User anonymous - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T11:08:57Z http://mathoverflow.net/feeds/user/3777 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14083/modular-forms-and-the-riemann-hypothesis Modular forms and the Riemann Hypothesis Anonymous 2010-02-03T23:44:12Z 2011-02-12T22:09:55Z <p>Is there any statement directly about modular forms that is equivalent to the Riemann Hypothesis for L-functions? </p> <p>What I'm thinking of is this: under the Mellin transform, the Riemann zeta function $\zeta(s)$ corresponds to a modular form $f$ (of weight 1/2). The functional equation of $\zeta(s)$ follows from the transform equation of $f$. So what is the property of $f$ that would be equivalent to the (conjectural) property that all the non-trivial zeros of $\zeta(s)$ lie on the critical line? Or perhaps is there any statement about some family of modular forms that would imply RH for $\zeta(s)$?</p> http://mathoverflow.net/questions/7957/books-well-motivated-with-explicit-examples/18177#18177 Answer by Anonymous for books well-motivated with explicit examples Anonymous 2010-03-14T16:12:00Z 2010-03-14T16:12:00Z <p>Milne's lecture notes contain many good, standard examples discussed in depth. For example, in Algebraic Number Theory, in the section about Frobenius elements, Milne proves quadratic reciprocity (which IMO is the "correct" proof of quadratic reciprocity). </p> http://mathoverflow.net/questions/17778/books-you-would-like-to-see-translated-into-english/18146#18146 Answer by Anonymous for Books you would like to see translated into English. Anonymous 2010-03-14T06:21:31Z 2010-03-14T06:21:31Z <p>The Collected Work of Carl Ludwig Siegel.</p> http://mathoverflow.net/questions/18100/theorems-with-unexpected-conclusions/18143#18143 Answer by Anonymous for Theorems with unexpected conclusions Anonymous 2010-03-14T05:47:14Z 2010-03-14T05:47:14Z <p>The Taniyama-Shimura conjecture (now proved, by Wiles and others): all elliptic curves over $\mathbb Q$ are modular. It's magical that one can give a "formula" for the numbers of points on the curve modulo $p$ using modular forms.</p> http://mathoverflow.net/questions/17778/books-you-would-like-to-see-translated-into-english/17883#17883 Answer by Anonymous for Books you would like to see translated into English. Anonymous 2010-03-11T16:58:32Z 2010-03-11T16:58:32Z <p>Don Zagier's German book about quadratic forms.</p> http://mathoverflow.net/questions/17516/the-ring-of-algebraic-integers-of-the-number-field-generated-by-torsion-points-on The ring of algebraic integers of the number field generated by torsion points on an elliptic curve Anonymous 2010-03-08T20:43:31Z 2010-03-09T21:56:53Z <p>(Warning: a student asking) Let $E$ be an elliptic curve over $\mathbf Q$. Let $P(a,b)$ be a (nontrivial) torsion point on $E$. Is there an easy description of the ring of algebraic integers of $\mathbf Q(a,b)$? I'm curious about the answer for general elliptic curves, but I'm not sure whether such an answer is possible.</p> <p>(This question is motivated by the nice description of the rings of integers of cyclotomic fields $\mathbf Q(\zeta_n)$) </p> http://mathoverflow.net/questions/10535/ways-to-prove-the-fundamental-theorem-of-algebra/16700#16700 Answer by Anonymous for Ways to prove the fundamental theorem of algebra Anonymous 2010-02-28T16:50:38Z 2010-02-28T16:50:38Z <p>There's a linear algebra proof by Harm Derksen: <a href="http://www.jstor.org/pss/3647746" rel="nofollow">http://www.jstor.org/pss/3647746</a>. You can also find the article posted here: <a href="http://math.berkeley.edu/~ribet/110/f03/derksen.pdf" rel="nofollow">http://math.berkeley.edu/~ribet/110/f03/derksen.pdf</a></p> http://mathoverflow.net/questions/13320/cool-problems-to-impress-students-with-group-theory/15057#15057 Answer by Anonymous for Cool problems to impress students with group theory Anonymous 2010-02-12T00:46:40Z 2010-02-12T00:46:40Z <p>There's a group-action proof of the famous Morley's theorem in Euclidean Geometry: <a href="http://www.ems-ph.org/journals/newsletter/pdf/2004-12-54.pdf" rel="nofollow">http://www.ems-ph.org/journals/newsletter/pdf/2004-12-54.pdf</a>. It's near the end of Alain Connes's article. This proof very convincingly shows that geometry is about symmetry which is what groups are about.</p> http://mathoverflow.net/questions/14468/how-much-of-scheme-theory-can-you-visualize/14917#14917 Answer by Anonymous for How much of scheme theory can you visualize? Anonymous 2010-02-10T17:07:46Z 2010-02-10T17:07:46Z <p>I've found this set of notes <a href="http://math.mit.edu/~poonen/papers/Qpoints.pdf" rel="nofollow">http://math.mit.edu/~poonen/papers/Qpoints.pdf</a> by Bjorn Poonen very helpful in learning arithmetic geometry. I'd read everything very carefully and do all the exercises there. Bjorn is a master of exposition.