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## Christopher Dean Sinclair

##### Associate Professor, University of Oregon
• Personal Data
• Education
• PhD (Mathematics) · Supervisor Jeff Vaaler · The University of Texas at Austin · May 2005
• BS (Mathematics) · The University of Arizona · August 1997
• Professional Experience
• Associate Professor · The University of Oregon · September 2013-
• Assistant Professor · The University of Oregon · August 2009-September 2013
• Research Member · Mathematical Sciences Research Institute · Fall 2010
• Instructor, Postdoctoral Fellow · The University of Colorado, Boulder · 2007-2009
• Visiting Postdoctoral Fellow · Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France · July 2007
• Visiting Postdoctoral Fellow · Max Planck Institut für Mathematik, Bonn, Germany · April-June 2007
• Postdoctoral Fellow · Simon Fraser University · 2005-2007
• Postdoctoral Fellow, Instructor · The University of British Columbia · 2005-2007
• Assistant Instructor · The University of Texas at Austin · 2002-2005
• Teaching Assistant · The University of Texas at Austin · 2000-2002
• Research Scientist · The University of Texas at Austin · Applied Research Laboratories · 1997-1999
• Research Interests
• Random matrix theory · Heights of algebraic numbers · Statistical physics
• Honors and Awards
• Simons Collaboration Grant for Mathematicians: #315685 · Random matrices and polynomials with logarithmic weights. 2014-2019.
• NSF grant: DMS-0801243 · Analysis program · Integrable Structure of Random Spectra Derived from Diophantine Geometry. 2008-2011.
• 2001-2002 Frank Gerth III Teaching Excellence Award
• 2003-2004 University of Texas at Austin Continuing Tuition Fellowship
• Refereed Publications
1. (with Maxim Yattselev). Root statistics of random polynomials with bounded Mahler measure · Advances in Math. Vol. 272: 124-199 · doi: 10.1016/j.aim.2014.11.022 · arXiv: math-ph/arxiv:1307.4128.
2. (with Brian Rider). Extremal laws for the real Ginibre ensemble; Ann. Appl. Probab. Vol. 24, No. 4: 1621-1651, 2014 · arXiv:math-ph/arxiv:1209.6085.
3. (with Brian Rider and Yuan Xu). A solvable mixed charge ensemble on the line: Global results; Probab. Theor. Relat. Fields Vol. 155: 127-164, 2013. · doi: 10.1007/s00440-011-0394-z · arXiv:math-ph/arxiv:1007.2246
4. (with Maxim Yattselev). Universality for ensembles of matrices with potential theoretic weights on domains with smooth boundary; J. Approx. Theor. Vol. 164: 682-708, 2012. · doi: 10.1016/j.jat.2012.02.001 · arXiv:math/arxiv:1108.3052
5. The partition function of multicomponent log-gases; J. Phys. A: Math. Theor. Vol. 45: 165002, 2012. · doi: 10.1088/1751-8113/45/16/165002 · arXiv:math-ph/arXiv:1201.0223
6. Ensemble averages when $\beta$ is a square integer; Monatsh. Math. Vol. 166, No. 1: 121-144, 2012 · doi: 10.1007/s00605-011-0371-8 · arXiv:math-ph/arxiv:1008.4362
7. (with Peter Forrester). A generalized plasma and interpolation between classical random matrix ensembles; J. Stat. Phys. Vol. 143, No. 2: 326-345, 2011. · doi: 10.1007/s10955-011-0173-3 · arXiv:math-ph/arxiv:1012.0597
8. (with Kathleen Petersen). Equidistribution of algebraic numbers of norm one in quadratic number fields; Int. J. Number Theory Vol. 7, Issue 7: 1841-1861, 2011 · doi: 10.1142/S1793042111004666 · arXiv:math.NT/arxiv:1004.0986
9. (with Kevin Hare and David McKinnon). Patterns and periodicity in a family of resultants; J. Théor. Nombres Bordeaux Vol. 21: 215-234, 2009 · doi: 10.5802/jtnb.667 ·download from archive.numdam.org
10. Correlation functions for $\beta=1$ ensembles of matrices of odd size; J. Stat. Phys. Vol. 136: 17-33, 2009 · doi: 10.1007/s10955-009-9771-8 · arXiv:math-ph/arxiv:0811.1276.
11. (with Alexei Borodin). The Ginibre ensemble of real random matrices and its scaling limits; Comm. Math. Phys. Vol. 291: 177-224, 2009 · doi: 10.1007/s00220-009-0874-5 · arXiv: math-ph/arxiv:0805.2986.
12. The range of multiplicative functions on $\mathbb{C}[x], \mathbb{R}[x]$ and $\mathbb{Z}[x]$. Proc. London Math. Soc. Vol. 96(3): 697-737, 2008 · doi: 10.1112/plms/pdm037 · arXiv: math.NT/0509591.
13. (with Jeffrey D. Vaaler). Self-inversive polynomials with all zeros on the unit circle. London Mathematical Society Lecture Note Series: Number Theory and Polynomials Vol. 352: 312-321 · Editors: James McKee and Chris Smyth · 2008 (refereed).
14. (with Kathleen L. Petersen). Conjugate reciprocal polynomials with all roots on the unit circle; Canad. J. Math. Vol. 60(5): 1149-1167, 2008 · doi: 10.4153/CJM-2008-050-8 · arXiv: math.NT/0511397.
15. Averages over Ginibre's ensemble of random real matrices; Int. Math. Res. Not. Vol. 2007: 1-15, 2007 · Article ID:rnm015, doi:10.1093/imrn/rnm015 · arXiv: math-ph/0605006.
16. The distribution of Mahler's measures of reciprocal polynomials. Int. J. Math. Math. Sci Vol. 52: 2773-2786, 2004 · doi:10.1155/S0161171204312469 · arXiv: math.NT/0311255.
• Other Publications/Manuscripts
1. (with Christopher Shum). A solvable two-charge ensemble on the circle· May 2014 · arXiv: math/arxiv:1404.5290.
2. (with Kathleen Petersen). Equidistribution of elements of norm 1 in cyclic extensions · Submitted for publication · April 2014 · arXiv: math/arxiv:1404.4648.
3. (with Alexei Borodin). Correlation functions of asymmetric real matrices; July 2007 · arXiv: math-ph/arxiv:0706.2670.
4. Multiplicative distance functions; PhD Thesis, The University of Texas at Austin, 2005
5. (with Lyn Pierce and Sara Matzner). An application of machine learning to network intrusion detection. 15th Annual Computer Security Applications Proceedings, 1999.
• Research in Progress
• (with Maxim Yattselev). Reproducing kernel asymptotics for measures supported on cusped hypocycloids.
• Towards a hyperpfaffian point process for random matrix ensembles when $\beta$ is a square integer.
• Annotated Bibliography

