User johnson-leung - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T01:06:00Z http://mathoverflow.net/feeds/user/3000 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108298/explicit-period-lattices-for-abelian-surfaces Explicit period lattices for abelian surfaces Johnson-Leung 2012-09-27T23:27:36Z 2012-09-28T02:41:09Z <p>Given an explicit description (as an intersection) of an abelian surface $A$ is there an algorithm for computing the period lattice of the surface? For the specific examples that I am interested in, the ideal of $A$ has been obtained by Weil restriction from the affine model of an elliptic curve.</p> http://mathoverflow.net/questions/17774/does-the-image-of-an-p-adic-galois-representation-always-lie-in-a-finite-extensio Does the image of an p-adic Galois representation always lie in a finite extension? Johnson-Leung 2010-03-10T23:03:35Z 2010-11-06T05:04:10Z <p>I have been looking at Serre's conjecture and noticed that there are two conventions in the literature for a p-adic representation $\rho:\mbox{Gal}(\bar{\mathbb Q}/\mathbb Q)\to \mbox{GL}(n,V).$ In some references (eg Serre's book on $\ell$-adic representations), $V$ is a vector space over a finite extension of $\mathbb Q_p$. However, in more recent papers (eg Buzzard, Diamond, Jarvis) $V$ is a vector space over $\bar{\mathbb Q_p}$. It is easy to show that the former definition is a special case of the latter, but I suspect, and would like to prove that they are actually the same. That is, I would like to show that the image of any any continuous Galois representation over $\bar{\mathbb{Q}_p}$ actually lies in a finite extension of $\mathbb Q_p$. </p> <p>Is this the case? </p> <p>I think that a proof should use the fact that $G_{\mathbb Q}$ is compact and that $\bar{\mathbb Q}_p$ is the union of finite extensions. I have tried to mimic the proof that $\bar{\mathbb Q}_p$ is not complete, but have not been able to find an appropriate Cauchy sequence in an arbitrary compact subgroup of GL($n,V$).</p> <p>(This is my first question, so please feel free to edit if appropriate. Thanks!)</p> http://mathoverflow.net/questions/38103/an-engineering-ph-d-teaching-math-in-college/38110#38110 Answer by Johnson-Leung for an engineering Ph.D. teaching math in college Johnson-Leung 2010-09-08T23:15:37Z 2010-09-08T23:15:37Z <p>I guess it probably depends on the hiring procedures at each University. I can only speak to ours. </p> <p>We give minimum and desired qualifications in our job ad. These have to be approved by HR and the Human Rights Compliance Officer at the university. We then screen all applications according to these criteria. If a candidate does not meet our minimum qualifications then it would be really tough to convince HR and the HRCO that we are doing our duty as an equal opportunity employer to bring them in for an interview. The desired qualifications allow for more interpretation on the part of the search committee, but even for this, there is a ridiculous amount of documentation and auditing that goes on to make sure that we are "fair". </p> <p>My main point is that I think that the openness on the part of the department to non-math PhDs would have to be at the ad-writing stage. By all means, your friend should call and ask the department, if they are open to a PhD in a related field. Just don't be surprised if they have already tied their own hands.</p> http://mathoverflow.net/questions/28496/what-should-be-learned-in-a-first-serious-schemes-course/28586#28586 Answer by Johnson-Leung for What should be learned in a first serious schemes course? Johnson-Leung 2010-06-18T00:05:55Z 2010-06-18T00:05:55Z <p>I think that base change is a very important and subtle idea which should certainly be included in a first course. In particular, one should discuss properties that are stable under base change and those that are not.</p> <p>In a similar vein, in discussing cohomology, the difference between the coefficients of the motive and the base should be emphasized. This was confusing to me as I learned the subject.</p> http://mathoverflow.net/questions/11467/how-seriously-should-a-graduate-student-take-teaching-evaluations/11475#11475 Answer by Johnson-Leung for How seriously should a graduate student take teaching evaluations? Johnson-Leung 2010-01-11T22:37:59Z 2010-01-11T22:37:59Z <p>You should worry about your teaching evaluations before you get them. In other words, you should put effort into your teaching as 1) it does matter at almost every institution, and 2) you are teaching people math, and that is important. It is also a good idea to discuss the evaluations with your students before you hand them out. Remind them that this feedback will help you to improve your teaching so you would appreciate them telling you what worked and what didn't. Often, only "angry" students will respond in the comment sections. By taking five minutes and reminding the students that the evaluations are a tool for communication, you should get much better feedback. </p> <p>As for worrying about them afterward, it is true that they are a secondary consideration at most research universities. However, it is not helpful to have anything on your application that puts you