# Funding Opportunities

## 2002 McCoy Award Recipients

**Dr. Roberto Colella**

Professor of Physics

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).

### Research

During the last ten years my research was focused 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 and the Phase Problem. Applications to Quasicrystals and to Resonant Scattering

**Abstract**

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**

Professor of Mathematics

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".

### Research

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.

- In the first period of his career (1972-1980) A. Eremenko worked in Lvov, in close contact with his advisor, A. A. Goldberg. The significant results of this period are: solution of the problem of D. Drasin and A. Weitsman (from Purdue University) about the set of asymptotic values of a meromorphic function of finite order and a series of papers (some joint with Goldberg) on asymptotic curves of entire functions. At the same time, by Goldberg's advise, A. Eremenko began to study analytic theory of differential equations. This led to an important result: the complete classification of first order algebraic differential equations that admit solutions meromorphic in the whole complex plane. This classification implies, in particular, that the order of growth of such solution can be only a half or a third of an integer, the result which completes the long line of investigation originated by Polya and Malmquist in 1920.
- In 1980, having difficulties with finding a job in Lvov, A. Eremenko moved to Kharkov. His first papers and his advisor's connections created him a sufficiently strong reputation to secure a permanent research position at the Institute of Low Temperature Physics and Engineering in Kharkov, one of the very best positions then available for a mathematician in Ukraine. This was a pure research position, so to maintain contact with graduate students, Eremenko taught part time at Kharkov University.
- His main collaborators of Kharkov period were younger colleagues, M. Sodin and M. Lyubich.
- In 1981 Eremenko joined Lyubich, then a graduate student, in his research on holomorphic dynamics. This old area was almost forgotten since 1920-s, but since 1982 it has seen explosive growth. It suddenly became one of the most popular areas of mathematics, due in part to computer graphics which brought wide public attention to the exciting images of the main objects of this theory, Julia and Mandelbrot sets. Eremenko and Lyubich singled out a class of transcendental entire functions, which they called Speiser class, with "good" dynamical properties and showed that entire functions which do not belong to this class may exhibit "pathological" behavior. This work was very influential: it is cited until now in almost every paper on dynamics of entire functions.
- The joint work of Eremenko and Sodin in 1980-s was centered at the new potential-theoretic method in value distribution theory of meromorphic functions and holomorphic curves. One of their principal results was an extension of the Second Main Theorem of value distribution theory to holomorphic curves in projective spaces and non-linear divisors. This result was conjectured by B. Shiffman in 1978. A lot of deep research was made in this area since then, but the ultimate form of the Second Main Theorem for holomorphic curves is still unknown, and the result of Eremenko and Sodin remains one of the top achievements in the subject.
- In 1980-s Eremenko and Sodin made the first major advance in the Littlewood's conjecture of 1959 on the mean spherical derivative of a polynomials and confirmed the main consequence of this conjecture about the distribution of values of entire functions. Building on this work, Lewis and Wu proved the Littlewood conjecture in 1989.
- In 1990 Eremenko moved to the US and started to work with John Lewis at the University of Kentucky. Together with Lewis they extended the potential-theoretic method of Eremenko and Sodin to higher dimensional analog of meromorphic functions, quasiregular maps. This led to a significant progress in the theory of quasiregular maps (Lewis, Rickman, Holopainen, Bonk, Heinonen and others).
- In 1991 Eremenko moved to Purdue University, where he became a full professor in 1993. In 1992 he solved two long-standing problems of value distribution theory, the Arakelian Conjecture of 1966 and the Small Ramification Problem which goes back to F. Nevanlinna (1929). Several outstanding mathematicians worked on these problems before, including D. Drasin and A. Weitsman at Purdue University whose contribution in 1970-80-s was very important.
- In 1995 Eremenko jointly with the German mathematician Bergweiler proved a general result about asymptotic values of meromorphic functions, which permitted them to confirm a conjecture of Hayman about distribution of values of derivatives of meromorphic functions. Their proof was based on an unexpected application of the ideas from holomorphic dynamics. This theorem became an indispensable tool in the study of value distribution of derivatives.
- In 1995 Eremenko found a counterexample to a conjecture of Henri Cartan of 1928 in complex hyperbolic geometry, about holomorphic maps from the unit disc to projective spaces of arbitrary dimension. The conjecture was known to be true in dimensions one and two, and Eremenko showed that it fails in all dimensions greater than two. Then he found a way to modify the conjecture preserving its main contents and proved this modified conjecture in dimension three. The modified conjecture remains open in higher dimensions.
- In the joint work of Bonk and Eremenko (1999-2000) the authors developed new geometric methods in the theory of meromorphic functions. This work brought Eremenko McCoy Award, and it is described in more detail in the Abstract of his lecture.
- In a series of papers (2000-2001) Eremenko and Gabrielov introduced the Wronski map from a Grassmann variety to a projective space and investigated its properties. This allowed them to solve an important problem of B. and M. Shapiro on rational functions with real critical points.
- Explicit computation of degree of the real Wronski map was qualified by the specialists as an outstanding contribution to enumerative real algebraic geometry and found applications to the pole placement problem in Control theory.
- In summer 2002, Bergweiler, Eremenko and Langley completed a long line of development in the theory of real entire functions by proving a 90 years old conjecture of A. Wiman on real entire functions with real roots.

### Meromorphic Functions and Intrinsic Geometry of Surfaces

**Abstract**

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.