The story begins with

The range of multiplicative functions on $\mathbb C[x], \mathbb R[x]$ and $\mathbb Z[x]$. Proc. London Math. Soc. Vol. 96(3): 697-737, 2008 · doi: 10.1112/plms/pdm037 · arXiv: math.NT/0509591.

While this wasn't my first publication, it is the beginning of the journey which got me to where I am today. This paper is a somewhat abbreviated version of my PhD thesis. My advisor (Jeff Vaaler) and one of his previous students (Shey-Jay Chern) had computed the volume of a certain set of coefficient vectors of polynomials to produce an asymptotic for the number of integer polynomials with fixed degree and bounded, but large height (a height is simply a measure of complexity of polynomials or other arithmetic objects; in this case the pertinent height is a quantity called Mahler measure). Their computation revealed the surprising fact that the volumes of interest were rational numbers which could be expressed as products of simpler rational numbers. The simplicity of this result seemed orthogonal to its lengthy and highly technical proof. For my thesis, I was tasked to find a similar formula for the related set of reciprocal polynomials (these are polynomials whose coefficient vectors read the same forwards as backwards, and they play a special role in the study of Mahler measure). I ultimately did this, but my own personal goal was to explain the simple product formulation arrived at previously by my advisor.

The final explanation was that Chern and Vaaler's volume, as well as my reciprocal analog, could both be expressed as the Pfaffian of related antisymmetric matrices. In this paper, I generalize this product formulation to volumes of polynomials formed from other heights besides Mahler measure and its reciprocal cousin, and explain the general Pfaffian mechanism responsible for it.

Stepping backwards in time a bit, we come to my first published paper

The distribution of Mahler's measures of reciprocal polynomials. Int. J. Math. Math. Sci Vol. 52: 2773-2786, 2004 · doi:10.1155/S0161171204312469 · arXiv: math.NT/0311255.

This paper is similar to the previous, except that the central result here is volume calculations for coefficient vectors of complex polynomials with bounded height, whereas the previous paper dealt primarily with real polynomials. In this paper, I describe the relevant volume of polynomials in terms of the determinant of a matrix of inner products associated to the relevant height. The central technical achievement here was the rediscovery of an 1883 formula of Andreief (which according to deBruijn was 'almost certainly' known by Cauchy) on integrals whose integrands are determinants. This determinantal formula predated the Pfaffian formulation in the The range of multiplicative functions on $\mathbb C[x], \mathbb R[x]$ and $\mathbb Z[x]$ (where the inner product is replaced by a certain skew-symmetric inner product).

The next paper in this line of research is

Averages over Ginibre's ensemble of random real matrices; Int. Math. Res. Not. Vol. 2007: 1-15, 2007 · Article ID:rnm015, doi:10.1093/imrn/rnm015 · arXiv: math-ph/0605006.

This paper was my first foray into random matrix theory, and (arguably) contains the central observation which lead to the solvability of Ginibre's ensemble of real random matrices. Ginibre's ensemble consists of $N \times N$ matrices with iid standard normal entries. Presented here is the calculation of the partition function of this ensemble in terms of the Pfaffian of a square matrix formed from a skew-symmetric inner product. In fact, this calculation is essentially identical to the volume calculation presented in my thesis (and at the time I wrote this paper, I saw it as a cheap way to get another publication on my CV). The main idea at for both the volume calculation for polynomials of bounded height, and the determination of the partition function for Ginibre's real ensemble is that, after a change of variables, both can be realized as a sum of integrals whose integrands contain factors of the absolute value of a Vandermonde determinant. The sum is over all possible numbers of real roots/eigenvalues. This decomposition into subdomains was the central stumbling block to the solvability of Ginibre's real ensemble. And, were it not for Chern and Vaaler's original observation of the product formulation for their volume calculation, Ginibre's ensemble might still be unsolved.

