User henry zorrilla - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T13:09:58Z http://mathoverflow.net/feeds/user/16533 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/129319/constructing-a-special-infinite-dimensional-vector-bundle/129382#129382 Answer by Henry Zorrilla for Constructing a special infinite-dimensional vector bundle Henry Zorrilla 2013-05-02T06:51:56Z 2013-05-02T06:51:56Z <p>I believe that the book, "The Structure of Classical Diffeomorphism Groups" by Augustin Banyaga had a very clear construction of the sought after tangent space. Also, "The Inverse Function Theorem of Nash and Moser" by Richard S. Hamilton develops a few of the tool need in order to work with Frechet manifolds. </p> http://mathoverflow.net/questions/108450/frechet-manifold-structure-of-cm-n Fréchet manifold structure of C(M, N) Henry Zorrilla 2012-09-30T07:16:46Z 2012-09-30T16:32:25Z <p>Let $F = C^{\infty}(M, N)$. I wish to give $F$ the structure of a Fréchet manifold. My plan was to emulate the construction of a smooth manifold. I know that for a finite dimensional smooth manifold M, $T_pM$ will be isomorphic to the model space (i.e if M is m dimensional, then $T_pM \cong \mathbb{R}^m$). Further more, I know that $T_pM$ can be identified with the equivalence classes of curves on M under the relation: For $\gamma, \gamma' \in C^\infty((-\epsilon,\epsilon), M)$ with $\gamma(o) = p$, $\gamma \sim \gamma'$ iff $\dot{\gamma(0)} = \dot{\gamma'(0)}.$ Define a path in $F$ to be a map $C:M\times[0,1] \rightarrow N$ with $C(x,t) = C_t(x)$ such that $C_0(x) = f$, $C_1(x) = g$, $C_{t_0}(x)$ is smooth for all $t_0 \in [0,1]$, and $C_{t}(x_0)$ is smooth for all $x_0 \in M.$ After a little work, ones sees that for $f \in F$, $T_{f}F = \Gamma_f(M, TN)$ which I have proved is (set)isomorphic to $\Gamma(M, f^*TN)$ where $f^*TN$ denotes the pullback bundle. </p> <p><strong>Question(s):</strong> (i) What is a good candidate for a semi-norm (or family of semi-norms) on $\Gamma(M, f^{*}TN)$ (ii) Is the metric structure on $\Gamma(M, f^*TN)\,$ dependent upon the choice of semi-norm (or family of semi-norms)? (iii) Does there exist any literature in which an explicit construction of even a trivial example of a Fréchet manifold can be found? (Hamilton's,"The inverse function theorem of Nash and Moser", I feel, comes closest to an example.)</p> <p>(I should mention also that this is a copy of a question on the math stackexchange website. <a href="http://math.stackexchange.com/questions/202048/frechet-manifold-structure-of-cm-n/%20%22question%22" rel="nofollow">http://math.stackexchange.com/questions/202048/frechet-manifold-structure-of-cm-n</a>)</p> http://mathoverflow.net/questions/103243/what-are-some-interesting-problems-in-the-intersection-of-algebraic-number-theory/103254#103254 Answer by Henry Zorrilla for What are some interesting problems in the intersection of Algebraic Number Theory and Algebraic Topology? Henry Zorrilla 2012-07-27T01:00:48Z 2012-07-27T01:00:48Z <p>Try "Primes and Knots" - By Toshitake Kohno &amp; Masanori Morishita.</p> http://mathoverflow.net/questions/108450/frechet-manifold-structure-of-cm-n Comment by Henry Zorrilla Henry Zorrilla 2012-09-30T22:20:31Z 2012-09-30T22:20:31Z @Mariano It did help me, but didn't answer my questions (or at least I could figure out how it did). http://mathoverflow.net/questions/103243/what-are-some-interesting-problems-in-the-intersection-of-algebraic-number-theory/103254#103254 Comment by Henry Zorrilla Henry Zorrilla 2012-07-27T02:17:53Z 2012-07-27T02:17:53Z There is paper titled &quot;Classical Knot Invariants and Elementary Number Theory&quot; written by K. Murasugi. The paper begins with a brief introduction to knot theory and then discusses a few knot invariants and their connection to number theory. It ends with a list of (open?) problems in knot theory related to number theory.