## Clark Honors College, Faculty in Residence application

When I was last on the academic job market in 2008, I was torn between positions at liberal arts colleges and research universities. I had offers from excellent liberal arts schools, including Claremont McKenna College and Bucknell College, but ultimately decided to come to UO so that I had an opportunity to supervise graduate students. I enjoy supervising undergraduate students as well, and have advised four CHC Honors theses, three departmental Honors theses, and several other undergraduate research/reading projects. Supervising students is my favorite aspect of the job. Beyond the usual reward one finds in sharing knowledge with others, getting to know our varied students—understanding their knowledge and skills, their likes and dislikes, and their dreams for the future—is the major driving force keeping me in academia.

I am applying for a Clark Honors College, Faculty in Residence position so that I can pursue the academic work I love in an environment where it is rewarded.

My research lies at the intersection of number theory, probability and mathematical statistical physics. This is a fascinating genre of mathematics research, with many opportunities for undergraduate research. The connection with physics allows intuition to be brought to bear on mathematical problems, which in turn allows undergraduates to make meaningful contributions to mathematical research—at least in the form of conjectures, and discovery of new phenomena.

I also enjoy reading mathematics broadly, and have experience supervising students on mathematics research that is either outside my educational background or applied to other domains of knowledge.

Besides supervision of research, I am also interested in undergraduate mathematics education, especially for students who may not ultimately pursue a degree in a quantitative/scientific field. Mathematics is simultaneously the language of the universe and a ubiquitous tool in modern life. Mathematics education tends to favor the latter, but it is in the former where the rich beauty of mathematics lies. The aesthetics of mathematics is often invisible to individuals who view it only as a tool. I would like to bring this aesthetic vision of mathematics to undergraduates (and others) who may not otherwise experience the sublime beauty of mathematics.

An example of a seminar I would like to offer would be the *Development of New Numbers*. Such a seminar could trace the history and necessity of new kinds of numbers (natural, integer, rational, algebraic, transcendental, real, complex, etc) as human knowledge has developed. I see such a seminar lying at the intersection of history, philosophy and mathematics, and I would interweave group exercises/projects to motivate the mathematics and inform the necessity (and beauty) of the development of new numbers.

Besides teaching, supervision and research, I also engage heavily in university service. Currently I am the Past President of the University Senate and the President of United Academics, as well as a member of many other committees (including chair of the Core Ed Council). I see some of my current service as fulfillment of certain projects/initiatives started as Senate President. My experience working on core education may be useful in any curricular redesign happening in CHC. While I expect to always be involved in university service, I also expect the level to subside from the current high-water mark. I enjoy the challenge of leadership, but I also wistfully dream of a time when I can fill my days reading, doing math, working with students and doing a *sensible* amount of service, and hopefully earning the rank of full professor.

Finally, I would like to underscore my commitment to the diversification of mathematics (and science more broadly). Much of this problem arises from enculturation of expectations by society at large, but many issues arise from an old guard of mathematicians who propagate racial and gender disparity via preferential treatment for men and microaggression towards others. These attitudes are incongruent with how I view myself as an educator and scholar, and I look forward to working in a unit that values the various backgrounds and experiences of our students, faculty and staff.

## Post-tenure Review Statement

## Research

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

### Projects in Progress

1*The 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.

2*Non-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.

3*Adèlic 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.

4*Pair 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.

5*Martingales 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:

- Implementing A System for the Continuous Improvement and Evaluation of Teaching
- Proposed Changes to Multicultural Requirement
- Resolution denouncing White Supremacy & Hate Speech on Campus
- Proposed Change to Admissions Policies Requiring Disclosure of Criminal and Disciplinary Hearing
- A Resolution in support of LGBTQAI Student Rights
- Declaring UO a Sanctuary Campus
- Reaffirming our Shared Values of Respect for Diversity, Equity, and Inclusion
- Student Sexual and gender-Based Harassment and Violence Complaint and Response Policy

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.

## Diversity and Equity

Notice

This post is part of my post-tenure review. If it seems self-serving, that is because it is.

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 postWomen & Men in Mathematicsfor 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.

## Math Pictures

The set of polynomials of degree 4 with all roots on the unit circle, and inscribed easily described regions.

Large resultants of families of polynomials generated as in Dobrowolski’s Lemma (which leads to the best known lower bound for Mahler measure as a function of degree).

The eigenvalues of a real random asymmetric matrix with iid normal entries.

I’m not sure what this is, but it’s interesting.