I mailed the Averages paper to several people whom I thought might be interested. One of these people was Brian Rider, which eventually got me a (second) postdoc at Colorado (my first postdoc was in Vancouver with Peter Borwein and David Boyd). I also mailed the paper to Percy Deift who happened to be at Caltech for the semester. Percy and Alexei Borodin responded by invited me to visit them to explain my work. This led to the following publication (and it's precursor manuscript)

(with Alexei Borodin). The Ginibre ensemble of real random matrices and its scaling limits; Comm. Math. Phys. Vol. 291: 177-224, 2009 · doi: 10.1007/s00220-009-0874-5 · arXiv: math-ph/arxiv:0805.2986; (with Alexei Borodin). Correlation functions of asymmetric real matrices; July 2007 · arXiv: math-ph/arxiv:0706.2670.

I view this work as something of a second PhD thesis; this one in random matrix theory. Alexei and Percy convinced me that the the Pfaffian formulation for the partition function must lead to a Pfaffian point process on the eigenvalues.

As before, the complexity introduced by the variable number of real eigenvalues presented the central algebraic complication. Historically, people had looked primarily at the correlations restricted to one of the subdomains formed by specifying the number of real eigenvalues (and then integrating out some number of real and some number of complex conjugate pairs of eigenvalues), thinking that this must simplify the problem. However, the situation is actually simpler when one sums these 'partial correlation functions' over all possible subdomains. This was the central algebraic observation presented in the above unpublished manuscript, and it was done by applying Eric Rains' Pfaffian version of the $\det(I + AB) = \det(I + BA)$ trick (introduced in this context by Craig Tracy and Harold Widom) to the the Pfaffian formula for the partition function for Ginibre's real ensemble.

I am convinced this is the 'right' way to derive the correlation functions for $\beta=1$ ensembles.

Besides the derivation of the Pfaffian point process, the paper with Alexei also centered on the scaling limits of the matrix kernel(s). This was a very busy time, since there were a number of competing research groups vying for some level of credit for solving Ginibre's real ensemble. In fact, once I produced the Pfaffian version for the partition function in the Averages paper, Peter Forrester and Taro Nagao quickly published the skew-orthogonal polynomials necessary to simplify the matrix kernel for the ensemble. This was useful, since it allowed Alexei and I to quickly write the entries in the matrix kernel(s) in terms of the partial sums of the exponential function. (It is worth reflecting on this fact; skew-orthogonal polynomials do not in general, satisfy a Christoffel-Darboux type relation which would allow the kernel asymptotics to easily be derived from the asymptotics of the polynomials. Somewhat miraculously, however, the entries of the matrix kernel for Ginibre's real ensemble were explicitly summable in terms of partial sums of the exponential and hyperbolic trigonometric functions, all of whose asymptotics had been well-studied). It should be remarked that there are actually several matrix kernels for this ensemble encoding information about the real/real, real/complex and complex/complex interactions between eigenvalues, and each of these has a different scaling limit in the bulk and at the edge. So, while the analysis was simplified by relying on previously known asymptotics, there was still a lot of work to do.

Alexei and I initially did the algebraic steps necessary to analyze Ginibre's real ensemble for even square matrices. The odd case, and indeed the odd case for all $\beta=1$ ensembles, is more complicated due to the fundamental fact that Pfaffians are only defined for even square (antisymmetric) matrices. The following paper contains a detailed derivation of the correlations for the odd case using Rains' Pfaffian identity and Tracy and Widom's basic outline.

Correlation functions for $\beta$=1 ensembles of matrices of odd size; J. Stat. Phys. Vol. 136: 17-33, 2009 · doi: 10.1007/s10955-009-9771-8 · arXiv:math-ph/arxiv:0811.1276.

The next paper in this line of reasoning came about during discussions with Brian Rider while I was a postdoc at Colorado.

(with Brian Rider and Yuan Xu). A solvable mixed charge ensemble on the line: Global results; Accepted for publication in Probab. Theor. Relat. Fields · doi: 10.1007/s00440-011-0394-z · arXiv:math-ph/arxiv:1007.2246.

