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Faculty Research

Scott Ahlgren
Ahlgren completed his Ph.D. in 1996 in the field of Diophantine equations under the direction of Wolfgang Schmidt. Much of his recent work has focused on the theory of modular forms to problems in number theory. For example, he has used this theory to answer several long-standing open problems on the arithmetic of the ordinary partition function. He has written several papers on the arithmetic properties of the Fourier coefficients of modular forms of half-integral weight; here there are applications to combinatorics, to the theory of elliptic curves, and to the study of the critical values of L-functions attached to modular forms. Ahlgren has written on a wide range of topics; he has authored or co-authored more than 35 research papers in various areas of number theory.

Bruce Berndt
Berndt received his Ph.D. in 1966 from the University of Wisconsin and spent a postdoctoral year at the University of Glasgow, in Scotland, before coming to the University of Illinois. He is an analytic number theorist with strong interests in several related areas of classical analysis. His primary interests are in theta functions, $q$-series, partitions, continued fractions, Eisenstein series, Dirichlet series, and character sums. Since early 1974, almost all of his research has been devoted to proving the claims left without proof by the famous Indian mathematician Ramanujan in his three notebooks and in his ``lost notebook.'' The three notebooks contain approximately 3300 results. With the help of several other mathematicians, he completed his work on the notebooks in 1998. An account of Berndt's work can be found in his five books, "Ramanujan's Notebooks, Parts I-V," published by Springer-Verlag in the years 1985, 1989, 1991, 1994, and 1998.

While studying Ramanujan's work over several years, it was natural for Berndt to develop a strong interest in Ramanujan as a human being as well as in the southeast Indian Tamil culture from which Ramanujan emerged. This interest led him and Robert A. Rankin to write "Ramanujan: Letters and Commentary" and "Ramanujan: Essays and Surveys", both jointly published by the American and London Mathematical Societies.

Since the mid 1990s, Berndt has been attempting to find proofs for many of the claims left by Ramanujan in his lost notebook, which was written in the last year of Ramanujan's life and contains approximately 650 assertions without proofs. Berndt and George Andrews, who found the lost notebook in 1976, are publishing volumes on the lost notebook analogous to those prepared by Berndt on the three earlier notebooks. The first two volumes appeared in 2005 and 2008.

Twenty-five students have completed doctoral theses under Berndt's direction. Currently, he advises about a half-dozen doctoral students. Most are focusing on material in the lost notebook or on research inspired by Ramanujan.

Florin P. Boca
Boca completed his undergraduate studies at the University of Bucharest (diploma 1986) and received his Ph.D. at UCLA (1993) under the supervision of Sorin Popa. He held positions at the Institute of Mathematics of the Romanian Academy (researcher since 1988) and University of Toronto (postdoctoral fellow 1993-1995), and was a EPSRC advanced research fellow in the UK between 1995-1997 (University of Wales Swansea) and 1998-2001 (Cardiff University).

Boca's interests lie in the areas of operator algebras, number theory, and ergodic theory. His research on some problems originating from operator algebras (the structure of non-commutative tori, subalgebras of rotation algebras, Bost-Connes systems and Araki-Woods factors) has substantial connections with number theory. After 1999 he became interested in the statistical properties of Farey fractions, fractional parts of polynomials, quadratic rationals, and in applications of Number Theory methods to the study of the angular distribution of "fat" lattice points which arise, for instance, in the model of the periodic Lorentz gas. His current interests range over problems in operator algebras, number theory, and ergodic theory.

Harold G. Diamond (Emeritus)
Diamond received his Ph.D. in 1965 from Stanford University, under the supervision of P. J. Cohen, and has been at Illinois since 1967, with visiting appointments at several American and European universities. He became an emeritus faculty member in 2002. His main area of work is multiplicative number theory, particularly elementary proofs of the prime number theorem, the theory of Beurling generalized numbers, and sieve theory. He is coauthor of a Carus monograph on algebraic number theory (with H. Pollard), a textbook on analytic number theory (with P. T. Bateman), and a research monograph on sieve theory (with H. Halberstam and W. F. Galway). Other interests include harmonic analysis and tauberian theorems, numerical computation, and mathematical problems.

He has served as an editor of the AMS Transactions and the Problem Section of the MAA Monthly. Eleven students completed Ph.D.'s under Diamond's direction.

