User slobodan simi&amp;#263; - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T21:25:33Z http://mathoverflow.net/feeds/user/5706 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75038/when-is-a-closed-differential-form-harmonic-relative-to-some-metric When is a closed differential form harmonic relative to some metric? Slobodan Simi&#263; 2011-09-09T21:34:09Z 2012-05-13T03:40:31Z <p>Let $\omega$ be a closed non-exact differential $k$-form ($k \geq 1$) on a closed orientable manifold $M$. </p> <p><strong>Question</strong>: Is there always a Riemannian metric $g$ on $M$ such that $\omega$ is $g$-harmonic, i.e., $\Delta_g \omega = 0$? </p> <p>Here $\Delta_g$ is the Laplace-deRham operator, defined as usual by $\Delta_g = d \delta + \delta d$, where $\delta$ is the $g$-codifferential. Note that non-exactness is important, since if $\omega$ were to be exact and harmonic, then by the Hodge decomposition theorem $\omega = 0$.</p> <p>For instance, if $\omega$ is a 1-form on the unit circle, then it is not hard to see that $\omega$ is harmonic with respect to some metric $g$ if and only if it is a volume form (i.e., it doesn't vanish). This observation generalizes to forms of top degree on any $M$.</p> <p>What can be said in general for forms which are not of top degree?</p> http://mathoverflow.net/questions/77425/failures-that-lead-eventually-to-new-mathematics/77510#77510 Answer by Slobodan Simić for Failures that lead eventually to new mathematics Slobodan Simić 2011-10-08T04:04:26Z 2011-10-08T04:04:26Z <p>In the early 1960's Smale published a paper containing a conjecture whose consequence was that (in modern language) "chaos didn't exist". He soon received a letter from Norman Levinson informing him of an earlier work of Cartwright and Littlewood which effectively contained a counterexample to Smale's conjecture. Smale "worked day and night to resolve the challenges that the letter posed to my beliefs" (in his own words), trying to translate analytic arguments of Levinson and Cartwright-Littlewood into his own geometric way of thinking. This led him to his seminal discovery of the <a href="http://www.scholarpedia.org/article/Smale_horseshoe" rel="nofollow">horseshoe map</a>, followed by the foundation of the field of hyperbolic dynamical systems. For more details, see Smale's popular article "Finding a horseshoe on the beaches of Rio", Mathematical Intelligencer 20 (1998), 39-44.</p> http://mathoverflow.net/questions/76580/famous-mathematicians-with-background-in-arts-humanities-law-etc/76689#76689 Answer by Slobodan Simić for Famous mathematicians with background in arts/humanities/law etc Slobodan Simić 2011-09-28T20:41:49Z 2011-09-28T20:41:49Z <p>Henri Poincaré was a mining engineer. His first job was at the Corps des Mines as an inspector of mines. He participated in the rescue of miners trapped after an explosion, himself descending the shaft into the mine to investigate the cause of the explosion! Check <a href="http://www.gap-system.org/~history/HistTopics/Poincare_mines.html" rel="nofollow">this link</a> for details.</p> http://mathoverflow.net/questions/75053/inner-products-on-differential-forms Inner products on differential forms Slobodan Simi&#263; 2011-09-09T23:45:34Z 2011-09-10T01:40:28Z <p>Given a Riemannian metric $g$ on a smooth manifold $M$, one defines an $L^2$-inner product on the space $\bigwedge^\ast(M)$ of differential forms by</p> <p>$$\langle \alpha, \beta \rangle_g = \int_M \alpha \wedge \ast_g \beta,$$</p> <p>where $\ast_g$ denotes the Hodge-star operator relative to $g$, and $\alpha, \beta$ are forms of the same degree.</p> <p><strong>Question</strong>: Does every inner product on $\bigwedge^\ast(M)$ as a graded vector space come from some metric $g$? How about inner products on $k$-forms $\bigwedge^k(M)$ for a single $k$, especially $0 &lt; k &lt; \dim M$?</p> http://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics/26071#26071 Answer by Slobodan Simić for Examples of common false beliefs in mathematics. Slobodan Simić 2010-05-26T21:39:37Z 2010-05-26T21:39:37Z <p>If $f$ is (Lebesgue) integrable on $\mathbb{R}$, then $f(x) \to 0$, as $x \to \infty$. False: there exists a <em>continuous</em> integrable function on $\mathbb{R}$ such that $\limsup_{\infty} f = \infty$ (an exercise in Stein and Shakarchi's <a href="http://www.amazon.com/Real-Analysis-Integration-Princeton-Lectures/dp/0691113866/ref=sr_1_2?ie=UTF8&amp;s=books&amp;qid=1274909846&amp;sr=1-2" rel="nofollow">Real Analysis</a>).</p> http://mathoverflow.net/questions/10282/alternative-undergraduate-analysis-texts/25745#25745 Answer by Slobodan Simić for Alternative Undergraduate Analysis Texts Slobodan Simić 2010-05-24T04:04:46Z 2010-05-24T04:04:46Z <p>V. A. Zorich's <a href="http://www.springer.com/mathematics/analysis/book/978-3-540-40386-9?changeHeader" rel="nofollow">Mathematical Analysis I</a> and <a href="http://www.springer.com/mathematics/analysis/book/978-3-540-40633-4?changeHeader" rel="nofollow">II</a> (Springer). It covers undergraduate material from an advanced viewpoint, contains lots of good physically oriented examples, and is quite comprehensive. </p> http://mathoverflow.net/questions/4994/fundamental-examples/24058#24058 Answer by Slobodan Simić for Fundamental Examples Slobodan Simić 2010-05-10T03:48:59Z 2010-05-10T03:48:59Z <p><a href="http://www.scholarpedia.org/article/Smale_horseshoe" rel="nofollow">Smale's horseshoe map</a> in dynamical systems.</p> http://mathoverflow.net/questions/76580/famous-mathematicians-with-background-in-arts-humanities-law-etc/76689#76689 Comment by Slobodan Simić Slobodan Simić 2011-09-28T21:38:53Z 2011-09-28T21:38:53Z Sorry, didn't realize that. http://mathoverflow.net/questions/75053/inner-products-on-differential-forms/75056#75056 Comment by Slobodan Simić Slobodan Simić 2011-09-23T19:14:34Z 2011-09-23T19:14:34Z Thanks, Brian and Paul - that's exactly what I wanted to know. Thanks also to Jos&#233; for an entirely different (at least for me) point of view. http://mathoverflow.net/questions/75038/when-is-a-closed-differential-form-harmonic-relative-to-some-metric Comment by Slobodan Simić Slobodan Simić 2011-09-23T19:09:41Z 2011-09-23T19:09:41Z Thanks to all who helped elucidate this question! I was mostly interested in the degrees $k = 1$ and $k = n-1$, but it would certainly be very interesting to see what can be said about the intermediate $k$'s. Perhaps a topic for a future Ph.D. thesis.