What if, instead of looking at Ginibre's real ensemble, where the eigenvalues are either real or complex conjugate pairs, we envisioned a situation where we forced the complex conjugate pairs together onto the real line? This would create a situation where the complex conjugate pairs are replaced with double eigenvalues on the line. This can be better visualized as a system of charged particles, some with charge 1 and some with charge 2, in the presence of the harmonic oscillator potential. The same trick which kept track of the variable number of real eigenvalues before, keeps track of the variable number of charge 1 particles, and the partition function can be written as the Pfaffian of an antisymmetric matrix formed from the sum of two skew-symmetric inner products: the inner product for the Gaussian Orthogonal Ensemble and the inner product for the Gaussian Symplectic Ensemble. The algebraic identities in place, we were soon stymied in trying to derive the skew-orthogonal polynomials that would (hopefully) allow for an in-depth analysis of the statistics of the particles. Sometime later I joined the faculty of the University of Oregon, and had the opportunity to talk with Yuan Xu about various problems I was interested in. I showed him the skew-symmetric inner product Brian and I had derived, and within a few days Yuan had derived the skew-orthogonal polynomials! Yuan also made the observation that the skew-orthogonal polynomials were in fact just one in a one-parameter family, and asked what I thought of this. After a few days of thinking, I realized that this parameter was a natural quantity, which really should have appeared at the start of the project; this quantity, the fugacity, determines in some sense how easy it is to exchange a particle of charge two for two particles of charge 1. Moreover, the partition function as a function of the fugacity, is also the probability generating function for the number of charge 1 particles.

The skew-orthogonal polynomials in hand, we were able to find a closed form for the density of the number of real particles, and a version of the Central Limit Theorem for this random variable in the limit as the total charge goes to infinity. We were also able to derive the global density of each kind of particle in this limit. We were unable to derive a closed form for the scaling limit of the kernel(s). However, my PhD student Chris Shum, has since derived the scaling limit for the kernel(s) in the analogous circular ensemble. By universality, the bulk kernel(s) should be the same in both cases.

The primary algebraic tool necessary to derive the Pfaffian formulation for the partition function of the two-component ensemble was the confluent Vandermonde determinant. The confluent Vandermonde determinant is applicable in a wider array of ensembles than just that one, and the next two papers are the beginning of a line of research that I am still engaged in.

The partition function of multicomponent log-gases; J. Phys. A: Math. Theor. Vol. 45: 165002, 2012. · doi: 10.1088/1751-8113/45/16/165002 · arXiv:math-ph/arXiv:1201.0223 ; Ensemble averages when $\beta$ is a square integer; Monatsh. Math. Vol. 166, No. 1: 121-144, 2012 · doi: 10.1007/s00605-011-0371-8 · arXiv:math-ph/arxiv:1008.4362.

The first of these two papers gives a hyperpfaffian formula for the partition function of ensembles when $\beta$ is a square integer. This very naturally generalizes the well-known $\beta=1$ and $\beta=4$ situations. From another viewpoint, this ensemble can also be thought of as an interacting particle system on the line (or circle) where the charged particles all have the same integer charge, at inverse temperature $\beta=1$. Slightly more explicitly, if the magnitude of the charge is $L$, and the particles are placed in a potential corresponding to some finite measure, then the partition function is given as the hyperpfaffian of an $L$-form whose coefficients are the integrals (with respect to the potential measure) of the various $L \times L$ Wronskians of a fixed family of polynomials. The hope, which heretofore has not been realized, is that a hyperpfaffian formula for the partition function will lead to a hyperpfaffian point process on the eigenvalues/particles. Some of the formulas in this work are rediscoveries of similar formulas (derived via a different method in a related but different context) by Jean-Gabriel Luque and Jean-Yves Thibon.

The second of these papers was an attempt to generalize the hyperpfaffian formula for the partition function for fixed integer charges to a formula for the partition function for systems composed of particles with different (positive integer) charge magnitudes, where the sum total of all charges in the system is a fixed integer, and the inverse temperature is $\beta=1$. This is a zero-current version of the grand canonical partition function. Here we allow fugacity variables which determine the expected ratios of the numbers of the various particles. The appropriate generalization of the hyperpfaffian in this situation is the Berezin integral, and the main result says that the partition function is the Berezin integral (with respect to the volume form) of the exponential of a sum of forms (in the sense of the Grassmann algebra) comprised of integrals of Wronskians of a family of polynomials (these are exactly the same forms which appear in the square $\beta$ paper).

In Ensemble averages when $\beta$ is a square integer I noticed that a formula in which the Vandermonde determinant can be expressed as a Pfaffian, allows for (among other things) a Pfaffian point process for $\beta=2$ ensembles. I mentioned this formula during a talk at MSRI during the 2010 special program on random matrices. Peter Forrester approached me shortly thereafter and indicated that this formula might be useful to answer a question he had considered previously, but had been unable to answer. This led to the paper

(with Peter Forrester). A generalized plasma and interpolation between classical random matrix ensembles; J. Stat. Phys. Vol. 143, No. 2: 326-345, 2011. · doi: 10.1007/s10955-011-0173-3 · arXiv:math-ph/arxiv:1012.0597.

The model system here is one with two species of particles restricted to the circle whose interaction energy is not proportional to the product of the charges. This provides a certain type of interpolation between $\beta=2$ and $\beta=4$ circular ensembles. The joint density of particles here is also of interest as a trial wave function in the anomalous quantum Hall effect. The Pfaffian point process for $\beta=2$ ensembles discovered in my previous paper combined with the well-known Pfaffian point process for $\beta=4$ ensembles allowed this generalized plasma model to be solved.

A final, and recent, paper in this line of research is the following written with a current postdoc, Max Yattselev, at the University of Oregon.