lower on the list. If your evaluations are bad, make sure that the person who writes your teaching letter speaks about the positive aspects of your teaching. If you have to send in the evaluations, then have them address directly the mitigating factors (if there were any) that caused the bad evaluations. </p> http://mathoverflow.net/questions/11404/what-is-an-euler-system-and-the-motivation-for-it/11449#11449 Answer by Johnson-Leung for what is an Euler system and the motivation for it? Johnson-Leung 2010-01-11T17:15:23Z 2010-01-11T17:15:23Z <p>These are good answers. I would just like to add that while the applications of Euler systems have been mainly p-adic, they are actually motivic (ie units if the motive is h<sup>0</sup>(Spec F)). One might hope that if one is able to attach an L-function to a geometric object, there is also an Euler system living in an appropriate motivic cohomology. A sort of cohomological Euler product if you will. At least that is motivation for the sweeping special values conjectures such as the Tamagawa Number Conjecture.</p> http://mathoverflow.net/questions/106265/vanishing-of-certain-mu-invariants-attached-to-abelian-extensions-of-imaginary/108299#108299 Comment by Johnson-Leung Johnson-Leung 2012-09-28T02:35:24Z 2012-09-28T02:35:24Z Well, I feel like I should just delete this answer, except that your comment is interesting. Are there rules for that? http://mathoverflow.net/questions/106265/vanishing-of-certain-mu-invariants-attached-to-abelian-extensions-of-imaginary/108299#108299 Comment by Johnson-Leung Johnson-Leung 2012-09-28T01:29:29Z 2012-09-28T01:29:29Z That's a good point. I haven't thought deeply about the $\mu$ invariant for a few years, but I don't remember anything else special about $p=3$. It would probably help to go through Gillard's paper and be certain where the restriction is necessary. http://mathoverflow.net/questions/108298/explicit-period-lattices-for-abelian-surfaces Comment by Johnson-Leung Johnson-Leung 2012-09-28T00:54:55Z 2012-09-28T00:54:55Z OK. I edited it to not imply that I have a projective model. Formally applying Weil restriction, I get the surface as the intersection of two affine varieties. http://mathoverflow.net/questions/108298/explicit-period-lattices-for-abelian-surfaces Comment by Johnson-Leung Johnson-Leung 2012-09-28T00:33:13Z 2012-09-28T00:33:13Z This is explicit: The Weil restriction of an elliptic curve over a quadratic extension is an abelian surface. Restriction of scalars of the ideal of the curve gives two equations in four variables. I'm not an algebraic geometer, but I think that makes it a complete intersection. http://mathoverflow.net/questions/38103/an-engineering-ph-d-teaching-math-in-college/38110#38110 Comment by Johnson-Leung Johnson-Leung 2010-09-08T23:17:57Z 2010-09-08T23:17:57Z Just saw your comment, Keith. I guess you already knew all of these things. I do think that a phone call beats an email for getting in touch with potential employers. Anyway, good luck to your friend! http://mathoverflow.net/questions/28496/what-should-be-learned-in-a-first-serious-schemes-course/28586#28586 Comment by Johnson-Leung Johnson-Leung 2010-06-18T12:46:02Z 2010-06-18T12:46:02Z For me, it was another one of those things that is easy once you understand what is going on, but very easy to screw up before that! http://mathoverflow.net/questions/28496/what-should-be-learned-in-a-first-serious-schemes-course/28586#28586 Comment by Johnson-Leung Johnson-Leung 2010-06-18T12:25:43Z 2010-06-18T12:25:43Z Well, I think that I'll elaborate with an example. If $X=$Spec$F$ is a variety over $\mathbb{Q}$ and $F$ is a number field then $H^0_B(X(\mathbb{C}),\mathbb{Q})$ is isomorphic to the group ring $\mathbb Q[G]$ where $G$ is the Galois group of $F$ over $\mathbb{Q|$. If I would like to decompose this into the irreducible representations of $\mathbb{Q}$, then I need to extend \em{coefficients} to a field $E$ over which the idempotents are defined. So I would be looking at $H^0_B(X(\mathbb{C}),E)$. If I wanted to look at a subgroup of $G$, I would need to change the base. http://mathoverflow.net/questions/17774/does-the-image-of-an-p-adic-galois-representation-always-lie-in-a-finite-extensio/17802#17802 Comment by Johnson-Leung Johnson-Leung 2010-03-11T23:51:55Z 2010-03-11T23:51:55Z This is the line of argument that I was attempting to make, but I accepted jnewton's answer because it came in first. I wanted to let you know that in your write-up you switch from $K_r$ to $G_r$ midway through the proof. Thanks again! http://mathoverflow.net/questions/17774/does-the-image-of-an-p-adic-galois-representation-always-lie-in-a-finite-extensio Comment by Johnson-Leung Johnson-Leung 2010-03-11T16:00:54Z 2010-03-11T16:00:54Z It seemed to me that it must be true and well-known, since basic results do not appear to hold otherwise (for example, the existence of a G-stable lattice in V). The ring of integers of $\bar{\mathbb Q}_p$ is not very nice! http://mathoverflow.net/questions/17774/does-the-image-of-an-p-adic-galois-representation-always-lie-in-a-finite-extensio Comment by Johnson-Leung Johnson-Leung 2010-03-11T02:00:15Z 2010-03-11T02:00:15Z Keith, that would be great...Thanks!