Speaker

:

Prof. Nessim Sibony    

Subject Area

:

Mathematics

Venue

:

Lecture Hall - I, Dept of Mathematics

Time

:

4 PM

Date  

:

December 22, 2008 January 02, 2009(Every Monday,Wednesday and Friday)

Title

:

Comparitive description of Circuits in Classical and Quantum Computers &  Estimation Of Density Operators in finite level Quantum Systems

Abstract. We will discuss two aspects : dynamics of maps and dynamics of holomorphic foliations. Holomorphic dynamics in several variables has interractions with ergodic theory, dynamical systems, algebraic geometry and number theory. It is indeed a special chapter in dynamics, but the
tools from complex analysis and geometry permit to answer questions which are out of reach in real dynamics. As a sample, we will discuss dynamics of holomorphic endomorphisms of complex projective spaces, polynomial automorphisms of Euclidian spaces and more generally dynamics of
meromorphic maps on compact Kahler manifolds. We will consruct in these cases Green currents, compute the entropy of the maps, introduce the measure of maximal entropy and develop some precise ergodic theory (exponential decay of correlation, central limit theorem). We will discuss
equidistribution problems. Let f be a meromorphic map of a compact Kahler manifold M. Let S be an analytic set of codimension p in M. The problem is to describe the distribution of (normalized) preimages of S under the iterates of f. Even the case of points is subtle. We will treat this
question for holomorphic endomorphisms of Pk and for polynomial automorphisms of Ck. We will introduce the theory of Super-potentials which provides a useful calculus on positive closed (p, p) currents for arbitrary p. It permits to obtain quantitative results on the speed of
equidistribution. The background is described in the notes with T.C Dinh: Dynamics in Several Complex Variables: endomorphisms of projective spaces and polynomial like mappings:ArXiv 08100811.v 

Dynamics of foliations: We will introduce foliations by Riemann Surfaces on Projective spaces. The main goal is to prove a unique ergodicity result for generic holomorphic foliations of P2. This is joint work with J.E Fornaess. The background is described in the recent survey by the authors
which appeared in the Journal of Geometric Analysis (April 2008).