(with Maxim Yattselev). Universality for ensembles of matrices with potential theoretic weights on domains with smooth boundary; J. Approx. Theor. Vol. 164: 682-708, 2012. · doi: 10.1016/j.jat.2012.02.001 · arXiv:math/arxiv:1108.3052

This paper is actually a return to some of the questions I originally asked in my thesis. The model considered here is an ensemble of charged particles (corresponding to a &beta=2$ensemble) restricted to the plane, and in the presence of an oppositely charged conducting region whose charge density is given by the equilibrium measure (in the sense of potential theory in the complex plane). Another model with the same statistics is the roots of a complex polynomial chosen at random from the unit ball of a height which corresponds to the exponentiated equilibrium measure of the conducting region as identified with a compact set in the complex plane. The main result in this paper is that, if the boundary of this conducting region is sufficiently smooth, then the limiting statistics of the particles (or roots) is essentially independent of the actual shape of the region. More specifically, we prove that if the domain is sufficiently smooth, the kernel in the scaling limit is (essentially) independent of the shape of the domain. The universal kernel is explicitly given and shown to be a generalization of the universal reproducing kernel for Bergman orthogonal polynomials derived by Doron Lubinsky. Max and I are currently investigating similar type universality results for domains with non-smooth boundaries (specifically, at corners and cusps). My other primary research direction diverged relatively early from the above sequence of publications. In 2004, I attended a workshop on the intersection of random matrix theory and number theory at the Newton Institute in Cambridge. During lunch one day, I sat next David Farmer, who asked me about my thesis problem. I told him that I was trying to compute the volume of a certain set of coefficient vectors of polynomials. He immediately asked me if I thought I could compute the volume of the set of (coefficient vectors of) polynomials of degree$N$with all roots on the unit circle. I immediately responded that I thought it shouldn't be too difficult, and that I would look into it. The problem was a bit more difficult than I originally anticipated, and I roped a fellow graduate student, Kathleen Petersen, into the project. In (with Kathleen L. Petersen). Conjugate reciprocal polynomials with all roots on the unit circle; Canad. J. Math. Vol. 60(5): 1149-1167, 2008 · doi: 10.4153/CJM-2008-050-8 · arXiv: math.NT/0511397, we compute the volume of this set, as well as show that the group of isometries is dihedral and that the set is homeomorphic to a ball, and geometrically has the structure of a colored simplex, where the colorings are given by cyclically ordered partitions of the integer$N$, which represent the various possible multiplicities for the roots. By the time we were ready to submit, we both landed postdoctoral positions in Canada (albeit at different institutions) so we thought the Canadian journal was an appropriate venue. A year or so after its publication, I returned to thinking about the set of polynomials of fixed degree and all roots on the unit circle, with the hope of producing easy-to-check necessary conditions and sufficient conditions for polynomials to have all roots on the unit circle. During this time I was invited for a couple weeks to a workshop on heights at the Schrodinger Institute in Vienna, where I ended up sharing an office with my thesis advisor. I told him what I was working on, and we quickly determined some very easy to check conditions, which in spite of their almost trivial proofs, appeared to be new. The next week I attended a conference on number theory and polynomials at the Heilbronn Institute in Bristol, and I presented our results there. Nobody there had seen these results before (the audience included many specialists on polynomials, including Andre Schinzel) so we thought the results should be published, and the conference proceedings seemed like a good choice. This resulted in (with Jeffrey D. Vaaler). Self-inversive polynomials with all zeros on the unit circle. London Mathematical Society Lecture Note Series: Number Theory and Polynomials Vol. 352: 312-321 · Editors: James McKee and Chris Smyth · 2008 (refereed). On several occasions, I have been contacted by engineers and scientists in other domains regarding this paper. Unfortunately, our conditions for a polynomial to have all roots on the unit circle, while easy to check, works only for a small proportion of polynomials which actually have all their roots on the unit circle (with the proportion going to 0 as the degree increases). Nonetheless, it is