Research

I study the distribution of algebraic numbers, mathematical statistical physics and roots/eigenvalues of random polynomials/matrices. 

Projects in Progress

1The distribution of values of the non-archimedean absolute Vandermonde determinant and the non-archimedean Selberg integral (with Jeff Vaaler). The Mellin transform of the distribution function of the non-archimedean absolute Vandermonde (on the ring of integers of a local field) is related to a non-archimedean analog of the Selberg/Mehta integral. A recursion for this integral allows us to find an analytic continuation to a rational function on a cylindrical Riemann surface. Information about the poles of this rational function allow us to draw conclusions about the range of values of the non-archimedean absolute Vandermonde.

2Non-archimedean electrostatics. The study of charged particles in a non-archimedean local field whose interaction energy is proportional to the log of the distance between particles, at fixed coldness $\beta$. The microcanonical, canonical and grand canonical ensembles are considered, and the partition function is related to the non-archimedean Selberg integral considered in 1. Probabilities of cylinder sets are explicitly computable in both the canonical and grand canonical ensembles.

3Adèlic electrostatics and global zeta functions (with Joe Webster). The non-archimedean Selberg integral/canonical partition function are examples of Igusa zeta functions, and as such local Euler factors in a global zeta function. This global zeta function (the exact definition of which is yet to be determined) is also the partition function for a canonical electrostatic ensemble defined on the adèles of a number field. The archimedean local factors relate to the ordinary Selberg integral, the Mehta integral, and the partition function for the complex asymmetric $\beta$ ensemble. The dream would be a functional equation for the global zeta function via Fourier analysis on the idèles, though any analytic continuation would tell us something about the distribution of energies in the adèlic ensemble.

4Pair correlation in circular ensembles when $\beta$ is an even square integer (with Nate Wells and Elisha Hulbert). This can be expressed in terms of a form in a grading of an exterior algebra, the coefficients of which are products of Vandermonde determinants in integers. Hopefully an understanding of the asymptotics of these coefficients will lead to scaling limits for the pair correlation function for an infinite family of coldnesses via hyperpfaffian/Berezin integral techniques. This would partially generalize the Pfaffian point process arising in COE and CSE. There is a lot of work to do, but there is hope.

5Martingales in the Weil height Banach space (with Nathan Hunter). Allcock and Vaaler produce a Banach space in which $\overline{\mathbb Q}^{\times}/\mathrm{Tor}$ embeds densely in a co-dimension 1 subspace, the (Banach space) norm of which extends the logarithmic Weil height. Field extensions of the maximal abelian extension of $\mathbb Q$ correspond to $\sigma$-algebras, and towers of fields to filtrations. Elements in the Banach space (including those from $\overline{\mathbb Q}^{\times}/\mathrm{Tor}$) represent random variables, and the set up is ready for someone to come along and use martingale techniques—including the optional stopping time theorem—to tell us something about algebraic numbers.

Instruction

I have three current PhD students and one current departmental Honors student. I have supervised two completed PhDs and six completed honors theses. You can find a list of current and completed PhD and honors students on my CV.

My teaching load has been reduced for the last five years (or so) due to an FTE release for serving on the Executive Council of United Academics. As President of United Academics, and Immediate Past President of the University Senate I am not teaching in the 2018 academic year. In AY2019, I am scheduled to teach a two-quarter sequence on mathematical statistical physics.

I take my teaching seriously. I prepare detailed lecture notes for most courses (exceptions being introductory courses, where my notes are better characterized as well-organized outlines). When practical and appropriate I use active learning techniques, mostly through supervised group work. I am a tough, but fair grader.

Service

Service encompasses pretty much everything that an academic does outside of teaching and research. This includes advising, serving on university and departmental committees, reviewing papers, writing letters of recommendation, organizing seminars and conferences, serving on professional boards, etc. The impossibility of doing it all allows academics to decide what types of service they are going specialize based on their interests and abilities.

I have spent the last three years heavily engaged in university level service. I currently serve as the president of United Academics of the University of Oregon, and I am the immediate-past president of the University Senate. Before that I was the Vice President of the Senate and the chair of the Committee on Committees. All of these roles are difficult and require a large investment of thought and energy. The reward for this hard work is a good understanding of how the university works, who to go to when issues need resolution, and who can be safely ignored.

