MixedMath - explorations in math and programinghttps://davidlowryduda.comDavid's personal blog.en-usCopyright David Lowry-Duda (2022) - All Rights Reserved.admin@davidlowryduda.comadmin@davidlowryduda.comTue, 02 Apr 2024 08:39:10 +0000Tue, 02 Apr 2024 08:39:10 +0000mixedmathapp/generate_rss.py v0.1https://cyber.harvard.edu/rss/rss.htmlhttps://davidlowryduda.com/static/images/favicon-32x32.pngMixedMathhttps://davidlowryduda.comFLT3 at LftCM2024https://davidlowryduda.com/flt3-at-lftcm2024David Lowry-DudaThis 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.https://davidlowryduda.com/flt3-at-lftcm2024Sat, 30 Mar 2024 03:14:15 +0000Quanta on Murmurationshttps://davidlowryduda.com/quanta-on-murmurationsDavid Lowry-Duda<p>Quanta wrote an article <strong>Elliptic Curve 'Murmurations' Found With AI Take Flight</strong> (<a href="https://www.quantamagazine.org/elliptic-curve-murmurations-found-with-ai-take-flight-20240305/">link to article</a>).</p> <p>The article describes some of the story behind the recent <em>murmurations in number theory</em> 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!</p> <p>It touches on <a href="/paper-modular-murmurations/">my recent work</a> with Bober, Booker, and Lee.<span class="aside">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.</span></p> <p>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 (<em>pop-math</em>). It's certainly the case that mathematicians aren't doing a particularly job covering our own work in an accessible way.</p>https://davidlowryduda.com/quanta-on-murmurationsTue, 05 Mar 2024 03:14:15 +0000Not quite 3 and a half yearshttps://davidlowryduda.com/three-years-countingDavid Lowry-Duda<h1>Publishing Record</h1> <p>I submitted a paper on 2 September 2020. It was accepted this week, on 17 February 2024.</p> <p><span class="aside">I'm continuing to include more of the tiny evaluations I do <em>all the time</em> with my computing environments. These may be easy manual calculations &mdash; but I've been known to make simple mistakes.</span></p> <div class="codehilite"><pre><span></span><code><span class="kn">from</span> <span class="nn">datetime</span> <span class="kn">import</span> <span class="n">date</span> <span class="kn">from</span> <span class="nn">dateutil.relativedelta</span> <span class="kn">import</span> <span class="n">relativedelta</span> <span class="n">submitted</span> <span class="o">=</span> <span class="n">date</span><span class="p">(</span><span class="mi">2020</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="n">accepted</span> <span class="o">=</span> <span class="n">date</span><span class="p">(</span><span class="mi">2024</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">17</span><span class="p">)</span> <span class="nb">print</span><span class="p">(</span><span class="n">accepted</span> <span class="o">-</span> <span class="n">submitted</span><span class="p">)</span> <span class="c1"># &gt;&gt; datetime.timedelta(days=1263)</span> <span class="nb">print</span><span class="p">(</span><span class="n">relativedelta</span><span class="p">(</span><span class="n">accepted</span><span class="p">,</span> <span class="n">submitted</span><span class="p">))</span> <span class="c1"># &gt;&gt; relativedelta(years=+3, months=+5, days=+15)</span> </code></pre></div> <p>That was 1263 days, or 3 years, 5 months, and 15 days. It wasn't quite long enough to include a leapday &mdash; it missed by two weeks.</p> <p>This beats my previous record (of 2 years and 3 months). I too am guilty: during the same time, I took a year to review a paper.</p> <p>This is an annoying aspect of academia and publishing.<sup>1</sup> <span class="aside"><sup>1</sup>At least this isn't one of those stoires where years pass and then the paper is <strong>rejected</strong>. I haven't had that happen, and I haven't done this as a reviewer. But it <em>does</em> happen too.</span></p> <p>I don't think I kept enough records to track my average time from submission to publication. Perhaps I should keep better track and report this on my <a href="/research/">Research</a> page too.