nice to know that a result, even one with an easy proof, may be useful in a wider setting! Not long afterward this paper was submitted, I was invited to a conference in Banff on the intersection of number theory and topology. My research isn't squarely in either of these areas, but as David Boyd was organizing the conference and he was one of my postdoctoral advisors, I got an invitation. Moreover, Kate Petersen was going to be there, and I thought it might be a good opportunity to continue talking about various topics we had previously been looking into. The closest thing to topology I had to talk about was the geometry of the set of degree$N$polynomials with all roots on the unit circle. After my talk, Ted Chinburg asked a question about how the roots of polynomials of low height, chosen from this set of polynomials, behaved. In fact, I don't remember the exact question Ted asked, but Kate and I started brainstorming about the question, which led us, eventually, to the following related question: Are algebraic numbers on the unit circle in an imaginary quadratic extension of$\mathbb Q$equidistributed with respect to the (Weil) height? The answer turns out to be yes. (with Kathleen Petersen). Equidistribution of algebraic numbers of norm one in quadratic number fields; Int. J. Number Theory Vol. 7, Issue 7: 1841-1861, 2011 · doi: 10.1142/S1793042111004666 · arXiv:math.NT/arxiv:1004.0986 In writing up these results, we realized the correct equidistribution question in this context is whether algebraic numbers$\gamma$of norm 1 in a quadratic extension are equidistributed when ordered with respect to the minimal norm of an algebraic integer$\alpha$such that$\gamma = \alpha^2/N(\alpha)$. This ordering is the same as that given by the Weil height, and allowed us to answer the analogous question for real quadratic extensions of$\mathbb{Q}$. We are currently working on extending this sort of equidistribution result to arbitrary number fields. Finally, we get to a paper that looks something like an outlier, in that it doesn't fit squarely into either of the research stories I've told. (In fact, it is somewhat related to both stories, but the connection isn't worth expounding upon here). The paper (with Kevin Hare and David McKinnon). Patterns and periodicity in a family of resultants; J. Théor. Nombres Bordeaux Vol. 21: 215-234, 2009 · doi: 10.5802/jtnb.667 ·download from archive.numdam.org, began in 2004 when I was still a grad student at Texas. I had to that point solved part of my thesis problem, but had yet to stumble across the Pfaffian formula that would eventually throw me in the direction of random matrix theory. When my first job search (with the intention of graduating in 2004) was unsuccessful, I decided that, in order to keep from having to quit mathematics in shame, that I would prove something big. I set my sights on Lehmer's conjecture, which can be very loosely expressed as: there are no 'almost' cyclotomic polynomials ('cyclotomicness' being determined by Mahler measure). Currently, there is a lower bound for the Mahler measure of non-cyclotomic integer polynomials due to Edward Dobrowolski, which basically says that a polynomial which is very nearly, but not, cyclotomic, must have very large degree. The central lemma necessary for this lower bound is as follows: Suppose$f(x)$is a monic irreducible non-cyclotomic degree$N$integer polynomial and$f_m(x)$is the monic integer polynomial whose roots are the$m$th power of those of$f(x)$. Dobrowolski's lemma says that if$p$is a prime, then$p^N$divides the resultant of$f$and$f_p$(and moreover, this resultant is non-zero). One path to the proof of Lehmer's conjecture would be to show that the resultants of$f$and$f_n$are always (or at least often) very large. In fact, all of the resultants of$f_n$and$f_m$for$m,n$up to some bound, figure into Dobrowolski's lower bound, and except for the resultants of$f$and$f_p$, only the trivial lower bound of$\left|\mathrm{Res}(f_m, f_n)\right| \geq 1$is used. Thus, I set out to show that some of the$\mathrm{Res}(f_m, f_n)$were very large. The central result of this paper is that, if$p$is a prime, and$m < n$, then$p^{N(m-1)}$divides the resultant of$f_{p^m}$and$f_{p^n}$. Unfortunately, the prime powers are too rarefied to improve Dobrowolski's lower bound. In order to get a feel for the size of$\mathrm{Res}(f_m, f_n)$, I looked at$p$-adic valuations for (that is the exponent of the largest power of$p$dividing) these resultants for various primes$p$and various integer polynomials. In all cases I looked at, this quantity appeared to be periodic in both$m$and$n\$. When visiting Kevin Hare at the University of Waterloo, I showed him some examples of this phenomenon, and we, together with David McKinnon, established this periodicity.