The zeros of Wronskians of consecutive Hermite (and other classical orthogonal) polynomials are pretty wild.

Pair correlation in the scaling limit near the real edge in Ginibre’s real ensemble.

## Women & Men in Mathematics

Mathematics has a diversity problem. This problem is far larger than just one of gender representation, but let’s start the conversation there. Only two of the thirty-five tenure-track professors in the Department of Mathematics at the University of Oregon are women. This is not an accident—the structure of the department, those empowered to make decisions, cultural factors (both as a discipline and the backgrounds of our faculty) and out-moded ideas of social gender roles have made our department the image of one from the 1960s (or earlier)!

I have heard tales of women students being told by male UO math professors that they were “good at math, for a girl”. Graduate students often ask faculty for advice, but the advice that “women should not have children during graduate school” is neither appropriate, welcoming, nor welcome. Off-hand musings that “the department was better when the wives [of faculty] organized and hosted social events” while questionably appropriate in 1960, are grossly inappropriate now. These are not hypothetical utterances—all have been said by UO math professors out loud, and in the presence of students.

Policies dictating the matriculation of graduate students through our program have been waived for male students—allowing them to continue unimpeded, having benefited from violating policy—while women in similar situations are summarily dismissed from the program in spite of comparable academic performance, and in spite of them following our policies to the letter. These decisions have been upheld by departmental leadership, even after the obvious structural bias has been pointed out.

In recent years, when reporting incidents of bias to departmental leadership—ostensibly empowered to uphold the institutional commitment to diversity—I have often been asked to ‘tone down’ my rhetoric, and steered toward the word ‘bias’ and away from ‘harassment’ or ‘discrimination.’ There are, apparently, different standards for reporting bias and discrimination, and it was more important to protect faculty from diversity trainings (which was explained to me by the same administrator as a waste of time and incapable of changing behavior) than to attempt to address the problem head-on. I have no doubt that the incidents I brought forward were not reported by those administrators to the appropriate offices on campus, though on several occasions I reported the incidents directly myself.

In faculty searches, committees routinely produce short-lists of candidates for interviews which are less diverse than the pools of applicants. Imagine a pool of 750 applicants (the approximate number we get for any tenure-track search) being winnowed down to 25—only three of which are women. (To be fair, faculty do not have easy access to demographics of individual applicants, so my assessment of these numbers is based on names and pronouns in application materials—not a good way to make such determinations but the best I can do with the information provided me).

If the department wanted to, out of 750 candidates for tenure-track positions, it could easily put forth 25 excellent woman for consideration. But the department does not want to. Indeed, it doesn’t even feel compelled to match the diversity of the candidate pools.

I should be clear, not everyone in the department (or the field) is a problem in this regard. There are excellent, inclusive, well-meaning individuals in the department who are supportive of *all* of our students and colleagues. There are individuals in the department who are working hard on initiatives that support diversity and students/colleagues from different backgrounds. The department has funded, and continues to support a local chapter of the Association of Women in Mathematics. Our graduate students by-and-large, are thoroughly modern when it comes to diversity and engage in more diversity work in total than our tenure-track faculty. To all those individuals, I say “Thank you”.

The problem is cultural. There is an ethos in mathematics that puts disproportionate weight on the opinions of those that are *good at math*. Somehow the expectation is that mathematical production is connected to good decision-making. **It is not**. All mathematicians have at least one bad decision in common, and alacrity in mathematical thinking does not translate to good decision-making on topics that involve tricky, non-idealized social considerations.

When it comes to diversity, the decorated but old-fashioned, stodgy and biased full professor should not be listened to disproportionately, or even at all. Tenure protects the opinions of those individuals, but we do not have to listen to them. Nor should we. Let’s hound them ceaselessly until they relent or we drive them to the dust bin of history.

Tenure is a platform from which to change the world. It is a privileged position that allows people to make unpopular or controversial statements. In a way, it’s an amped-up First Amendment that protects faculty not only from government interference in their speech, but also protects them from university interference in their speech. However, like the First Amendment, tenure does not protect professors from public outrage, nor does it keep the rest of the academic (and general) community from pointing out obvious moral failings in their arguments. If tenure is to survive in the modern academic world, then it is incumbent on us to ensure that those with the privilege are using it to change the world *for good*. We do this by loudly and vociferously speaking out against those holding us back from an excellent, diverse faculty representing the backgrounds and interests of our students.

## The Math Blob Grows!

Click to take a closer look.

## Non-archimedean absolute values and completions

#### Absolute Values

We start by defining absolute values on $\mathbb Q$, but much of this extends naturally to absolute values on other fields. An *absolute value* on $\mathbb Q$ is a function $\mathbb Q \rightarrow [0,\infty)$ such that:

- positive definiteness: $|x| = 0$ iff $x = 0$,
- multiplicativity: $|x y| = |x| |y|$ for all $x, y \in \mathbb Q$
- triangle inequality: $|x + y| \leq |x| + |y|$ for all $x,y \in \mathbb Q$.

If, in place of (3), $|\cdot|$ satisfies the *strong* triangle inequality: $|x + y| \leq \max\{|x|, |y|\}$, then we say $| \cdot |$ is a *non-archimedean* absolute value.

##### Example

Let $p$ be a prime, and define the standard $p$-adic absolute on $\mathbb Q$ by

$$

\left|p^{\ell} \frac{a}b\right| = p^{-\ell}.

$$

Here we assume that $a$ and $b$ are rational integers not divisible by $p$. $| \cdot|_p$ is a non-archimedean absolute value. Note that if $n$ is a rational integer, then $|n| \leq 1$.

#### Completions

Given an absolute value $|\cdot|$ on $\mathbb Q$, we say a sequence $(x_n)$ is *Cauchy* if given $\epsilon > 0$, there exists $N \in \mathbb N$ so that $n, m > N$ implies $|x_n – x_m| < \epsilon$. Similarly, we say that $(x_n)$ converges to zero if given $\epsilon > 0$ there exists $N \in \mathbb N$ so that $n > N$ implies $|x_n| < \epsilon$. Using the standard operations on sequences, the set of Cauchy sequences $\mathcal C$ is a (commutative) ring, and the set of sequences which converge to zero, $\mathcal Z$ is a maximal ideal. It follows that $\mathcal C / \mathcal Z$ is a field, and we call this the completion of $\mathbb Q$ with respect to the absolute value $| \cdot |$. We may embed $\mathbb Q$ into $\mathcal C / \mathcal Z$ by the diagonal embedding $x \mapsto (x,x,\cdots)$. The fact that $\mathbb Q$ is dense in $\mathcal C / \mathcal Z$ allows us to extend $| \cdot |$ to a unique absolute value on $\mathcal C / \mathcal Z$.

The completion of $\mathbb Q$ with respect to the usual absolute value is $\mathbb R$.

The completion of $\mathbb Q$ with respect to $| \cdot |_p$ is called the field of $p$-adic numbers and denoted $\mathbb Q_p$.

#### The Topology of $\mathbb Q_p$

A basic neighborhood in $\mathbb Q_p$ is a set of the form $B(x, \epsilon) = \{y \in \mathbb Q_p : |y – x|_p \leq \epsilon\}$. Since $| \cdot |_p$ takes values in the discrete set $\{p^{-n} : n \in \mathbb Z\}$, every basic neighborhood is both open and closed, since for instance

$$

\{y : |y|_p \leq p^{-n} \} = \{y : |y|_p < 2p^{-n} \}.

$$

Let’s investigate the ball $B(1, p^{-1})$ by looking at what integers it contains. Clearly $1 \in B(1, p^{-1})$. The condition $|y – 1| \leq p^{-1}$ is equivalent to $y \equiv 1 \bmod p$. Similarly, $B(1, p^{-n})$ contains all the integers $y$ such that $y \equiv 1 \bmod p^n$. For instance, when $p = 7$, the integers $1$ and $344$ are close, a distance of $7^{-3}$ from each other, since $344 – 1 = 343 = 7^3$.

For a slightly more complicated ball, consider $B(1/2, p^{-2})$. This is by definition

$$

\{y \in \mathbb Q_p : |y – 1/2|_p \leq p^{-2} \}.

$$

From the definition of $| \cdot |_p$, if $y$ is a rational number so that $|y – 1/2|_p = p^2$, then there exist integers $a$ and $b$, both relatively prime to $p$, so that $y – 1/2 = p^2 a/b$. More generally, if $|y – 1/2|_p \leq p^2$ then there exist integers $a$ and $b$, with $b$ relatively prime to $p$, so that $y – 1/2 = p^2 a/b$. It follows that the rational numbers in $B(1/2, p^{-2})$ are of the form $1/2 + p^2 r$ where $r$ is a rational number which contains no factors of $p$ in the denominator when written in lowest terms.

