Jekyll2020-06-25T08:38:59+00:00https://mjungmath.github.io/feed.xmlMichael JungThis is my personal homepage where I give insights into my ongoing work and ambitions.Michael JungBonsai Update April 20202020-04-22T00:00:00+00:002020-04-22T00:00:00+00:00https://mjungmath.github.io/blog/bonsai/replanting<p>Today, I replanted two trees into proper bonsai pots. On the one hand, I replanted one of my <em>pomegranates</em>. The pomegranate certainly needs a thicker trunk to represent an actual bonsai. However, the tree has a wonderful crown, so I decided to give it a try in a nice pot:</p>
<p><img src="/assets/images/bonsai/2020-04/punica_inside.jpg" alt="pomegranate" /></p>
<p>The trunk of my <em>ficus tree</em>, on the other hand, has developed magnificently with a glimpse of a nebari:</p>
<p><img src="/assets/images/bonsai/2020-04/ficus_outside_nebari.jpg" alt="ficus benjamini nebari" /></p>
<p>The whole picture looks as follows:</p>
<p><img src="/assets/images/bonsai/2020-04/ficus_inside.jpg" alt="ficus benjamini" /></p>
<p>As you can see, the crown looks a little bit wild at the moment, but I want to give the tree some time to get used to its new pot. Moreover, I am confident that it will grow into the direction I desire very soon.</p>Michael JungToday, I replanted two trees into proper bonsai pots. On the one hand, I replanted one of my pomegranates. The pomegranate certainly needs a thicker trunk to represent an actual bonsai. However, the tree has a wonderful crown, so I decided to give it a try in a nice pot:Alternative Algorithm for Characteristic Forms2020-04-20T00:00:00+00:002020-04-22T00:00:00+00:00https://mjungmath.github.io/blog/sagemanifolds/char-class-alternative<p>Today, I want to introduce a slightly different algorithm, that I am currently working on, to implement the computation of characteristic forms in Sage. This new approach aims for two purposes:</p>
<ol>
<li>I anticipate to gain a speed-up especially in high dimensions.</li>
<li>I want to understand transgression forms even better and hope to lay foundations for them.</li>
</ol>
<p class="notice--warning"><strong>Note:</strong> If you are not familiar with my previous work, take a peek at my <a href="/assets/files/masters_thesis.pdf">master’s thesis</a>. A <em>very</em> short review is coming soon.</p>
<h2 id="review-and-outline">Review and Outline</h2>
<p>So far, the algorithm follows the Chern–Weil approach straightforwardly:</p>
<ol>
<li>Compute the curvature matrix.</li>
<li>Insert the curvature matrix in an invariant polynomial (e.g. trace, determinant, Pfaffian) composed with an holomorphic function.</li>
</ol>
<p>However, due to the composition with the holomorphic function, the computation of high powers of matrices over the algebra of mixed differential forms is necessary. Hence, the computational cost swiftly scales with the base space’s dimension and vector bundle’s rank. To avoid this kind of computation completely, my new approach uses <em>Chern roots</em> instead. The idea goes as follows: we compute the <em>additive/multiplicative sequence</em> of a given polynomial, then we compute the Chern/Pontryagin form in the usual manner and insert it into the sequence. This approach involves significantly less computations with mixed differential forms.</p>
<h2 id="chernpontryagin-forms">Chern/Pontryagin Forms</h2>
<p>Let <script type="math/tex">E \to M</script> be a vector bundle and <script type="math/tex">\nabla</script> a connection on <script type="math/tex">E</script>. Recall that the <em>Chern form</em> of <script type="math/tex">\nabla</script> for complex vector bundles is given by</p>
<script type="math/tex; mode=display">c(E,\nabla) = \det\left(1 + \frac{\Omega^\nabla}{2 \pi \mathrm{i}}\right)</script>
<p>wheras the <em>Pontryagin form</em> of <script type="math/tex">\nabla</script> for real vector bundles is obtained by</p>
<script type="math/tex; mode=display">p(E,\nabla) = \det\left(1 + \frac{\Omega^\nabla}{2 \pi}\right).</script>
<p>Here, <script type="math/tex">\Omega^\nabla</script> denotes the curvature form matrix associated to <script type="math/tex">\nabla</script>. Obviously, the computation does not involve any kind of powers of <script type="math/tex">\Omega^\nabla</script>. This is beneficial for the computational intensity.</p>