Suffice it to say that none of this led to a proof of Lehmer's conjecture, though I occasionally return to this problem to see if any new progress can be made. Nonetheless, not getting a job that year ended up being a positive development, in the sense that I redoubled my efforts, and in the end derived the Pfaffian formula that began me down the research path that I am on.

• Plenary Lectures/Colloquia
1. Mathematics in the computer age: exploration and exposition · The 2014 SFU Symposium on Mathematics and Computation · Burnaby, British Columbia · August 2014
2. Mahler measure and 2D electrostatics · Colloquium · Indiana University-Purdue University Indianapolis · January 2014
3. What's the fuss over random matrix theory? · Joint University of Oregon/Oregon State University Colloquium · October 2012
4. Random Matrices, Pfaffian Point Processes and Beyond · Colloquium · Florida State University · October 2010
5. Random Matrix Theory: Asymmetric ensembles and their applications · Department of Pure Mathematics Colloquium · University of Waterloo · Waterloo, Ontario · March 2010
6. The Geometry of Polynomials with all Roots on the Unit Circle · Department of Mathematics Colloquium · Willamette University · Salem, Oregon · November 2009
7. Patterns and Periodicity in a Family of Resultants · Discovery and Experimentation in Number Theory · Plenary Lecture · Fields Institute/IRMACS · Toronto/Vancouver · September 2009
• Invited Lectures
1. TBA · AMS Western Sectional Meeting · Salt Lake City · April 2016
2. Random Polynomials with Bounded Mahler Measure · The Algebra, Analysis and Number Theory of Algebraic Numbers· Banff International Research Station· October 2015
3. Random Polynomials with Bounded Mahler Measure · International Conference on Physics and Number Theory · Insituto Nacional de Matematica Pura e Aplicada, Rio de Janeiro · June 2015
4. Dobrowolski's Lower Bound for Mahler Measure · Number Theory Seminar · Oregon State University · November 2014
5. Equidistribution of Elements of Norm 1 in Number Fields · Number Theory Seminar · Oklahoma State University · October 2014
6. Solvable ensembles of random polynomials · Constructive Functions 2014 · Nashville · May 2014
7. Equidistribution of Elements of Norm 1 in Number Fields · Pacific Northwest Number Theory Conference · Vancouver · May 2014
8. Equidistribution of Elements of Norm 1 in Number Fields · Number Theory Seminar · Oregon State University · October 2013
9. Tate's Thesis · Basic Notions Seminar · University of Oregon · October 2013
10. Dobrowolski's Lower Bound · Growth and Mahler measures in geometry and topology · Mittag-Leffler Institute · Stockholm · July 2013
11. Kernel asymptotics for the Mahler ensemble of random polynomials · 2012 Winter Meeting of the Canadian Mathematical Society; Session on Mathematical Physics - Random Matrices and Integrable Systems · Montréal · December 2012
12. Partition Functions for Multi-component Log Gases · Clifford Analysis Seminar · Universiteit Gent · Gent, Belgium · July 2012
13. Hyperpfaffian point processes when $\beta$ is a square integer? · New Directions Short Course: Advances in Random Matrix Theory · Institute for Mathematics and its Applications · Minneapolis · June 2012
14. Universality of Ensembles of Matrices with Potential Theoretic Weights · XII Latin American Congress of Probability and Mathematical Statistics · Viña del Mar, Chile · March 2012
15. Universality of Ensembles of Matrices with Potential Theoretic Weights · International Symposium in Approximation Theory · Vanderbilt University · May 2011
16. Counting Integer Polynomials with Bounded Mahler Measure · Algebra-Geometry-Combinatorics Seminar · San Francisco State University · November 2010
17. Random Polynomials and Point Processes · Number Theory Seminar · University of Texas · November 2010
18. Random Matrices and Heights of Polynomials · Number Theory Seminar · Florida State University · October 2010
19. Two Charge Ensembles on the Line · Random Matrix Theory Seminar · Mathematical Sciences Research Institute · October 2010
20. Random matrix theory when $\beta$ is an even square · Probability Seminar · University of Colorado · April 2010
21. Random Matrix Theory · Probability Seminar · Oregon State University · November 2009
22. Pfaffian Point Processes and Random Matrices · Northwest Probability Seminar · Seattle · October 2009
23. Real Ginibre: Correlations, Kernels and the Largest Point · Random Matrices, Inverse Spectral Methods and Asymptotics · Banff International Research Station · October 2008
24. Correlation Functions for Ginibre's Real Ensemble · Mathematical Physics Seminar · University of Bristol · June 2008
25. Repulsion of Quadratic Algebraic Numbers on the Unit Circle · The Mathematical Interests of Peter Borwein · Simon Fraser University · Burnaby, British Columbia · May 2008
26. Repulsion of Quadratic Algebraic Numbers on the Unit Circle · Special Session on Diophantine Problems and Discrete Geometry · AMS Western Section Meeting · Claremont McKenna College, California · May 2008
27. The Geometry of Polynomials with all Roots on the Unit Circle · Low Dimensional Topology and Number Theory · Banff International Research Station · October 2007
28. Correlation Functions of Ensembles of Real Asymmetric Matrices · Interactions of Random Matrix Theory, Integrable Systems and Stochastic Processes · 2007 Joint Summer Research Conferences, Snowbird, Utah · June 2007
29. Averages over Ensembles of Real Asymmetric Matrices · California Institute of Technology · December 2006
30. Heights of Polynomials and Random Matrix Theory · Number Theory Seminar · University of Arizona · October 2006