</p>https://davidlowryduda.com/three-years-countingWed, 21 Feb 2024 03:14:15 +0000Examining Excess in the Schmidt Boundhttps://davidlowryduda.com/schmidt-experimentDavid Lowry-DudaThis 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.https://davidlowryduda.com/schmidt-experimentTue, 13 Feb 2024 03:14:15 +0000Bounds on partial sums from functional equationshttps://davidlowryduda.com/bounds-on-partial-sums-from-feDavid Lowry-DudaThis 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.https://davidlowryduda.com/bounds-on-partial-sums-from-feTue, 16 Jan 2024 03:14:15 +0000Paper: Towards a Classification of Isolated $j$-invariantshttps://davidlowryduda.com/paper-isolated-j-invariantsDavid Lowry-Duda<p>I'm happy to announce that a new paper, "Towards a classification of isolated $j$-invariants", now appears <a href="https://arxiv.org/abs/2311.07740">on the arxiv</a>. This was done with my collaborators Abbey Bourdon, Sachi Hashimoto, Timo Keller, Zev Klagsbrun, Travis Morrison, Filip Najman, and Himanshu Shukla.</p> <p>There are many collaborators because this was borne out of a workshop at CIRM from earlier this year, and we all attacked this problem together. This is the first time I have collaborated with any of these collaborators (though several of us are now involved in another project that will eventually appear).</p> <p>The modular curve $X_1(N)$ is moduli space and an algebraic curve (defined over $\mathbb{Q}$ for us) whose points parametrize elliptic curves with a point of order $N$. We study <em>isolated</em> points, which are morally points on $X_1(N)$ that don't come from infinite families.</p> <p>Perhaps the simplest form of infinite families comes come from a rational map $f: X_1(N) \longrightarrow \mathbb{P}^1$ of degree $d$. By Hilbert's irreducibility theorem, $f^{-1}(\mathbb{P}^1(\mathbb{Q}))$ contains infinitely many closed points of degree $d$.</p> <p>Similarly, to any closed point $x$ of degree $d$ one can associate the rational divisor $P_1 + \cdots + P_d$, where $P_j$ are the points in the Galois orbit associated to $x$. This gives a natural map $\Phi_d: X_1(N)^{(d)} \longrightarrow \mathrm{Jac}(X_1(N))$. If $\Phi_d(x) = \Phi_d(y)$ for some point $y$, one can show that there exists a nonconstant function $f : X_1(N) \longrightarrow \mathbb{P}^1$ of degree $d$ again. Thus positive rank abelian subvarieties of the Jacobian also give infinite families of points.</p> <p>Roughly, we say that a closed point $x$ is <strong>isolated</strong> if it doesn't come from either of the two constructions ($\mathbb{P}^1$, which we call $\mathbb{P}^1$ isolated &mdash; and abelian subvariety, which we call AV-isolated) above. Further, a closed point $x$ of degree $d$ is called <strong>sporadic</strong> if there are only finitely many closed points of degree at most $d$.</p> <p>If $x \in X_1(N)$ is an isolated point, we say $j(x) \in X_1(1) \cong \mathbb{P}^1$ is an <strong>isolated $j$-invariant</strong>. In this paper, we seek to answer a question of Bourdon, Ejder, Liu, Odumodo, and Viray.</p> <blockquote> <p>Question: Can one explicitly identify the (likely finite) set of isolated $j$-invariants in $\mathbb{Q}$?</p> </blockquote> <p>Our main result is decision algorithm. Given a $j$-invariant, our algorithm produces a finite list of potential (level, degree) pairs such that one only needs to verify that degree $\mathrm{degree}$ points on $X_1(\mathrm{level})$ are not isolated.</p> <p>Stated differently, our algorithm has one-sided error. It either reports that an element is not isolated, or it reports that it <em>might</em> be isolated and gives a list of places containing the data where isolated points must come from.</p> <p>In principle, this sounds like it might be insufficient. But we ran our algorithm on every elliptic curve in the LMFDB and the outputs of our algorithm are always the empty set &mdash; except for $4$ exceptions where we know the $j$-invariants are isolated.</p> <p>Concretely, we know that for any non-CM elliptic curve over $\mathbb{Q}$ with an isolated $j(E) \in \mathbb{Q}$ and with</p> <ul> <li>conductor up to $500000$,</li> <li>or with conductor that is $7$-smooth,</li> <li>or of prime conductor $p < 3 \cdot 10^8$,</li> </ul> <p>then $j(E) \in \{ -140625/8, -9317, 351/4, -162677523113838677 \}$. These latter correspond to $\mathbb{P}^1$ isolated points on $X_1(21), X_1(37), X_1(28)$, and $X_1(37)$ (respectively).