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Dr. Roberto Colella was born in Milan (Italy) in 1935. He received his academic education at the University of Milan, where he obtained his doctoral degree in 1958. In 1961, he joined the staff of Euratom Nuclear Research Center at Ispra (Italy) as a Research Scientist in the Solid State Division, with Dr. A. Merlini. In 1967 he came to the U.S. as a postdoctoral research associate at Cornell University, Department of Materials Science and Engineering, with Prof. B. W. Batterman, where he stayed until 1970. He joined the Physics Faculty of Purdue University on September 1, 1971, as an assistant professor. He was tenured and promoted to associate professor on July 1, 1975, and became a full professor on July 1, 1977.
During the academic year 1991-92 Dr. Colella was Visiting Professor at the University of Paris-Sud (Laboratoire de Physique des Solides-Orsay, France), and at the University of Paris VI and VII, Place Jussieu, Paris (France).
During the last ten years my research was focussed on the properties of quasicrystals, and, more recently, on resonant scattering. The problem explored in the case of quasicrystals was: is there a center of inversion? Are quasicrystals centrosymmetric ? The answer is: no, but the deviations from centrosymmetry are very small. The problem is treated in papers 84, 93, 98, 99 and 101 (papers 93 and 101 present the most exhaustive discussion of this issue). The method used to test for centrosymmetry is based on the idea of determining phases of x-ray reflections using multibeam diffraction, a technique we have developed at Purdue (papers 17 and 67).
Another research project was the study of diffuse scattering by a non-periodic structure such as a quasicrystal. The sound velocity in a quasicrystal is the same in all directions, and in a periodic medium this condition leads to isotropic diffuse scattering along certain planes in reciprocal space. Instead, we found oval iso-intensity contours, the signature of particular excitations called "phasons", typical of a quasicrystal. (The paper has just been submitted to Phys. Rev.). On a different line of research, we have performed absolute measurements of x-ray reflections, the first ones ever performed on a quasicrystal. From such measurements it is possible to go back to the actual atomic locations, and verify how good (or bad) are the models currently used for quasicrystals. It is an attempt to answer the very first question raised soon after the discovery of quasicrystals in 1984: "Where are the atoms ?"
More recently my interests have shifted on the properties of resonant scattering. A preliminary study, based on the space-group forbidden (600) reflection in germanium, has appeared recently as a Rapid Communication in Phys. Rev. (paper 106). The experimental results presented in paper 106 have been the subject of a theoretical analysis performed by a group lead by Prof. G.A. Sawatzky, of the University of Groningen, in the Netherlands, (I.S. Elfimov et al., Phys. Rev. Letters, Vol. 88, 7 January 2002, 015504-1). At the present time I am collaborating with a group at Cornell, trying to perform a 3-beam experiment on LaMnO3, a manganate with colossal magneto-resistance, interesting as a spintronic device, namely, source of spin polarized electrons to be used for information storage and transport. The 3-beam experiment we are trying to perform should be able to resolve an ambiguity in the way the lobes of d-electrons are oriented in the crystal. There are two possible orientations, leading to the same intensities of certain forbidden reflections. The two possible orientations, however, generate diffracted beams with different phases. Here is a case in which a phase determination can be applied to resolve a controversy in an interesting physical problem. The experiment is in progress, and we expect to get some results soon.
Multiple Bragg Scattering, a situation in which two or more Bragg reflections are excited at the same time, is a source of phase information. Applications to quasicrystals will be presented. The general problem of centrosymmetry, or lack of it, will be discussed, along with experimental results. Resonant scattering is a new technique used to get information about "orbital ordering". The crystal potential gives rise to preferred orientations of aspherical degenerate orbitals, responsible for chemical bonding. This is commonly referred to as "orbital ordering". One of the effects of orbital ordering is to excite forbidden reflections when the energy of the x-rays corresponds to an absorption edge. The phases of these forbidden reflections may be very useful for a complete and accurate description of orbital ordering. Applications will be shown for Ge and LaMnO3.
