DOC PREVIEW
Research Statement

This preview shows page 1-2 out of 5 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 5 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 5 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 5 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Research StatementChristopher D. SinclairOctober 2006AbstractMy research lies at the intersection of number theory and random matrix theory. I am interestedin the connection between heights of polynomials and ensembles of asymmetric random matrices.While there are known links between number theory and random matrix theory (e.g. statisticalsimilarities between the statistics of the zeros of the Riemann zeta functions and the eigenvaluesof certain ensembles of random matrices), my research demonstrates a new connection betweenthe two fields.1 Number Theory and Volumes of PolynomialsGiven a polynomial in C[x], its Mahler measure is the absolute value of the leading coefficienttimes the product of the absolute values of its roots outside the unit circle. Mahler measure is aheight (meas ure of complexity) of polynomials, which for integer polynomials can be interpretedas ‘cyclotomicness’: the Mahler measure of an irreducible integer p olynomial is 1 if and only if thepolynomial is cyclotomic. It is still unknown if 1 is a limit point of Mahler measures of integerpolynomials; this is Lehmer’s Problem [11]. There are other open questions regarding the range ofMahler m eas ure restricted to integer polynomials [1].1.1 Counting Reciprocal Polynomials with Bounded Mahler MeasureA reciprocal polynomial is one whose coefficient vector is invariant under reversal of order. In 1971,Smyth proved that if the Mahler measure of an irreducible integer polynomial is less than 1.3 . . .,then the polynomial is necessarily recipro c al (x − 1 is the sole exception). It is known that forevery positive integer N and T > 0, there are a finite number of reciprocal integer polynomials ofdegree at most N and Mahler measure at most T . We will call this number MN(T ). In my thesis,I use geometry of numbers techniques to give asymptotic estimates for MN(T ) by com puting thevolume of coefficient vectors of reciprocal real polynomials of degree at most N and Mahler measureat most T [18]. We will call this number VN(T ). It turns out that VN(T ) = TN+1VN(1). Thusif we regard Mahler measure as a distance function (in the sense of the geometry of numbers) onthe coefficient vectors of degree N polynomials, the asymptotic estimates for MN(T ) reduce to thecomputation of VN(1) — the volume of the unit star body (ball).I proved that VN(1) is an explicitly computable rational number and moreover that this num-ber is most easily described as the product of N simpler rational numbers. This follows resultsof S.-J. Chern and J. Vaaler which show that the analogous volume of (not necessarily reciprocal)polynomials is also a rational number most easily expressed as a product [2]. On first inspection,these product formulations seem like a mere curiosity; however, they turn out to have implicationsin random matrix theory. My method for computing VN(T ) and producing e stimate s for MN(T )Research Statement, Christopher D. Sinclair 2drastically simplifies Chern and Vaaler’s calculations and generalizes to a large class of multiplica-tive heights. These generalizations of Mahler measure were the primary focus of my thesis [18].Applications of this theory to number theory are recorded in [19]. A similar analysis may also becarried out for volumes of complex polynomials; I have done this for complex reciprocal polynomialsin my paper [17].1.2 Conjugate Reciprocal Polynomials and the Link with Random MatrixTheoryA conjugate reciprocal polynomial is one whose coefficient vector is invariant under reversal followedby complex conjugation. David Farmer asked me if I could compute the volume of degree N monicconjugate recipro c al polynomials with all roots on the unit circle (call this set WN). David and hisco-authors (F. Mezzadri and N. Snaith) were interested in zeros of conjugate reciprocal polynomialsas models for zeros on the critical line of certain L-functions [7].In a joint paper with Kathleen Pe terse n (Queen’s University), I show that the volume of WNis equal to the volume of the (N − 1)-ball of radius 2 [13]. This is done by reducing the volumecalculation to an integral done by Dyson in one of his first papers on random matrix theory [4].We also show that WNis homeomorphic to an (N − 1)-ball and that the group of isometries isisomorphic to the dihedral group of order 2N .In a recent manuscript with Jeff Vaaler (University of Texas at Austin), we provide sufficientconditions for a degree N conjugate reciprocal polynomial to have all roots on the unit circle [21].These results rely on the identification of special subsets of WNwhich are defined by geometricconditions. This follows work of Schinzel [16], and Lakatos and Losonczi [10].2 Ensembles of Asymmetric Random MatricesIn 1965 Ginibre introduced three ensembles of random matrices whose entries are chosen inde-pendently with Gaussian density from R, C and Hamilton’s Quaternions [8]. (In the language ofrandom matrix theory an ensemble is a set of matrices together with a probability measure). Thestudy of the eigenvalue statistics of Ginibre’s real ensemble (GinOE) is difficult because the eigen-values come in two flavors: real and complex conjugate pairs. Quantities of interest often fractureinto unwieldy sums over all possible numbers of real and complex conjugate pairs of eigenvalues.Due to this complication, there are many basic questions surrounding GinOE which remain open.2.1 Averages over Ginibre’s Real EnsembleEnsemble averages are key to the calculation of many quantities associated to GinOE. Connectingheights of polynomials with random matrix theory, I proved that VN(1) can be written as theaverage of a function over GinOE. This is important since (as I me ntioned in Section 1.1), VN(1) ismost easily expressed as a product. The method which produces this simple product formulationalso allows for a product formulation for more general ensemble averages over GinOE [20]. Inaddition, this product formulation is (seemingly) independent of the decomposition of the space ofeigenvalues. This allows us to write ensemble averages over GinOE using the exact same mechanismused for writing averages over the ‘classical’ ensembles of Dyson and Wigner.Establishing this connection is significant since the correlation functions (and other basic quan-tities) for the classical ensembles can be computed from the averages of certain functions over theseResearch Statement, Christopher D. Sinclair 3ensembles (see [22]). There is hope that this discovery will


Research Statement

Download Research Statement
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Research Statement and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Research Statement 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?