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  <h2>Dr. Professor Anthony Dukan "Big Tony D" Dender, B.A. B.A. MASt M.A. Ph.D.</h2>

  <p><big><br>
  <b>Office:</b> <a href="#" class="popup-link" onclick="openPopup('house')">High Regency of Swirles Court upon Wellington</a>, Flat 422<br>
  <b>Email:</b> <span id="email"></span><br>
  <b>Telephone:</b> <span id="phone"></span></big><br>
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  <p>&nbsp;
  <big><br></big></p>

  <p>
  <p class="about-blurb">
  First Croatian to win NYU's French Award. Research focus: effects of Ricci curvature and mean geodesic flow on the presidency of William McKinley. Generalized James Garfield's proof of the Pythagorean theorem to a hyperbolic setting.
</p>
  </p><big>
  <a href="/Curriculum.pdf" target="_blank">Curriculum Vitae</a>
  </big><br></p>
  <br>
  <hr>
  <br><br>

  <p><h2>Craftsmanship</h2>
<div class="math-section">
  <div class="entry">
    <img src="/mind-thumb.png" alt="Scary lines approaching a peaceful dotted line">
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      <div class="entry-title">"A Vision Into My Mind"</div>
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      <div class="entry-title">"A Glimpse Into My Soul"</div>
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        <a href="/Soul.pdf" target="_blank"><button>PDF</button></a>
        <button onclick="openPopup('soul')">Explanation</button>
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  <p>&nbsp;
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  <p><big>
  Special content:<br>
  <a href="/passports.html" target="_blank">Passport Calculator</a><br>
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    mind: {
      mathy: `<div style="font-size: 1.05em; text-align: left; max-width: 680px; margin: 0 auto;">
      <p><b><i>A Vision Into My Mind</i></b> is the name I've given to a slightly disorganized paper of mine. This paper attempts to expand various fundamental definitions and concepts in metric geometry to a setting where geodesics may be sparse or nonexistent. Think &Qopf;<sup>2</sup>.</p>

      <p>The fundamental concept in this paper is that of an <b>undergeodesic</b> in a metric space, which is defined as an isometric embedding from <i>any subset</i> of an interval into the space whose domain contains both endpoints of that interval.</p>

      <p>The document is divided into seven sections. Many parts of this document, especially the later sections, are incomplete. Here is a brief description of each of the paper's sections:</p>

      <ol>
        <li>This section defines undergeodesics and gives some of their basic properties, while also giving motivating examples.</li>
        <li>This section introduces the extremely important notion of a <b>maximal</b> undergeodesic. The size of the maximal undergeodesics in a space is a useful way to track how close to being geodesic the space is. The other important concept introduced in this section is that of undergeodesic <b>uniqueness</b>, which extends the familiar concept of geodesic uniqueness.</li>
        <li>This section describes both when undergeodesics are "close" to being geodesics, and when a metric space is "close" to being a geodesic metric space. The nuances involved in defining this notion of "close" are central to much of the work in this paper. Some fundamental examples are also given here.</li>
        <li>This section introduces the notion of the <b>geodesic saturation</b> of a metric space, and gives conditions under which this geodesic saturation of a space can be viewed as the <i>smallest</i> geodesic space containing it.</li>
        <li>This section attempts to generalize various fundamental results in the study of <b>CAT(&kappa;) spaces</b> to an undergeodesic setting.</li>
        <li>This section attempts, so far without success, to generalize the concept of the <b>visual boundary</b> of a space to an undergeodesic setting.</li>
        <li>This section compiles ideas and questions that do not fit into any earlier sections and are not sufficiently developed to necessitate their own section.</li>
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      casual: `<div style="font-size: 1.05em; text-align: left; max-width: 680px; margin: 0 auto;">
       <p><b><i>A Vision Into My Mind</i></b> is the name I've given to a slightly disorganized paper of mine. It's a collection of mathematical work inspired by the following question: "Can a space be flat even if you can't draw a straight line on it?"</p>

<p>A <a href="https://en.wikipedia.org/wiki/Geodesic" target="_blank">geodesic</a> is a concept that generalizes the notion of a straight, unbroken line. On a flat surface, geodesics are exactly the same as straight lines. On a very bumpy surface, though, you can see what a geodesic would look like by imagining a small car driving over the surface, without steering: the line it traces as it drives would (ideally) be a geodesic. That path, even if curvy, is the "straightest possible" for the terrain it's on.</p>