<sup>1</sup> <span class="aside"><sup>1</sup>An appendix to our paper by Derickx and Mark van Hoeij shows that the last $j$-invariant is actually isolated, not merely sporadic.</span></p> <p>More generally, we are led to conjecture that these (and the CM $j$-invariants) are all of the isolated points on $X_1(N)$.</p> <p>Broadly, our algorithm works by first considering the Galois image of elliptic curves (thanks to the code of David Zywina and the efforts of David Roe and others for making this broadly accessible). An earlier result of Bourdon and her collaborators<sup>2</sup> <span class="aside"><sup>2</sup>Abbey proposed this problem at the workshop based on the observation that her result might make things tenable. She was right!</span></p> <p>allows one to radically narrow the focus of the Galois image to a minimal set of points of interest. We do this narrowing. We also show that no elliptic curve with adelic image of genus $0$ gives an isolated point, which allows us to ignore many potential leafs of computation.</p> <p>The details are interesting and we try to be as explicit as possible. The code for this project is also available, and can be found on my github at <a href="https://github.com/davidlowryduda/isolated_points">github.com/davidlowryduda/isolated_points</a>.</p>https://davidlowryduda.com/paper-isolated-j-invariantsWed, 15 Nov 2023 03:14:15 +0000Paper: Congruent number triangles with the same hypotenusehttps://davidlowryduda.com/paper-congruent-triangles-same-hypotenuseDavid Lowry-DudaThis 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.https://davidlowryduda.com/paper-congruent-triangles-same-hypotenuseTue, 14 Nov 2023 03:14:15 +0000Bringing back the blogrollhttps://davidlowryduda.com/bringing-back-the-blogrollDavid Lowry-Duda<p>It used to be <em>very</em> common for sites to have a page that said "blogroll" at the top and it linked to a bunch of blogs and sites that the author liked. People used to find other sites like this, and these sorts of links powered early search engines.</p> <p>I stopped having a blogroll after I switched away from Wordpress and didn't think too much about it.</p> <p>But things changed. RSS worked<sup>1</sup> <span class="aside"><sup>1</sup>RIP Google Reader, 2013. It died so that Google+ could... also die a quiet death.</span> and google search/Twitter were semi-reliable ways to find new things.</p> <p>Now Twitter is dead, email newsletters are awful<sup>2</sup> <span class="aside"><sup>2</sup>though almost all of them were almost always awful. The medium is inferior to RSS in almost every way except a very important one: it's easier to monetize.</span> , and Google search has lost its utility to the combination of earnest actors paying top dollar to appear at the top and SEO optimzer spam flooding the web.<sup>3</sup> <span class="aside"><sup>3</sup>And <a href="https://en.wikipedia.org/wiki/Goodhart%27s_law">Goodhart's Law</a> strikes again!</span></p> <p>The web is full of walled gardens, hiding the beautiful things within behind large, bland, stone walls.</p> <p>The fundamental problem of internet content discovery is hard again! It's now rather hard to find places with consistently good content. I love longform media. It requires thought and intention. And it requires time.</p> <p>So I'm bringing back my blogroll (and more generally, a list of sites, links, resources, and books that I find interesting). It is at the <a href="/blogroll">top of every page</a>. Check it out!</p> <p>If you have a site, bring out your blogroll. Help the small web flourish.</p>https://davidlowryduda.com/bringing-back-the-blogrollMon, 13 Nov 2023 03:14:15 +0000Blogroll (and interesting links)https://davidlowryduda.com/blogrollDavid Lowry-DudaThis 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.https://davidlowryduda.com/blogrollWed, 01 Nov 2023 03:14:15 +0000Paper: Murmurations of modular forms in the weight aspecthttps://davidlowryduda.com/paper-modular-murmurationsDavid Lowry-DudaThis 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.https://davidlowryduda.com/paper-modular-murmurationsSat, 21 Oct 2023 03:14:15 +0000