• Curriculum Vitæ
• Reprints and Preprints

• 2006 Research Statement
• 2005 Research Statement
• 2004 Research Statement
• 2003 Research Statement

• From a talk I gave at Advances in Number Theory and Random Matrix Theory at the University of Rochester.
• From a talk I gave at the 10th Annual Pacfic Northwest Number Theory Conference in Redmond WA.
• From a talk I gave at the Special Session on Mahler Measure and Heights at the Joint Meetings of the AMS
• From a talk I gave at the Mahler Measure in Mobile Conference
  

“I study numbers.”

“I study polynomials with integer coefficients.”

“I use calculus to study measures of complexity of polynomials with integer coefficients.”

“I study the eigenvalues of matrices whose entries are chosen randomly, and connections with zeros of polynomials.”

“The Mahler's measure of a polynomial is the absolute value of the leading coefficient times the product of the moduli of the roots outside the unit circle. I study the Mahler's measure of polynomials with integer coefficient. I also study the eigenvalues of matrices whose entries are real or complex and chosen randomly with Gaussian probability.”

“The Mahler's measure is a measure of the cyclotomicness of an integer polynomial. It is an open question (Lehmer's problem) to determine if the Mahler's measures of irreducible non cyclotomic integer polynomials are bounded away from 1. The range of values of Mahler's measure restricted to degree N real polynomials can be computed by finding the average of a certain function over the ensemble of asymmetric N × N whose entries are chosen independently with Gaussian density.”

“We may generalize Mahler's measure to create the class of multiplicative distance functions by creating multiplicative functions on C[x] which are distance functions (in the sense of the geometry of numbers) on finite dimensional subsets of C[x]. These functions share many properties of Mahler's measure. One way of creating a multiplicative distance function is to look at Mahler's measure on C[G(x)], where G(x) is a fixed Laurent polynomial. The study of such an object leads to information about algebraic numbers with geometric symmetries in the Galois group determined by G. One project I am interested in is to find Schanuel type estimates for the number of algebraic numbers of fixed degree whose conjugates satisfy certain specified geometric symmetries and whose Mahler measure is bounded but large.”

“I am currently studying the range of multiplicative distance functions on Z[x] by using geometry of numbers techniques. In particular associated to each degree N there is a distribution function which gives the Lebesgue measure of the set of monic polynomials of degree N with distance bounded by T. The Mellin transform of this distribution function encodes information about the range of multiplicative distance function into a function which is analytic in a half plane. These analytic functions demonstrate a great deal of structure which suggests that the multiplicativity condition is a powerful condition on distance functions. In particulare the analytic functions associated to Mahler's measure on C[G(x)] for the Laurent polynomial G(x) = x + t/x (0 ≤ t ≤ 1) has an analytic continuation to a rational function with poles at integers and a high order zeros at the origin. Moreover when t is rational the coefficients of this rational function are also rational. The residues of the poles and certain special values of these rational functions tell us about the range of values of the multiplicative distance function.”

“An interesting class of multiplicative distance functions are those formed from the equilibirum potential of a simply connected compact subset of C. As such, if G is a Laurent polynomial which is also a conformal map from the complement of the unit disk onto its image, then Mahler's measure restriced to C[G(x)], suitably interpreted, can be written in terms of the equilibrium of the compact set K where K is the complement of the image of the complement of the closed unit disk under G. This provides an analog of Jensen's formula for multiplicative distance functions formed by composing Mahler's measure with certain rational functions. Since potentials have a physical significance, the associated analytic functions turn out to be the canonical partition function associated to N particle systems in the presence of a charged conducting region represented by K.”

“ I am interested in the eigenvalue statistics of ensembles of asymmetric random matrices. Thes ensembles arise in my work, since the Mellin transforms which yield information about the range of multiplicative distance functions can be represented as averages over Ginibre's ensembles of random matrices whose entries are chosen independtly with Gaussian density. The special form of the Mellin transforms has allowed me to find an elementary description of ensemble averages over Ginibre's ensembles in terms of the Pfaffian of skew-symmetric Gram-like matrices formed from a skew inner product associated to the function being averaged. This simple form for ensemble averages opens the door for the calculation of quantities of interest to random matrix theorists: correlation functions, cluster functions, etc. For Ginibre's ensemble of real matrices, a closed form for these quantities has remained out of reach. ”

“The fact that the moment functions of certain multiplicative distance functions turn out to be rational functions with poles at integers is due to the appearance of Vandermonde factors in the Jacobian of the change of variables from coefficients of polynomials to roots of polynomials. It is exactly the integer powers that arise in the Vandermonde matrix which yields the expectation of poles at rational numbers in the moment functions. The fact that these poles are in fact integers suggests that there is much to be learned about the exact relationship between these moment functions and the Vandermonde factors. In particular it would be quite interesting to investigate the family of orthogonal and skew-orthogonal polynomials associated to one of these multiplicative distance functions whose moment functions are rational functions.”