MixedMath - explorations in math and programing

code4math research community: call for contributors

David Lowry-Duda — Fri, 18 Oct 2024 03:14:15 +0000

The code4math community is organizing some events for next year. One of these is a monthly community call and presentation (more on that later).

We are also currently in the process of proposing an AIM Research Community.

If this is something that you think you would like to be involved in but currently aren't, let me or one of the other organizers know.

Fourier expansions at different cusps

David Lowry-Duda — Fri, 27 Sep 2024 03:14:15 +0000

This post is larger than 10000 bytes, which is above the limit for this RSS feed. Perhaps it is long or has embedded images or code. Please view it directly at the url.

Explicit equations for cubic surfaces

David Lowry-Duda — Fri, 20 Sep 2024 03:14:15 +0000

This post is larger than 10000 bytes, which is above the limit for this RSS feed. Perhaps it is long or has embedded images or code. Please view it directly at the url.

Notes

David Lowry-Duda — Tue, 10 Sep 2024 03:14:15 +0000

The notes below are approximately sorted into areas.

Notes on Number Theory

Explicit equations for cubic surfaces, 2024.
The note
discussion page
Some details on Hecke algebras, 2024.
The note
discussion page
Bounds on partial sums from functional equations, 2024.
The note
discussion page
Computing Petersson inner products, 2023.
The note
Visualizing Modular Curves, a short technical note on implementation details for modular curve visualizations, 2022.
The note
discussion page
A note on Gaussian Integers, and a note on Gaussian primes, supplements for an elementary number theory course.

Hecke Algebra details

David Lowry-Duda — Tue, 10 Sep 2024 03:14:15 +0000

This post is larger than 10000 bytes, which is above the limit for this RSS feed. Perhaps it is long or has embedded images or code. Please view it directly at the url.

A note on sums over primes

David Lowry-Duda — Wed, 24 Jul 2024 03:14:15 +0000

This post is larger than 10000 bytes, which is above the limit for this RSS feed. Perhaps it is long or has embedded images or code. Please view it directly at the url.

Maass forms in the LMFDB

David Lowry-Duda — Wed, 05 Jun 2024 03:14:15 +0000

Rigorous Maass forms are now in the $L$-functions and modular forms database LMFDB. This is something I've been working on for a while, and it's nice to actually make the data available. This naturally follows certain talks I've given before, include this one and this one, as well as various chalk talks over the last couple fo years.

The computations of the actual Maass forms uses work of Kieran Child, Andrei Seymour-Howell, and me. A variety of underlying techniques are used, including rigorous implementations of the Selberg trace formula, rigorous Hejhal's algorithm, and certification strategies.¹ ¹See Certification of Maass cuso forms of arbitrary level and character by Child and Dimensions of the spaces of cusp forms and newforms on $\Gamma_0(N)$ and $\Gamma_1(N)$ by Booker, Strömbergsson, and Venkatesh. I haven't talked much about these before. And this all builds on the earlier heuristic approaches of Hejhal, later refined significantly by Strömberg, Lemurell, and Then. (And all data for $\mathrm{GL}(3)$ or $\mathrm{GL}(4)$ forms comes from completely different techniques of Farmer, Lemurell, and others).

For now, the Maass forms are on the LMFDB Beta. If nothing breaks (and I hope it won't), then they'll soon be on the main LMFDB too.

For the rest of this note, I want to describe a couple interesting facets of the newly available database.

Consecutive Maass Forms

One of our promises in the LMFDB is to not miss any Maass forms. For any given level, we have every Maass form with spectral parameter $0 < r < R$ for some $R$ (that currently depends strongly on the level). This guarantee comes largely from the trace formula: we can guarantee that we haven't missed any forms.² ²Or rather, any missing forms come from an implementation mistake, not a theoretical mistake.

Thus when you look at all Maass forms in some data range, we know that they are all there. This should be a big help with experimentation.