Alex Eremenko grew up in Ukraine (former Soviet Union). He received his Master degree in Mathematics from Lvov University in 1976, and PhD from Rostov-on-Don University (Russia) in 1979. There was an additional scientific degree in the Soviet Union, the Doctor of Sciences, which A. Eremenko received from the Novosibirsk Institute of Mathematics (Russia) in 1987.
Since 1980 he worked as a researcher at the Institute of Low Temperature Physics and Engineering of Ukrainian Academy of Science in Kharkov and taught part time at Kharkov University.
In 1990 A. Eremenko had a visiting position at the University of Kentucky, and in 1991 at Purdue University. In 1992 he decided to move to the United States and accepted an offer of a permanent position from Purdue University. Since then he has been Professor of Mathematics here.
Since 1992, A. Eremenko had held visiting positions at the Imperial College (London), University of Paris-12, Technion (Haifa, Israel) and Carl Albrecht University (Kiel, Germany).
His research has been supported by NSF, US-Israel Binational Science Foundation, Lady Davis Foundation (Israel) and Alexander von Humboldt Foundation (Germany).
In 2001 he was awarded Humboldt Prize (Germany), and in 2002 was an invited speaker at the International Congress of Mathematicians (Beijing).
A. Eremenko published approximately 100 scientific papers. His scientific interests are in the areas of the theory of analytic functions, differential equations, potential theory, holomorphic dynamics, geometry of surfaces, real algebraic geometry and control theory.
He serves on the editorial board of the journal "Computational Methods and Function Theory".
A. Eremenko started his research in Mathematics as an undergraduate of Lvov university in 1972. His style in research formed under the influence of A.A. Goldberg. In the later years he was also much influenced by his collaborators, especially Mario Bonk, Andrei Gabrielov, Walter Hayman, Misha Lyubich and Misha Sodin. Eremenko is a "problem-solver". His main contribution consists of solution of several hard problems, which in some cases resisted the efforts of mathematicians for many years. Most of his work is related to analytic functions of one complex variable and their applications, but he also wrote on other subjects, such as ordinary differential equations, potential theory, holomorphic dynamics, real algebraic geometry, control theory and statistics.
Here is a list of his most important contributions, roughly in chronological order.
Meromorphic functions of one complex variable constitute an important class which contains most of the functions encountered in applications of mathematics to physics and engineering. For example, it contains rational, exponential, trigonometric, elliptic and gamma functions, as well as the so-called special functions of mathematical physics.
Meromorphic functions can be considered as mappings from regions in the plane to a sphere of radius one, called the Riemann sphere. Among all such mappings they are characterized by the geometric property that the infinitesimal length distortion is the same for all directions at any given point. Mappings with such property are called conformal. Conformality provides the basis of many applications, for example, in the fluid dynamics.
A new universal property of conformal mappings from the plane to the sphere was recently discovered in joint work of Mario Bonk (University of Michigan) and Alex Eremenko (Purdue University). It is related to the existence of the inverse branches of such mappings. An inverse branch of a meromorphic function f is a continuous function g in a region on the Riemann sphere such that the composition of f and g is the identity map of this region. The theorem of Bonk and Eremenko establishes the best possible lower bound for the radius of a disc on the Riemann sphere in which an inverse branch exists. This estimate is given by an absolute constant which is approximately equal to 1.255 radian.
The function for which the lower bound is achieved turns out to be the Weierstrass elliptic P-function corresponding to a hexagonal lattice. It maps conformally an equilateral triangle in the plane onto an equilateral triangle on the sphere whose angles are 120 degrees each. The Reflection Principle implies that this function has hexagonal symmetry. There are several unsolved extremal problems in geometric function theory where the expected extremal function has hexagonal symmetry. Bonk--Eremenko theorem is apparently the first rigorously proved result of this type.
The result is obtained by a geometric method, based on the investigation of Gaussian curvature of surfaces, which provides an interesting link between analysis and geometry. The geometric construction developed by Bonk and Eremenko has already found some other applications in geometry of surfaces and in the theory of dynamical systems.
The geometric optimization problem involved is somewhat similar to covering and packing problems which are important in many applications of mathematics, such as the problem on the maximal number of disjoint discs of equal radii on the sphere, or the dual problem of covering the sphere by a minimal number of discs of prescribed equal radii.