Quantum resonances describe metastable states created by phenomena such as tunnelling, radiation, or trapping of classical orbits. Mathematically they are elegantly defined as poles of meromorphically continued operators such as the resolvent or the scattering matrix: the real part of the pole gives the rest energy or frequency, and the imaginary part, the rate of decay. With that interpretation they appear in expansions of linear and non-linear waves. And they can be found in other branches of mathematics and science: as poles of Eisenstein series and zeta functions in geometric analysis, scattering poles in acoustical and electromagnetic scattering, Ruelle resonances in dynamical systems, and quasinormal modes in the theory of black holes. In my talk I will present some basic concepts and illustrate recent mathematical advances with numerical and experimental examples.

Self assembly is the idea of creating a system whose component parts spontaneously assemble into a structure of interest. In this talk I will outline our research program aimed at creating self-assembled structures out of very small spheres, that bind to each other on sticking. The talk will focus on

(i) some fundamental mathematical questions in finite sphere packings (e.g. how do the number of rigid packings grow with N, the number of spheres);

(ii) algorithms for self assembly (e.g. suppose the spheres are not identical, so that every sphere does not stick to every other; how to design the system to promote particular structures);

(iii) physical questions (e.g. what is the probability that a given packing with N particles forms for a system of colloidal nanospheres); and

(iv) comparisons with experiments on colloidal nanospheres.

Many-body scattering problem is solved asymptotically when the size of the particles tends to zero and the number of the particles tends to infinity.
A method is given for calculation of the number of small particles and their boundary impedances such that embedding of these particles in a bounded domain, filled with known material, results in creating a new material with a desired refraction coefficient.
iThe new material may be created so that it has negative refraction, that is, the group velocity in this material is directed opposite to the phase velocity.
Another possible application consists of creating the new material with some desired wave-focusing properties. For example, one can create a new material which scatters plane wave mostly in a fixed given solid angle. In this application it is assumed that the incident plane wave has a fixed frequency and a fixed incident direction.
An inverse scattering problem with scattering data given at a fixed wave number and at a fixed incident direction is formulated and solved. Acoustic and electromagnetic (EM) wave scattering problems are discussed.

A.G.Ramm's vita, list of publications and some papers can be printed from the Internet address http://www.math.ksu.edu/~ramm