User chris judge - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T17:53:47Z http://mathoverflow.net/feeds/user/7120 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83453/example-of-hyperbolic-3-fold-with-no-embedded-incompressible-subsurfaces Example of hyperbolic 3-fold with no embedded incompressible subsurfaces Chris Judge 2011-12-14T18:46:37Z 2011-12-14T18:59:05Z <p>Kahn-Markovic show that every hyperbolic 3-fold contains an immersed $\pi_1$ injective surface. Are there any known examples of hyperbolic 3-folds that do not contain a embedded $\pi_1$ injective surface? </p> http://mathoverflow.net/questions/10514/teichmuller-theory-introduction/72037#72037 Answer by Chris Judge for Teichmuller Theory introduction Chris Judge 2011-08-03T21:13:34Z 2011-08-03T21:13:34Z <p>McMullen's notes (http://www.math.harvard.edu/~ctm/home/text/class/harvard/275/09/html/base/rs/rs.pdf)</p> http://mathoverflow.net/questions/72003/convergence-of-elliptic-operators Convergence of elliptic operators Chris Judge 2011-08-03T16:15:29Z 2011-08-03T17:32:24Z <p>Let $A_t$ be family of second order, positive, elliptic differential operator mapping Sobolev $H^2$ of a compact smooth manifold (or bounded domain) to L^2. Suppose that the coefficients of $A_t$ converge uniformly in $C^k$ for every $k$ to the coefficients of a second order, positive, elliptic differential operator $A$. $A$ is invertible (with domain L^2 and range H^2) and so we may consider the sequence $A_t \circ A_0^{-1}$ of operators from $L^2$ to $L^2$. Does this family converge to the identity in the $L^2$ operator norm? Why or why not?</p> http://mathoverflow.net/questions/29700/generators-for-congruence-group-gamma2 Generators for congruence group $\Gamma(2)$ Chris Judge 2010-06-27T11:58:47Z 2010-06-27T17:16:05Z <p>Is the congruence group $\Gamma(2)$ generated by the upper triangular matrix $(1, 2; 0, 1)$ and the lower triangular matrix $(1, 0; 2, 1)$ or does on need to also throw in the negation of the identity? To be specific, how do I check that the negation of the identity is not a word in the above matrices? </p> http://mathoverflow.net/questions/34658/is-there-a-whitney-embedding-theorem-for-non-smooth-manifolds/34661#34661 Comment by Chris Judge Chris Judge 2012-05-31T19:48:21Z 2012-05-31T19:48:21Z But these notes appear to consider smooth case, no? http://mathoverflow.net/questions/83453/example-of-hyperbolic-3-fold-with-no-embedded-incompressible-subsurfaces Comment by Chris Judge Chris Judge 2011-12-15T10:03:51Z 2011-12-15T10:03:51Z Thanks Ian. This is very helpful. http://mathoverflow.net/questions/72003/convergence-of-elliptic-operators/72017#72017 Comment by Chris Judge Chris Judge 2011-08-03T20:57:08Z 2011-08-03T20:57:08Z I guess ellipticity has nothing to with it. $\int |(A-B) u|^2 \leq C \sum_{\alpha} \int |\partial^{\alpha} u|^2$ where $C= \sup |a_{\alpha}- b_{\alpha}|^2$ where $\alpha$ is a multi-index and the $a$'s and $b$'s are the coefficients. http://mathoverflow.net/questions/72003/convergence-of-elliptic-operators/72017#72017 Comment by Chris Judge Chris Judge 2011-08-03T20:36:26Z 2011-08-03T20:36:26Z Is ellipticity necessary? (Part of the `why' part of question...) http://mathoverflow.net/questions/29700/generators-for-congruence-group-gamma2/29709#29709 Comment by Chris Judge Chris Judge 2010-06-27T17:18:06Z 2010-06-27T17:18:06Z Quite beautiful! Thank you for adding this. http://mathoverflow.net/questions/29700/generators-for-congruence-group-gamma2/29702#29702 Comment by Chris Judge Chris Judge 2010-06-27T17:01:36Z 2010-06-27T17:01:36Z Wadim, exercise 6 does not answer the question.