Math Pictures

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

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.


Post-tenure Reviews

As I wrote in Women & Men in Mathematics, tenure is a platform to change the world. I encourage all newly-tenured individuals to pause and reflect on what is important to them—to take a break from the academic conveyer belt, assess their skills and passions, and determine for themselves how best to change the world.

For many, this will involve a short pause to recoup from the trauma of hoop-jumping, before hopping back on the academic conveyer. Others may reflect and decide that their passion lies in one of the other official job-duties of tenured professors: teaching and service. Still others may decide to jump fields, angle for an administrative position or use their accumulated academic and political capital to work on social issues. Academics are legion, and there are lots of ways to change the world!

Post-tenure reviews work great for those happy to hop back on the academic conveyer belt, and those whose passion truly lies in their scholarship. But what is their use for individuals who want to reinvent themselves? How do we reconcile the narrow standards put forward by department heads and associate deans, with the lofty dreams and ideals of those who have moved beyond them? How does a university harness the passion and energy of these diverted academics without relegating them to the insulting, false category of dead wood?

I do not have an answer for these questions, except to say that the answer does not lie a hastily written summary of activity ending in the words Meets Expectations, Exceeds Expectations or the dreaded Does Not Meet Expectations.

The diverted academic has transcended your expectations, and any assessment based on narrow, metrics-driven criteria is more apt to drive them farther from the academic milieu than bring them back to the fold.

One possible path forward is to use a post-tenure review to identify all faculty activities that support the university mission (hint: that’s most of them) and to determine what resources are available to support that work.

Occasionally, though I suspect much less frequently than associate deans would have you believe, individuals are not fulfilling the required parts of the job. Independent of their dreams and ideals, most tenured faculty still have to teach and sit on committees. This is real work, and it should be done well. For those individuals falling down on this portion of their job, a suggested course correction is necessary and appropriate.

I understand the motivation for post-tenure reviews. I just wish they reflected the diversity of opinions on what it means to be an academic in the modern age. Instead, as currently envisioned, they drive us to the least-common-denominator of expectations as pushed by metrics-chasing administrators more worried about rankings than actually changing the world for good.


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.


My calendar

My schedule is always changing, but this will give you an idea of my availability. Email me if you would like to set up an appointment. This information is incomplete as I also have events on other shared calendars not displayed here.

The main calendar for United Academics (with event details hidden) is below. I am obligate to participate in many, but not all, of the event on the calendar.


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:

  1. positive definiteness: $|x| = 0$ iff $x = 0$,
  2. multiplicativity: $|x y| = |x| |y|$ for all $x, y \in \mathbb Q$
  3. 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.


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 $|n| \leq 1$.


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