MixedMath - explorations in math and programing

Now page

David Lowry-Duda — Tue, 10 May 2022 03:14:15 +0000

What I'm doing Now

Last updated 10 May 2022.

This is a now page. It is a written version of what I might say if we met in person and you asked me what I'm up to. The date at the top is important: it helps the reader determine if this is a now page or a then page.

Research Travel

PCMI 2022 Program: Number Theory Informed by Computation from July 17th to August 3rd.
MathFest from August 3rd to August 6th.
ANTS-XV from August 8th to August 12th.
For the following two weeks, I'll be in various places around the UK.

Other than that, I'm typically in either Boston or Providence before September.

Research

I'm currently actively working on several projects.

I'm working to develop methods to explicitly and rigorously compute Maass forms. I'm currently implementing a rigorous form of Hejhal's algorithm for weight $0$ Maass forms in arb.
At the same time, I'm currently writing down interrelated things about Maass forms, including implementation notes for my computational work, the theory behind rigorous evaluation, and work towards an explicit and computable trace formula for forms of nontrivial character on non-squarefree level.
I'm working on a paper related to half-integral weight modular forms and their computation. This is joint with Mehmet Kiral, Li-Mei Lim, and Thomas Hulse.
With my frequent collaborators Chan Ieong Kuan and Alex Walker, I'm investigating discrete reduction problems for the divisor function.

Teaching

I'm not currently teaching.

Slides from a talk on Improved Bounds for Number Fields

David Lowry-Duda — Mon, 09 May 2022 03:14:15 +0000

At a meeting of the Algebraic Geometry, Number Theory, and Computation Simons Collaboration, I gave a short talk surveying the results and ideas of Improved bounds on number fields of small degree, joint work with Anderson, Gafni, Hughes, Lemke Oliver, Thorne, Wang, and Zhang.

In many ways this is a follow-up to an earlier talk I gave to the Simons Collaboration about work towards a quantitative form of Hilbert's irreducibility theorem. Both of these results grew out of an AIM workshop.

The slides for this talk are available here.

Simplified proofs and reasoning in "Improved Bounds on Number Fields of Small Degree"

David Lowry-Duda — Sat, 30 Apr 2022 03:14:15 +0000

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Paper: Counting number fields of small degree

David Lowry-Duda — Thu, 28 Apr 2022 03:14:15 +0000

Counting number fields of small degree

Recently, my collaborators Theresa C. Anderson, Ayla Gafni, Kevin Hughes, Robert J. Lemke Oliver, Frank Thorne, Jiuya Wang, Ruixiang Zhang, and I uploaded a preprint to the arxiv called "Improved bounds on number fields of small degree". This collaboration is a continuation¹ ¹though with a few different cast members of our previous work on quantitative Hilbert irreducibility, which will appear in IMRN.

In this paper, we improve the upper bound due to Schmidt for estimates on the number of number fields of degree $6 \leq n \leq 94$. Actually, we improve on Schmidt for all $n \geq 6$, but for $n \geq 95$ Lemke Oliver and Thorne have different, better bounds.

Schmidt proved the following.

For $n \geq 6$, there are $\ll X^{(n+2)/4}$ number fields of degree $n$ and having discriminant bounded by $X$.

We prove a polynomial improvement that decays with the degree.

For $n \geq 6$, there are \begin{equation*} \ll_\epsilon X^{\frac{n + 2}{4} - \frac{1}{4n - 4} + \epsilon} \end{equation*} number fields of degree $n$ and having discriminant bounded by $X$.

Towards the end of this project, we learned that Bhargava, Shankar, and Wang were also producing improvements over Schmidt in this range. On the same day that we posted our paper to the arxiv, they posted their paper, in which they prove the following.

For $n \geq 6$, there are \begin{equation*} \ll_\epsilon X^{\frac{n + 2}{4} - \frac{1}{2n - 2} + \frac{1}{2^{2g}(2n-2)} + \epsilon} \end{equation*} number fields of degree $n$ and having discriminant bounded by $X$, where $g = \lfloor \frac{n-1}{2} \rfloor$.

In both our work and in BSW, the broad strategy is based on Schmidt's approach. For a monic polynomial \begin{equation*} f(x) = x^n + c_1 x^{n-1} + \cdots + c_n, \end{equation*} we define the height $H(f)$ to be \begin{equation*} H(f) := \max( \lvert c_i \rvert^{1/i} ). \end{equation*} Then Schmidt showed that to count number fields of discriminant up to $X$, it suffices to count polynomials of height roughly up to $X^{1/(2n - 2)}$.

The challenge is that most of these polynomials cut out number fields of discriminant much larger than $X$. The challenge is then to count relevant polynomials and to identify irrelevant polynomials.

Remarkably, the broad strategy in out work and in BSW for identifying irrelevant polynomials is similar. For a prototypical polynomial $f$ of degree $n$ and of height $X^{1/(2n-2)}$, we should expect the discriminant of $f$ to be approximately $X^{n/2}$. We should also expect the field cut out by $f$ to have discriminant roughly this size. Recalling that we are counting number fields of discriminant only up to $X$, this means that a relevant polynomial of this height must be exceptional in one of two ways:

either the discriminant of $f$ is unusually small, or
the discriminant of the number field cut out by $f$ is much smaller than the discriminant of $f$.