Iwan Duursma
Duursma received master's degrees in aerospace engineering at Delft University and in mathematics at the University of Amsterdam. He received his Ph.D. in 1993 under the direction of Jack van Lint and Ruud Pellikaan at Eindhoven University, all in the Netherlands. Part of his thesis work was the formulation and proof of the Feng-Rao algorithm for the decoding of a general geometric Goppa code. Among his current interests in coding theory are properties of codes over rings, in particular self-dual codes over p-adic rings; weight distributions for asymptotically good codes; and relations with zeta functions. In cryptography his interests include most of the algebraically formulated protocols and, in particular, those that use number theory (such as RSA) or elliptic curves (such as elliptic curve digital signature schemes). His interests extend to other classes of curves and their Jacobians.

Kevin Ford
Ford has degrees from California State University, Chico (B.S., 1990) and the University of Illinois at Urbana-Champaign (Ph.D.,1994). He has held positions at the Institute for Advanced Study, the University of Texas and the University of South Carolina before returning to UIUC in Fall, 2001.

His research interests cover a variety of topics in elementary, analytic, combinatorial and probabilistic number theory. They include Waring-type problems, Weyl sums, the distribution of values of arithmetic functions, sieve theory, zeros of the Riemann zeta function, irregularities in the distribution of primes and almost-primes in arithmetic progressions (prime race type problems), covering systems of congruences, the distribution of divisors of integers, and configurations of prime numbers. Ideas and techniques from probability theory play an important role in many recent investigations. Some highlights of his research are large improvements in estimates for mean values of Weyl sums and Vinogradov's mean value theorem, determining accurately how many values Euler's function assumes in an interval [1, x], settling two conjectures of Sierpinski (one jointly with Sergei Konyagin) concerning the values taken by the sum of divisors function and Euler's totient function, showing that the sum-of-divisors function and Euler's function have infinite intersection (joint with Carl Pomerance and Florian Luca), establishing the best known quantitative zero-free region for the Riemann zeta function, determining accurately the number of distinct products in an N by N multiplication table (an old Erdos problem), and settling conjectures of Erdos, Graham and Selfridge on covering systems (joint with M. Filaseta, S. Konyagin, C. Pomerance and G. Yu).

A.J. Hildebrand
Hildebrand earned a Ph.D. in 1983 from the University of Freiburg, Germany, and a Doctorat d'Etat in 1984 from the University of Paris-Sud, Orsay, France, and spent a year at the Institute for Advanced Study in Princeton before joining the Illinois faculty in 1986.

Trained as a number theorist, he is interested also in problems in analysis, probability theory, and combinatorics, and, in particular, in problems that lie at the interface of these areas with number theory. Most of his research falls into the areas of analytic number theory, which investigates problems of number theory by methods of analysis, and probabilistic number theory, which studies number theoretic problems of a statistical nature.

Hildebrand has taught special topics courses on asymptotic methods of analysis, exponential sums, combinatorial number theory, and probabilistic number theory, and has supervised six PhD students.

At the undergraduate level, Hildebrand has a long-standing interest in and involvement with the local mathematical contest scene. For over twenty years, he has served as local coordinator of the William Lowell Putnam Competition and a coach of the Illinois Putnam team, and has organized training sessions, practice contests, and related activities.

Leon McCulloh (Emeritus)
McCulloh received his PhD in 1959 from the Ohio State University under H.B.Mann, and he came to the University of Illinois in 1961. He held visiting appointments at Indiana University and the University of Hawaii in 1963 and 1967, respectively, and a short term appointment at the University of Bordeaux in 1983. He has also visited King's College London, the University of Regensburg, and Cambridge University on sabbatical leaves. He has supervised nine PhD students.

His research, starting with his thesis on integral bases in relative Kummer extensions of number fields, has been a logical progression from that beginning. It has centered on relative integral and normal integral bases, and the related notions of realizable Steinitz and Galois module classes. These topics were the basis for the thesis topics of four of his students. He developed a generalization of the notion of Stickelberger relations to class groups of integral group rings and found connections with realizable Galois module classes and with class number formulas for integral group rings. (He owes a heavy debt to Steve Ullom for pointing out the connection between normal integral bases and Stickelberger relations in the work of Hilbert.) In 1987 he published a characterization "in Stickelberger terms" of the realizable Galois module classes of (tame) abelian extensions, in particular showing that they form a subgroup of the classgroup. He has since partially generalized these results to nonabelian extensions but a full generalization is not yet in sight. There are many related open problems and conjectures which he feels provide an important and fruitful area for research.

Bruce Reznick
Reznick's degrees are from Caltech (BS, 1973) and Stanford (Ph.D., 1976). He has been on the faculty of the University of Illinois since 1979. He was a Sloan Foundation fellow from 1983--1986, and received the Prokasy Award for Excellence in Undergraduate Teaching from his College in 1997.

His main research interests originate from questions about the structure of polynomials in several variables and their representations as sums of powers of polynomials and then spread out. The subjects ultimately range from concrete realizations of Hilbert's 17th problem for positive definite forms to Hilbert identities; from spherical designs and quadrature formulas to the linear algebra of polynomials and the Linfty norm of the Laplacian on spaces of forms of fixed degree; from polynomial solutions of Diophantine equations to concrete versions of theorems in abstract real algebraic geometry; from questions about the lattice point structure of polytopes with lattice point vertices to structure theorems for recursively defined sequences. So far, these have encompassed questions in number theory, algebra, analysis and combinatorics.

Armin Straub (Doob Postdoc)
Armin Straub received his Ph.D. in 2012 from Tulane University under the direction of Victor Moll. His research interests are in special functions, especially hypergeometric and modular ones, and their many connections to number theory, combinatorics and computer algebra. Recently he has for instance studied short planar random walks from a number theoretical perspective.

Kenneth B. Stolarsky
Stolarsky did his undergraduate studies at Caltech and received his Ph.D. at the University of Wisconsin (Madison) in 1967 under the direction of Marvin Knopp. His research interests include number theory, geometry, classical analysis, and combinatorics, and he has supervised six Ph.D. theses in these areas. Stolarsky is particularly interested in Diophantine approximation and has taught many graduate courses in this and the related areas of transcendence theory and the geometry of numbers. His undergraduate teaching has centered on introductory differential equations, and he has also done much editorial work for the problems section of the American Mathematical Monthly. He retired in 2010.

Stephen Ullom
Ullom received his Ph.D. in 1968 from the University of Maryland; his thesis advisor was Sigekatu Kuroda. He spent his NSF Postdoctoral Fellowship (1968--69) at the University of Karlsruhe and King's College, University of London, his mentors being H. W. Leopoldt and A. Frohlich respectively. In 1969--1970 he was a visiting member of the Institute for Advanced Study. He came to the University of Illinois in 1970 as an Assistant Professor and was promoted to Professor in 1978. Ullom has spent sabbatical leaves at King's College London and Cambridge University (Bye Fellow of Robinson College) with a shorter visit to the University of Arizona. He has supervised five Ph.D. students, the last two jointly with Nigel Boston.

In his thesis Ullom studied the Galois module structure of ideals in a local or global field which are invariant under the Galois group. He found the first example where the ring of integers is not projective over the integral group ring, but some ideals are projective. He generalized Hilbert's theorem on the connection between Galois module structure and Stickelberger elements to proper ideals. Frequently in collaboration with Irv Reiner, Ullom studied class groups of integral group rings, particularly on their arithmetic properties. He proved a conjecture of Kervaire and Murthy on the class groups of cyclic p-groups by using Iwasawa theory applied to group rings. His work on Swan modules, a particularly simple form of projective module, showed most noncyclic groups G have projective nonfree modules over the group ring ZG. Tate cited Ullom's 1977 survey article on class groups in his book on Stark's conjecture. Ullom's student, Steve Watt, extended Frohlich's results on Galois groups of p-extensions with restricted ramification over the rationals to the case of imaginary quadratic base field. Ullom and Watt used this result to give a criterion for when an abelian p--extension L of an imaginary quadratic field K has class number prime to p (assuming the genus number for L/K is prime to p).

Ullom wrote a paper with N. Boston on Galois deformations associated to elliptic curves with complex multiplication. An important consequence is that for the Fermat curve there are infinitely many primes p such that the deformation ring is not simply a ring of formal power series in several variables.

Marcin Mazur and Ullom investigated the Galois module structure of units modulo torsion in certain real abelian fields of two power degree. There are many fascinating connections here between Galois module structure and classical topics such as genus field, central class field, and sign of the fundamental unit in a quadratic field. At present Ullom is also investigating some problems on the structure of Galois groups of extensions of number fields with restricted ramification.

Alexandru Zaharescu
Zaharescu received his Ph.D. in 1995 from Princeton University under the direction of Peter Sarnak. He has held positions at the Massachusetts Institute of Technology, McGill University, the Institute for Advanced Study, and since 2000 he is a faculty member at UIUC. He is currently working on several projects concerned with the zeros of the Riemann zeta function and more general L-functions, and on some problems on the distribution of various sequences of interest in number theory.