Labels

One advantage of having consecutive Maass forms is that we can label Maass forms in a systematic way. For a (cuspidal, Hecke, new) Maass form on $M_k(\Gamma_0(N), \chi)$, we assign it the label N.k.a.m.d, where

$N$ is the level,
$k$ is the weight,
$N.a$ is the Conrey label of the Dirichlet character $\chi$,
$m$ is the spectral index, and
$d$ is a disambiguation index if multiple Maass forms share the same earlier data. (This currently never happens).

The spectral index is the index of the spectral parameter $R$ in the list of all spectral parameters for a given weight, level, and character, starting from $1$ if $R > 0$. The special case when the spectral index $m = 0$ is reserved for Maass forms induced from Hecke characters. These are not currently in the LMFDB, but they will be at some point in the future.

In addition, if $k = 0$, $a = 1$, and $d = 1$, (which accounts for every Maass form currently in the LMFDB), then we assign the short label N.m.

Labels in the LMFDB

Labels can be a big deal in the LMFDB. We don't like changing labels, ever. Once we give a form a name, we want that name to never change. This means that there can be big label discussions.

This time I tried designing what I thought was a robust label format, implementing it, and then asking for confirmation. This worked well! David Roe had the idea of adding the short label, which makes things look nicer.

Plots of Maass Forms

With new objects in the LMFDB, there comes new portraits.

These are similar to the portraits we made for classical modular forms. But the Maass forms we currently have are all real-valued, and thus only take on two separate hues. (And unlike the classical modular forms, we included hints at contours to indicate a bit more).

I overengineered portrait creation. I actually have a lot to say about it, and I'll write that down another time.

Check out the Maass Forms

So go forth and check them out! Right now they're on beta. If you manage to break anything, let me know and I'll fix it.

Happy mathing!

Another year, another TeXLive reinstallation

David Lowry-Duda — Mon, 15 Apr 2024 03:14:15 +0000

Every year, TeX Live updates in a breaking way. This year it was on 13 March 2024.

I don't notice until I need to do something somewhat uncommon with my TeX distribution, such as compiling a new template or style file. Today, I'm writing a funding proposal and was (re)compiling a similar proposal I wrote a couple of years ago. It happens to use the extdash package, which I don't have installed. But trying to install it now (via tlmgr) throws an error saying that my distribution is 2023, and now it's 2024, so it's time to upgrade.

LaTeX distributions come in two forms: the main binaries and packages. Packages are updated continually and can be updated throughout the year. The main binaries are updated once each year. But once a binary is updated, the package manager (associated to a previous binary) will no longer allow package updates.

The effect is that a new texlive installation is required each year. There are some update scripts, but these are unsupported and still essentially require downloading as much material as a full install.

I haven't heard a compelling reason why this behavior is tolerated (let alone desirable). Further, following instructions typically leads with several side-by-side installations of tex (although each installation is HUGE and this is also unacceptable).

The only benefit of this is that keeping older versions allows snapshot recompilation of older documents, or snapshot recompilation using packages with backwards incompatible updates (such as moderncv, which breaks all the time). But I don't like having that bloat, so I end up completely removing old versions and installing a fresh texlive each year.

I sometimes want bleeding-edge latex things and don't use my linux distribution's package manager for this. Instead, I do it myself through the texlive package manager tlmgr. These are my notes on updating texlive from year to year.

(In particular, these are the steps that I went through just now so that I can compile what I really wanted to compile).

Remove the previous installation

I keep the texlive source in $HOME/src/texlive.¹ ¹You can probably guess where I keep source files for other things that I build and manage myself). Looking now, I see that I have 598832 bytes of stuff there, meaning that the complete size of my texlive distribution for all latex that I've compiled in the last year is approximately 585MB. This is about 1/10th the size of the standard TeXLive Full installation on Ubuntu, which is one of the reasons why I prefer to manage the installation myself.² ²I began to use tlmgr to manage texlive when I was using a chromebook as my main driver, and I needed to be able to fit texlive on my tiny harddrive. I learned a lot about resource constraints then.

I note for interest that 261MB of my texlive is dedicated to documentation for installed packages. Another 96MB is for fonts.

Regardless, I remove the entirety of my $HOME/src/texlive at once.