<p>The behavior of geodesics on a surface can tell you a lot about that surface, especially in the realm of <a href="https://en.wikipedia.org/wiki/Metric_geometry" target="_blank">metric geometry</a>, where geodesic information <i>defines</i> whether a surface is considered <i>flat</i> or <i>curved</i> in a mathematical sense.</p>

<p>In order to understand the point of my paper, imagine a very porous object: for example, a thin slice of <a href="https://en.wikipedia.org/wiki/Swiss_cheese_(North_America)" target="_blank">Swiss cheese</a>. As you can easily test at home with a sharpie and a suitable cheese, if you try to draw a straight and unbroken line on such a slice, you'll quickly find yourself blocked by an inconvenient hole. And if your cheese is extremely (say, <a href="https://en.wikipedia.org/wiki/Totally_disconnected_space" target="_blank">infinitely</a>) hole-ridden, you may find that you simply can't get anywhere at all. There are no geodesics, and there is no geodesic information to extract.</p>

<p>Does this mean that your slice of cheese is neither flat nor curved? To a metric geometer, perhaps the answer is <a href="https://en.wikipedia.org/wiki/CAT%28k%29_space" target="_blank">yes</a>, at least by the usual definitions. But I say <i>no</i>: a thin slice of Swiss cheese is <i>flat</i>, no matter how many holes it has. The original goal of this paper was to define a language to formalize that conclusion, although I've explored many fun tangents along the way.</p>`
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    soul: {
      mathy: `<div style="font-size: 1.05em; text-align: left; max-width: 680px; margin: 0 auto;">
<p><b><i>A Glimpse Into My Soul</b></i> is the long-awaited sequel and companion piece to <em>A Vision Into My Mind</em>, whose main purpose is to refine and develop the ideas of the fourth section of that paper.</p>

<p>After rehashing some preliminaries and lemmas from <em>A Vision</em>, this paper focuses on making clear the relationship between an almost geodesic space and its geodesic saturation, eventually establishing the category of full metric spaces as a reflective subcategory of the category of almost full metric spaces via the saturation construction. There are a couple of technical lemmas involved in this section that I am, as of yet, unable to prove, although I am reasonably sure that they are correct.</p>

<p>Two of the later sections (6 and 8) attempt to clarify notions from <em>A Vision</em> in the more restrictive setting of almost full spaces. Section 7 proposes a slightly odd definition of <em>homotopy</em> in almost full spaces, which attempts to detect "large" holes in infinitely hole-filled objects: I am yet to see if this results in anything interesting.</p>

<p>The anticipated release date for this paper is about a month after I get the motivation to start working on it again.</p>`,
      casual: `<div style="font-size: 1.05em; text-align: left; max-width: 680px; margin: 0 auto;">

      <p><b><i>A Glimpse Into My Soul</b></i> is the long-awaited sequel and companion piece to <em>A Vision Into My Mind</em>, whose main purpose is to refine and develop the ideas of the fourth section of that paper.</p>

<p>While much of the work here is technical, one idea that might be of interest is my proposed <a href="https://en.wikipedia.org/wiki/Homotopy_theory" target="_blank">homotopy theory</a> for extremely hole-ridden spaces. Homotopy is, crudely, a way of detecting holes in spaces: it could be used to calculate, for example, whether a straw has one or two holes, or whether your slice of swiss cheese has 35 or 36.</p>

<p>Things become more complicated in the case of those weird spaces with infinitely many holes. In these cases, naive ideas of homotopy often either <a href="https://en.wikipedia.org/wiki/Fundamental_groupoid" target="_blank">fail completely</a> or look <a href="https://mathoverflow.net/questions/67549/fundamental-group-of-r2-q2" target="_blank">extremely horrible</a>. I propose an altered definition of homotopy, used only for these "extremely holey" spaces, which could perhaps detect whether their infinitely tiny holes are large or small. It's all pretty speculative.</p>

<p>The anticipated release date for this paper is about a month after I get the motivation to start working on it again.</p>`
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