I know what academic initiatives are underway, being involved in several of them. I am spearheading, with the new Core Education Council, the reform of general education at UO. I am working on the New Faculty Success Program—an onboarding program for new faculty—with the Office of the Provost and United Academics. I am currently on the Faculty Salary Equity Committee and its Executive Committee. I have been a bit player in many other projects and initiatives including student evaluation reform, the re-envisioning of the undergraduate multicultural requirement, and the creation of an expedited tenure process to allow the institution alacrity when recruiting imminent scholars. This list is incomplete.

Next year, with high probability, I will be the chair of the bargaining committee for the next collective bargaining agreement between United Academics and the University of Oregon (this assumes I am elected UA president). I will also be working with the Core Ed Council to potentially redefine the BA/BS distinction, with a personal focus on ensuring quantitative/data/information literacy is distributed throughout our undergraduate curriculum. I will also be working to help pilot (and hopefully scale) the Core Ed “Runways” (themed, cohorted clusters of gen ed courses) with the aspirational goal of having 100% of traditional undergraduates in a high-support, high-engagement, uniquely-Oregon first-year experience within the next 3-5 years.

As important as the service I am doing, is the service I am not doing. I do little to no departmental service (though part of this derives from the CAS dean’s interpretation of the CBA) and I avoid non-required departmental functions (for reasons). I do routinely serve on academic committees for graduate/honors students, etc. I decline most requests to referee papers/grants applications, and serve on no editorial boards. The national organizations for which I am an officer are not mathematical organizations, but rather organizations dedicated to shared governance.

Diversity & Equity

The two principles which drive my professional work are truth and fairness.

I remember after a particularly troubling departmental vote, a senior colleague attempted to assuage my anger at the department by explaining that “the world is not fair.” I hate this argument because it removes responsibility from those participating in such decisions, and places blame instead on a stochastic universe. And, while there is stochasticity in the universe, we should be working toward ameliorating inequities caused by chance, and in instances where we have agency, making decisions which do not compound them.

I do not think the department does a very good job at recognizing nor ameliorating inequities. Indeed, there are individuals, policies and procedures that negatively impact diversity. See my recent post Women & Men in Mathematics for examples.

My work on diversity and equity issues has been primarily through the University Senate and United Academics. As Vice-president of the UO Senate, I sat on the committee which vetted the Diversity Action Plans of academic units. I also worked on, or presided over several motions put forth by the University Senate which address equity, diversity and inclusion. Obviously, the work of the Senate involves many people, and in many instances I played only a bit part, but nonetheless I am proud to have supported/negotiated/presided over the following motions which have addressed diversity and equity issues on campus:

Besides my work with the Senate, I have also participated in diversity activities through my role(s) with United Academics of the University of Oregon. United Academics supports both a Faculty of Color and LGBTQ* Caucus which help identify barriers and propose solutions to problems affecting those communities on campus. United Academics bargained a tenure-track faculty equity study, and I am currently serving on a university committee identifying salary inequities based on protected class and proposing remedies for them.

I have attended in innumerable rallies supporting social justice, and marched in countless marches. I flew to Washington D.C. to attend the March for Science. I’ve participated in workshops and trainings on diversity provided by the American Federation of Teachers, and the American Association of University Professors.

I recognize that I am not perfect. I cannot represent all communities nor emulate the diversity of thought on campus. I have occasionally used out-moded words and am generally terrible at using preferred pronouns (though I try). I recognize my short-comings and continually work to address them.

There are different tactics for turning advocacy into action, and individuals may disagree on their appropriateness and if/when escalation is called for. My general outlook is to work within a system to address inequities until it becomes clear that change is impossible from within. In such instances, if the moral imperative for change is sufficient then I work for change from without. This is my current strategy when tackling departmental diversity issues; I work with administrative units, the Senate and the union to put forth/support policies which minimize bias, discrimination and caprice in departmental decisions. I ensure that appropriate administrators know when I feel the department has fallen down on our institutional commitment to diversity, and I report incidents of bias, discrimination and harassment to the appropriate institutional offices (subject to the policy on Student Directed Reporters).

Fairness is as important to me as truth, and I look forward to the day where I can focus more of my time uncovering the latter instead of continually battling for the former.

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