University of Cagliari (Italy)
1996

Doctor of Philosophy (Ph.D.)

Physics
University of Cagliari (Italy)
1994

Laurea

Physics
University of Cagliari (Italy)
1990

Mathematical Biology

Dynamical Systems/Fractals Theory

MATH243 (Dynamical Systems I, graduate course)

MATH164 (Intro to Numerical Analysis)

Provost funds, 2015

Funded proposal to buy a small High-Power Computational Cluster

New Faculty Award, Howard University, 2015

New Faculty Award, Howard University, 2014

Streams and graphs of Dynamical Systems

Streams and graphs of Dynamical Systems

While studying gradient dynamical systems (DSs), Morse introduced the idea of encoding the qualitative behavior of a DS into a graph. Smale later refined Morse's idea and extended it to Axiom-A diffeomorphisms on manifolds. In Smale's vision, nodes are indecomposable closed invariant subsets of the non-wandering set with a dense orbit and there is an edge from node N to node M if the unstable manifold of N intersects the stable manifold of M. Since then, the decomposition of the non-wandering set was studied in many other settings, while the edges component of Smale's construction has been often overlooked. In the same years, more sophisticated generalizations of the non-wandering set were elaborated first by Auslander in 60s, by Conley in 70s and later by Easton and other authors. In our language, each of these generalizations involves the introduction of a closed and transitive extension of the non-wandering relation, that is closed but not transitive. In the present article, we develop a theory that generalizes at the same time both these lines of research. We study the general properties of closed transitive relations ("streams") containing the space of orbits of a discrete- or continuous-time semi-flow and we argue that these relations play a central role in the qualitative study of DSs. All most studied concepts of recurrence currently in literature can be defined in terms of our streams. Finally, we show how to associate to each stream a graph encoding its qualitative properties.

Graph and backward asymptotics of the tent map

Graph and backward asymptotics of the tent map

The tent map family is arguably the simplest 1-parametric family of maps with non-trivial dynamics and it is still an active subject of research. In recent works the second author, jointly with J. Yorke, studied the graph and backward limits of S-unimodal maps. In this article we generalize those results to tent-like unimodal maps. By tent-like here we mean maps that share fundamental properties that characterize tent maps, namely unimodal maps without wandering intervals nor attracting cycles and whose graph has a finite number of nodes.

Backward asymptotics in S-unimodal maps

Backward asymptotics in S-unimodal maps

While the forward trajectory of a point in a discrete dynamical system is always unique, in general a point can have infinitely many backward trajectories. The union of the limit points of all backward trajectories through x was called by M.~Hero the "special α-limit" (sα-limit for short) of x. In this article we show that there is a hierarchy of sα-limits of points under iterations of a S-unimodal map: the size of the sα-limit of a point increases monotonically as the point gets closer and closer to the attractor. The sα-limit of any point of the attractor is the whole non-wandering set. This hierarchy reflects the structure of the graph of a S-unimodal map recently introduced jointly by Jim Yorke and the present author.

On the Numerical Solution of the Far Field Refractor Problem

On the Numerical Solution of the Far Field Refractor Problem

The far field refractor problem with a discrete target is solved with a numerical scheme that uses and simplify ideas from Caffarelli, Kochengin and Oliker. A numerical implementation is carried out and examples are shown.