31. Averages over Ginibre's Real Ensemble of Random Matrices · Applied Analysis Seminar · University of Arizona · October 2006
32. Heights of Polynomials and Random Matrix Theory · Number Theory Seminar · University of Waterloo · June 2006
33. Random Matrix Theory and Heights of Polynomials · Conference on Advances in Number Theory and Random Matrix Theory · University of Rochester · June 2006
34. Conjugate Reciprocal Polynomials with all Roots on the Unit Circle · 10th Annual Pacific Northwest Number Theory Conference · Redmond, Washington · February 2006
35. Partition Functions in 2D Electrostatics · The Joint Meetings of the AMS: Mahler Measure and Heights · San Antonio · January 2006
36. Multiplicative Distance Functions on $\mathbb{C}[x]$ · University of York · York, England · October 2003
37. The Distribution of Mahler's Measures of Reciprocal Polynomials · Mahler's Measures of Polynomials Conference · Simon Fraser University · Burnaby, British Columbia · June 2003
38. Heights of Polynomials, Asymptotic Estimates and the Mellin Transform · Mahler's Measures of Polynomials Conference · Simon Fraser University · Burnaby, British Columbia · June 2003
39. The Distribution of Mahler's Measures of Reciprocal Polynomials · The Many Aspects of Mahler's Measure · Banff International Research Station · March 2003
• Selected Other Lectures and Presentations
1. What is Mathematics Really? · Freshman Honors Colloquium · University of Oregon · November 2009
2. Ginibre's Real Ensemble and its Scaling Limits · Random Matrices, Related Topics and Applications · Centre de Recherches Mathématiques, Montréal · August 2008
3. Conjugate Reciprocal Polynomials with all Roots on the Unit Circle · Number Theory and Polynomials · Heilbronn Institute For Mathematical Research · Bristol, England · April 2006
4. Multiplicative Distance Functions · Mahler Measure in Mobile · University of South Alabama · January 2006
5. The Distribution of Mahler's Measure of Reciprocal Polynomials · West Coast Number Theory · San Francisco State University · December 2002
• Nate Wells · Graduate Student · 2014-
• Dongmin Roh · Undergraduate, Departmental Honors Thesis · 2015-2016 (expected)
• Graham Simon · Undergraduate, Honors College · 2015-2016 (expected)
• Zach Chalmers · Undergraduate, Honors College · 2011-2012
• Christopher Shum · Graduate Student · 2009-2013
• Ryo Moore · Undergraduate, Honors College · 2010-2011
• Maxim Yattselev · Postdoc · 2010-2013
• Mathematics Department Committees
• Search (math-bio) 2014-2015
• Moursund/Niven Lecture 2013-2014
• Executive 2010-2011, 2011-2012
• Library 2012-2013
• Teaching Effectiveness 2012-2013
• Postdoc Search 2012-2013
• Calculus 2012-2013
• Search 2010-2011, 2011-2012
• University Committees
• United Academics Executive Committee 2015-2017
• United Academics Organizing Committee 2015-2017
• Intercollegiate Athletics Committee 2015-2017
• Faculty Research Award Committee 2015-2017
• University Library Committee 2013-2015
• University Senate 2011-2013
• Lesbian, Gay, Bisexual & Transgender Concerns Committee 2011-2013
• Rich Higgins, Economics PhD · Spring 2014
• Chris Shum, Mathematics PhD · Spring 2013
• Aaron Montgomery, Mathematics PhD · Spring 2013
• Blair Ahlquist, Mathematics PhD · Summer 2010
• Aaron Heuser, Mathematics PhD · Spring 2010
• Ralph Hutchison, Mathematics MS · Spring 2010
• Journals Refereed for
• Annals of Applied Probability
• Annals of Probability
• Electronic Journal of Probability
• The International Journal of Computer Mathematics
• Journal of Approximation Theory (book review)
• Journal of Differential Geometry and Applications
• Journal of Physics A: Mathematical and Theoretical
• Journal of Statistical Physics
• Probability Theory and Related Fields
• The Rocky Mountain Journal of Mathematics
• The Ramanujan Journal
• Courses Taught
• Stochastic Processes · (UO) Math 684-5 · Fall 2015, Winter 2015.
• Business Calculus I · (UO) Math 241 · Fall 2014
• Statistical Models and Methods · (UO) Math 343 · Spring 2014
• Stochastic Processes · (UO) Math 467/567 · Winter 2014, Winter 2015, Winter 2016
• Introduction to Proof · (UO) Math 307 · Winter 2014
• Business Calculus I · (UO) Math 241 · Fall 2013
• Fourier Analysis on Number Fields · (UO) Math 607 · Winter 2013
• Algebraic Number Theory · (UO) Math 607 · Fall 2012
• Introduction to Analysis I, II · (UO) Math 413/513, Math 414/515 · Fall 2012, Winter 2013
• Statistical Models/Methods · (UO) Math 410/510 · Spring 2012
• Theory of Probability II · (UO) Math 673 · Spring 2012
• Theory of Probability I · (UO) Math 672 · Winter 2012
• Real Analysis I · (UO) Math 616 · Fall 2011
• Mathematical Statistics I · (UO) Math 461/561 · Spring 2011
• Elementary Analysis · (UO) Math 315 · Winter 2010, Spring 2011
• Mathematical Statistics II · (UO) Math 465/565 · Winter 2010, Winter 2011
• Mathematical Statistics I · (UO) Math 464/564 · Fall 2009
• Multivariable Calculus I · (UO) Math 281 · Fall 2009
• Mathematics of Coding and Cryptography · (CU) Math 4440 · Spring 2009
• Complex Analysis (graduate) · (CU) Math 6350 · Fall 2008
• Introduction to Probability · (CU) Math 4510 · Summer, Fall 2008
• Algebraic Number Theory (graduate) · (CU) Math 6180 · Spring 2008
• Introduction to Linear Algebra · (CU) Math 3130 · Fall 2007
• Calculus III · (CU) Math 2400 · Fall 2007
• Calculus I for science and engineering majors · (UBC) Math 100 · Fall 2006
• Calculus II for business majors · (UBC) Math 105 · Summer 2006
• Introduction to Number Theory · (UBC) Math 312 · Fall 2005
• Precalculus · (UT) Math 305G · Fall 2004, Spring 2005
• Emerging Scholars Workshop · (UT) Math 210E · Fall 2002, Spring 2003
• Calculus I for business majors · (UT) Math 403K · Fall 2000, Spring 2002
• Differential and Integral Calculus · (UT) Math 408C · Fall 2001