#### The General Setup

Suppose $K$ is a number field of degree $d$ over $\mathbb Q$ and suppose $\mf O$ is the ring of integers of $K$. Given a prime ideal $\mf p \subseteq \mf O$ we can define an absolute value $| \cdot |_{\mf p}$ by fixing $q > 0$ and setting

\[

|x|_{\mf p} = q^{-n} \qquad \mbox{where} \qquad x \in \mf p^n \setminus \mf p^{n+1}.

\]

The completion of $K$ with respect to $| \cdot |_{\mf p}$ is denoted $K_{\mf p}$.

The ring of integers in $K_{\mf p}$ is defined to be $\mf o_{\mf p} = \{x \in K_{\mf p} : |x|_{\mf p} \leq 1\}$. The ideal $\mf m_{\mf p} \subseteq \mf o_{\mf p}$ given by $\mf m_{\mf p} = \{x \in \mf o_{\mf p} : |x| < 1 \}$. The units in $\mf o_{\mf p}$ are given by $U_{\mf p} := \{x \in \mf o_{\mf p} : |x|_{\mf p} = 1 \}$.

The field $\mf o_{\mf p} / \mf m_{\mf p}$ is a finite field of size $p^f$ for some rational prime $p$ and positive integer $f.$ It is convenient to take $q$ in the definition of $| \cdot |_{\mf p}$ to be $q := p^f.$

Often times, when we are working with a fixed non-archimedean absolute value, we drop the subscript $\mf p$ from various quantities. When in this situation we assume $(K, |\cdot|)$ is a field complete with respect to $|\cdot|$ and with integers $\mf o$, maximal ideal $\mf m$ and units $U$. Unless specifically indicated otherwise we take $q = [\mf o : \mf m]$. The maximal ideal is principal, and we set $\varpi$ to be a generator or *uniformizer* for the ideal: $\mf m = \varpi \mf o$. Clearly then $|\varpi| = 1/q$.

## $p$-adic electrostatics

Suppose $k$ is a local field with ring of integers $\mathfrak o$. Let $\mf m \subseteq \mf o$ denote the unique maximal ideal, and define $q = [\mf o: \mf m]$. The maximal ideal is principal and we fix a generator, or *uniformizer,* $\varpi$ so that $\mf m = \varpi \mf o$. There is a unique absolute value $| \cdot |$ on $k$ with

$$

| \varpi | = \frac{1}{q}.

$$

In this context $\mf o = \{ \alpha : |\alpha| \leq 1\}$ and $\mf m = \{\alpha : |\alpha| < 1\}$. As a locally compact abelian group, $k$ has a Haar measure $\mu$ which can be made unique by specifying $\mu(\mf o) = 1$.

Imagine two charged particles, $p$-adic electrons if you will, identified with $\alpha, \alpha’ \in \mf o$ and whose interaction energy is given by

$$

E(\alpha, \alpha’) = -\log|\alpha’ – \alpha|.

$$

This vaguely corresponds with our intuition as to how electrons ($p$-adic or not) should behave in the sense that the energy is minimized when the distance between $\alpha$ and $\alpha’$ is maximal, and the energy is infinite if the electrons are on top of each other, that is when $\alpha = \alpha’$. Since we have restricted $\alpha$ and $\alpha’$ to $\mf o$, the farthest apart they can be is $| \alpha – \alpha’| = 1$, and in this case the interaction energy is 0. This observation is central in understanding how these particles behave when we introduce thermal fluctuations to the mix.

Moving from $2$ particles to multiple particles, there are three main systems or *ensembles* we will consider.

- The Microcanonical Ensemble
- The system contains $N$ particles at a specified energy $E$.
- The Canonical Ensemble
- The system contains $N$ particles at a specified temperature $T$. Energy can be exchanged with a heat bath, and is now variable.
- The Grand Canonical Ensemble
- The system is now in contact with a heat bath and a particle reservoir so that $E$ and $N$ are both variable. The temperature $T$ and a new quantity,
*the chemical potential*, are fixed and control the average energy and particle number.

### The Microcanonical Ensemble

In this setting, the energy of the $N$ particles located at the coordinates of $\boldsymbol \alpha \in \mf o^N$ is given by

$$

E(\bs \alpha) = -\sum_{m < n} \log|\alpha_n – \alpha_m|.

$$

All states with the same energy are assumed to be equally probable, and the main problem for the micro canonical ensemble is the determination of the measure of the set with prescribed energy.

For reasons that will become apparent, it is sometimes useful to deal with the exponentiated energy

$$

e^{-E(\bs \alpha)} = \prod_{n