Acquire and run the new installer

Go to tug.org and download the texlive internet installer. This year, this is a 5.5MB file install-tl-unx.tar.gz .

Move it to a freshly created $HOME/src/texlive and unpack it.

cd into the new directory and run the installer (which is a little perl script).

I now customize many of the options. For people who read this that aren't me, I emphasize that my usecase might not be the same as your usecase. Now is an obvious time to deviate from my procedure.

Verify that it detects the correct platform, GNU/Linux on x86_64 for me.
Change the installation scheme from scheme-full (which takes 8.253GB this year) to custom .
Go into the Collections submenu. I default to having too little and then install things later with tlmgr as necessary later. Thus I first deselect all, and then I install exactly three collections: Essential programs and files, LaTeX fundamental packages, and XeTeX and packages.
Set the installation directories. I use $HOME/src/texlive and I do not separate by year. I also set TEXMFHOME to ~/.texmf (i.e. I change it to be a hidden file instead of polluting my home directory visibly).
I set tex to use letter size instead of A4 by default, because I live in the US. I note that I keep the "install font/macro doc tree" option, which downloads documentation and which ultimately doubles the size of my installation. I actually read the documentation sometimes. I think this is unusual.
Set the installation to proceed.

This year, this apparently uses 557MB of disk space.

This led to texlive installing 181 packages (and their commented source and documentation) and took less than 10 minutes.

Afterwards, the installer will display a very important message about setting MANPATH, INFOPATH, and PATH. In principle I would alter these in my .bashrc, but in practice my previous .bashrc points to these new spots since I'm overwriting where my texlive installation directory.

Remove texlive installer

TeXLive is now installed, so I can remove $HOME/src/texlive/. The exact name is different, depending on the date.

Install utility packages

I use various utilities frequently. I install these with

which tlmgr  # to make sure that new tlmgr is detected
tlmgr install latexmk lacheck chktex latexdiff pdfcrop \
              pdfjam texdiff texdoc

These are utility packages that are less common. I use lacheck and chktex for latex linting. I use texdoc to see documentation for packages. I use latexmk to handle recompilation. I use the others for various scripts I've written. All of these are tiny.

Installation took 31 seconds.

Install specific packages I use

I have four fundamental types of papers that I often compile.

A generic research paper that I might post to the arxiv.
Personal notes
Public notes
Beamer presentations

I have different templates and packages that I use for each of these. I know that I'll make all four again this year and I know exactly which packages they need. I install those now.

tlmgr install setspace mathtools booktabs wrapfig changebar \
              xcolor lipsum tocloft fancyvrb enumitem threeparttable \
              beamer beamertheme-metropolis adjustbox pgfopts \
              xkeyval collectbox times minted ragged2e multirow

I made this list a couple of years ago by checking what was necessary to compile a couple specific documents.

I install any other package as necessary throughout the year. Right now, my installation uses 706MB.³ ³Wow, that's 150 MB more than my entire distribution last year, including various packages installed throughout the year for ad hoc reasons! I wonder what changed so much. I see that 334MB, which is 73MB more than before.

FLT3 at LftCM2024

David Lowry-Duda — Sat, 30 Mar 2024 03:14:15 +0000

This post is larger than 10000 bytes, which is above the limit for this RSS feed. Perhaps it is long or has embedded images or code. Please view it directly at the url.

Quanta on Murmurations

David Lowry-Duda — Tue, 05 Mar 2024 03:14:15 +0000

Quanta wrote an article Elliptic Curve 'Murmurations' Found With AI Take Flight (link to article).

The article describes some of the story behind the recent murmurations in number theory phenomena that I've been giving talks about. I think it's a pretty well-written article that gives a reasonable overview. Check it out!

It touches on my recent work with Bober, Booker, and Lee.If they'd waited a couple of weeks, then they might have been able to include forthcoming work with Booker, Lee, Seymour-Howell, and Zubrilina! But we're a couple of weeks away for that, I think.

And as far as I see, Quanta is the only outlet (in English) that covers recent research developments in math for a non-specialist audience (pop-math). It's certainly the case that mathematicians aren't doing a particularly job covering our own work in an accessible way.