<h2 id="multiplicative-sequences">Multiplicative Sequences</h2>
<p>Let <script type="math/tex">f(x)</script> be a polynomial in <script type="math/tex">x</script>. Consider the invariant polynomial <script type="math/tex">P\colon \mathrm{Mat}(n \times n, \mathbb{C}) \to \mathbb{C}</script> with</p>
<script type="math/tex; mode=display">P(X) = \det\left(f(X)\right)</script>
<p>for any <script type="math/tex">X \in \mathrm{Mat}(n \times n)</script>. Now, we want to express <script type="math/tex">P</script> in terms of <a href="https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial"><em>elementary symmetric functions</em></a>. Let <script type="math/tex">(x_1, \ldots, x_n)</script> be the eigenvalues of <script type="math/tex">X</script> including multiplicities. Then we have</p>
<script type="math/tex; mode=display">P(X) = \prod^n_{k=1} f(x_i)</script>
<p>and evidently obtain that this is a symmetric polynomial in the <script type="math/tex">x_i</script>. Due to the fundamental theorem of elementary symmetric functions, we can write <script type="math/tex">P</script> as a polynomial in the elementary symmetric functions <script type="math/tex">e_i</script>:</p>
<script type="math/tex; mode=display">P = Q(e_1, \ldots e_n).</script>
<p>The proof of this theorem comes with an algorithm [1]. However, the computation with symmetric polynomials is already realized in Sage via the <a href="http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/sf/sfa.html"><code class="language-plaintext highlighter-rouge">SymmetricFunctions</code></a> class using the backend <a href="http://www.algorithm.uni-bayreuth.de/en/research/SYMMETRICA/">Symmetrica</a> written in C. That is perfect for our purposes.</p>
<p class="notice--primary"><strong>Note:</strong> For the particular polynomial <script type="math/tex">P</script>, one can do even better. See [2] for details. However, the computational cost is, in any case, negligible compared to computations with mixed differential forms in high dimensions.</p>
<p>Notice that the <script type="math/tex">i</script>-th homogeneous component of the Chern/Pontryagin class represents the <script type="math/tex">i</script>-th elementary symmetric function by definition.<sup id="fnref:1"><a href="#fn:1" class="footnote">1</a></sup> This connection becomes clear when we take a closer look at the defining equation of the elementary symmetric functions:</p>
<script type="math/tex; mode=display">\det(1+t X) = \sum^n_{i=0} t^i\,e_i(X).</script>
<p>Thus, when we insert the homogeneous components of the Chern/Pontryagin class into the polynomial <script type="math/tex">Q</script>, then we obtain the multiplicative characteristic class associated to <script type="math/tex">f(x)</script>.</p>
<p>Of course, we can proceed similarly for additive classes: we simply replace the determinant by the trace and the product by a sum.</p>
<h2 id="summary">Summary</h2>
<p>The concrete proceeding goes as follows now:</p>
<ol>
<li>Compute the additive/multiplicative sequence of the function associated to the class using <code class="language-plaintext highlighter-rouge">SymmetricFunctions</code>.</li>
<li>Compute the Chern/Pontryagin form in the usual manner.</li>
<li>Insert the homogeneous components of the Chern/Pontryagin form into the additive/multiplicative sequence.</li>
</ol>
<p>In a subsequent post, I will show the outlines of my implementation and provide some examples with computation times.</p>
<h2 id="references">References</h2>
<p>[1] Ben Blum-Smith and Samuel Coskey — <a href="https://arxiv.org/pdf/1301.7116.pdf">The Fundamental Theorem on Symmetric Polynomials: History’s First Whiff of Galois Theory</a>. 2016.</p>
<p>[2] Oleksandr Iena — <a href="https://orbilu.uni.lu/bitstream/10993/21949/2/ChernLib.pdf"><em>On Symbolic Computations with Chern Classes</em></a>. 2016.</p>
<p>[3] H. Blaine Lawson and Marie-Louise Michelsohn — <em>Spin Geometry</em>. 1989.</p>
<div class="footnotes">
<ol>
<li id="fn:1">
<p>Notice that the real case needs special care. See [3, 225 ff.] for details. <a href="#fnref:1" class="reversefootnote">↩</a></p>
</li>
</ol>
</div>Michael JungToday, I want to introduce a slightly different algorithm, that I am currently working on, to implement the computation of characteristic forms in Sage. This new approach aims for two purposes: