The subject of enumerative geometry goes back at least to the middle
of the 19th centuary. It deals with questions of enumerating geometric
objects, e.g.
(a) how many lines pass through 2 points or
through 1 point and 2 lines in 3-space?
(b) how many conics in 2-space are tangent to k lines and
pass through 5-k points?

There has been an explosition of activity in this field over the past
twenty years, following the development of Gromov-Witten invariants in
sympletic topology and string theory. The idea of counting parameterizations
of curves in order to count curves themselves has led to solutions of
whole sets of long-standing classical problems. At the same time, string
theory has generated a multitude of predictions for the structure of
GW-invariants, as well as for the behavior of certain natural families
of Laplacians. It has in particular suggested that there is a diality
between certain symplectic and complex manifolds and that in some cases
GW-invariants see some geometric objects, that are yet to be fully
discovered mathematically.

In this talk I hope to give an indication of what enumerative geometry
is about and of the shift in the paradigm that has occured over the past
two decades.

In this talk I will give a survey of Monte Carlo methods for solving linear systems, and present results of recent research on a new estimator. I will also present numerical results comparing a sequential Monte Carlo method with the standard deterministic solvers. I will then describe quasi-Monte Carlo methods, and present results of current research made on a new modification of the Halton sequence.

Broadband, coherent array imaging can be made quite robust in random media by using interferometric
algorithms that tend to minimize the effect of random inhomogeneities. I will introduce and describe these algorithms in detail, and I will
show the results of several numerical simulations that assess their effectiveness.

Consider a rational map φ on the projective line, from which we form a (discrete) dynamical system via iteration, and let K be a number field. A fundamental question in arithmetic dynamics is the uniform boundedness conjecture of Morton and Silverman, which states that there is a constant independent of φ (depending only on its degree) giving an upper bound for the number of K-rational preperiodic points of φ. This is a deep conjecture, and no specific case of it is known. I have proposed a specific version of the conjecture: that in the case of a degree-2 rational map and K = Q, the upper bound is 12.

In this talk, which assumes no previous knowledge of arithmetic dynamics, I will describe why this question is so difficult and sketch work that has been done to date, including giving justification for my refined uniform boundedness conjecture. The techniques used so far, which have clear limitations, involve constructing algebraic curves parameterizing maps $\phi$ together with points of period n for varying n (so-called dynamic modular curves).

The problem of classification of smooth dynamical systems had been a reach source of motivation for beautiful constructions and conjectures for
several decades. The history of these conjectures, as well as the current "state of the art" of the subject will be described. Some of the notions to
be covered are: structural stability, Hadamard-Perron Theorem, invariant manifolds, Morse-Smale systems, Smale horseshoe, Kupka-Smale systems, homoclinic picture, hyperbolic sets, Anosov diffeomorphisms, Axiom A diffeomorphisms, Spectral Decomposition Theorem, homoclinic tangencies, Newhouse phenomena, heterodimensional cycles, Palis' Conjectures.

The purpose of the talk is to provide a very general description; no proofs or technical details will be given.