In both out work and in BSW, those $f$ with unusually small discriminant are bounded straightforwardly and lossily.

The heart of the argument is in the latter case. Here, the ratio of the two discriminants is the square of the index $[\mathcal{O}_K : \mathbb{Z}[\alpha]]$, where $\alpha$ is a root of $f$. Thus we bound the number of polynomials whose discriminants have large square divisors.

In establishing bounds for polynomials with particularly squarefull discriminants that our ideas and those in BSW significantly diverge.

In our work, we study the problem locally. That is, we study the behavior of $\psi_{p^{2k}}$, the characteristic function for monic polynomials of degree $n$ over $\mathbb{Z}/p^{2k}\mathbb{Z}$ having discriminant congruent to $0 \bmod p^{2k}$. As in our work on quantitative Hilbert irreducibility, we translate this problem into a sieve problem with local weights coming from Fourier transforms $\widehat{\psi_{p^{2k}}}$ after passing through Poisson summation, and we study the Fourier transforms using a variety of somewhat ad-hoc techniques.

In BSW, they reason differently. They use recent explicit quantitative Hilbert irreducibility work from Castillo and Dietmann to replace the fundamental underlying sieve. To do this, they translate the task of counting relevant polynomials into the task of counting distinguished points in spaces of $n \times n$ symmetric matrices — and then show that Castillo and Dietmann's work bounds these points.

Even though the number field count in BSW is stronger than our number field count, we think that our methods and ideas will have other applications. Further, we've noticed remarkable interactions between local Fourier analysis and discriminants of polynomials.

Comments on this site

David Lowry-Duda — Sat, 02 Apr 2022 03:14:15 +0000

I am on the fourth iteration of a comment system for this site.

Comment Graveyard

Initially I used Wordpress default, which is okay-ish. But there are problems with formatting and writing mathjax-able math in comments.¹ ¹More generally, post-rendering Wordpress content with javascript is a security nightmare and is often hard. This is one reason why this site is no longer using Wordpress.

Then I used Disqus. Disqus works by running externally hosted javascript.² ²Also a security nightmare. As one might worry, they began to inject ads into comment sections. I do not run ads and that was unacceptable. I've learned a lesson about external dependencies.

Then I used a comment system built on top of Wordpress. This was slightly better, but written in PHP.

Simple Comments

The new comment system is email-based. Plain email. I drew inspiration from tdarb and lowtechmagazine (who have precisely the same comment "system").

I know that requiring an email adds a small amount of friction in the comment process. I don't know how this will affect comment spam,³ ³which was wildly common in previous iterations. but I think it might balance out.

Currently, I enable a significant amount of markup in comments. The comments are processed with markdown and allow mathjax (assuming that mathjax is enabled on the page). This is because I use the same preprocessing on comments as I do on pages for this site.

Plaintext Email

David Lowry-Duda — Mon, 28 Mar 2022 03:14:15 +0000

Some email clients and email marketing groups have popularized email usage patters which are considered poor form for developer emails, technical emails, or on mailing lists.

Plaintext Email

Many email clients compose emails with HTML, enabling rich text formatting. Rich text formatting hinders development-oriented email conversations as it can break simple tasks like copy-pasting code snippets.

HTML emails are mainly used for marketing (or to include tracking pixels, i.e. special images hosted on a server that tracks information about the receiver upon loading them). HTML emails are one of the most common vectors for phishing, they're less accessible, and they're viewed inconsistently among receivers.

If you're sending an email, consider preferring plaintext. If you're sending a technical email or an email concerning programming, you should very strongly prefer plaintext.

For more on plaintext email, see useplaintext.email.

Slides from a talk: Computing and verifying Maass forms

David Lowry-Duda — Thu, 24 Mar 2022 03:14:15 +0000

Today, I'm giving a talk on ongoing efforts to compute and verify Maass forms. We describe Maass forms, Hejhal's algorithm, and the idea of how to rigorously improve weak initial estimates to high precision estimates.

The slides for my talk are available here.

Slides from a talk: How computation and experimentation inform research

David Lowry-Duda — Wed, 23 Mar 2022 03:14:15 +0000

Today I'm giving a talk at BYU on computation, experimentation, and research. The first half of the talk examines the historic role of computation and experimentation. Several examples are given. In the second half, I take a more personal angle and begin to describe how I've incorporated computational and experimental ideas into my own work.

The slides from my talk are available here.

colormapplot - like phasematplot, but with colormaps

David Lowry-Duda — Fri, 11 Feb 2022 03:14:15 +0000

I am happy to announce that an enhanced version of phasemagplot is now available, which I refer to as colormapplot. (See also the announcement post for phasemagplot).

This is available at davidlowryduda/phasemagplot on github as a sage library. See the github page and README for examples and description. The docstring from within sage should also be of use.

As a general rule, the interface is designed to mimic the complex plotting interface from sage as closely as possible. The primary difference here is that there is an optional cmap keyword argument. This can be given any matplotlib-compatible colormap, and the resulting image will be given with that colormap.

This is capable of producing colormapped, contoured images such as the following.

Zeros of Dirichlet Series II

David Lowry-Duda — Mon, 24 Jan 2022 03:14:15 +0000

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