</p> http://mathoverflow.net/questions/10419/depressed-graduate-student/14705#14705 Answer by Anonymous for Depressed graduate student. Anonymous 2010-02-08T23:05:06Z 2010-02-08T23:05:06Z <p>I went through a depressed period in grad school. I felt I couldn't understand modern algebraic geometry (even after a full year of working through Hartshorne's book!) and thought there was no hope of doing serious research for me (I was interested Number Theory). I dropped out and tried to do other things. After a couple years I found that I loved solving math problems the best. I went back to grad school and had the great fortune of working under a wonderful adviser who always gave me the help I needed to move ahead. I enjoyed and treasured every minute talking with my adviser. Eventually I was able to finish the PhD and graduated. I can say without too much exaggeration that it was my adviser who saved me.</p> <p>So your friend may just need to search for the right professors/advisers to help him.</p> http://mathoverflow.net/questions/14083/modular-forms-and-the-riemann-hypothesis/14170#14170 Answer by Anonymous for Modular forms and the Riemann Hypothesis Anonymous 2010-02-04T18:35:44Z 2010-02-04T18:35:44Z <p>@Hansen and @Anonymous: your answers are appreciated. I want to know why people almost never discuss this question, so even the answer that the question is not a good one is appreciated, provided it also gives a reason, like you did.</p> <p>As Emerton suggested, I want to know whether RH could be stated for eigenforms directly, instead of the L-functions. I'm no expert in this field, but it seems to me that analytic properties of modular forms are easier to understand (than those of L-functions), so why not expressing RH in the space of modular forms and working with them? </p> <p>@Anonymous: do you know of any readily accessible source for statements about families of modular forms that imply RH for zeta? I don't have access to MathSciNet.</p> http://mathoverflow.net/questions/20112/interesting-results-in-algebraic-geometry-accessible-to-3rd-year-undergraduates/20143#20143 Comment by Anonymous Anonymous 2010-04-02T15:31:00Z 2010-04-02T15:31:00Z KConrad's answer is great! http://mathoverflow.net/questions/19957/how-do-you-approach-your-childs-math-education/19976#19976 Comment by Anonymous Anonymous 2010-03-31T20:55:42Z 2010-03-31T20:55:42Z That would be &quot;conservation of mathematical ideas&quot;, but no I didn't count like that :-) http://mathoverflow.net/questions/19957/how-do-you-approach-your-childs-math-education/19964#19964 Comment by Anonymous Anonymous 2010-03-31T19:55:38Z 2010-03-31T19:55:38Z I respect your views and thanks for sharing them here. I will try to keep them in my head while raising my kids (which I'm doing now). There are things about school I'm very worried about. I have opportunities to know what normal kids are learning in school and from what I have seen, they often get bad teachers and they know it. But they can't do nothing about it except to bear it out, hoping they will get better teachers in college. (BTW, it's not my kids I'm talking about here, but other kids). So I kind of know the value of keeping a child in the system, but what if the system doesn't work? http://mathoverflow.net/questions/19957/how-do-you-approach-your-childs-math-education/19976#19976 Comment by Anonymous Anonymous 2010-03-31T19:26:12Z 2010-03-31T19:26:12Z It's funny that I was just pushing my son on the swing the other day and I also stopped when I reached 100 (counting out loud). Maybe it's a special case of &quot;conservation of energy&quot;? :-) http://mathoverflow.net/questions/19957/how-do-you-approach-your-childs-math-education/19964#19964 Comment by Anonymous Anonymous 2010-03-31T19:20:52Z 2010-03-31T19:20:52Z &quot;The one thing I never do though, is to try and push my kids ahead in the UK mathematical curriculum. I leave that for the schools. The last thing I want them to be is bored at school because they &quot;know it all&quot;&quot;. This seems like a good idea. However, if your child is capable of &quot;knowing it all&quot;, would this policy then hold him back? http://mathoverflow.net/questions/18593/what-are-the-worst-notations-in-your-opinion/18627#18627 Comment by Anonymous Anonymous 2010-03-18T21:01:36Z 2010-03-18T21:01:36Z Writing $a\dot b$ every time would make group presentation theorems a night mare to read. http://mathoverflow.net/questions/18593/what-are-the-worst-notations-in-your-opinion/18619#18619 Comment by Anonymous Anonymous 2010-03-18T20:53:49Z 2010-03-18T20:53:49Z Writing x\y for &quot;x divides y&quot; seems much more horrible than &quot;x|y&quot;. Maybe I'm just too used to the latter. http://mathoverflow.net/questions/18288/ants-on-a-rod-puzzle/18294#18294 Comment by Anonymous Anonymous 2010-03-15T21:21:17Z 2010-03-15T21:21:17Z The hats-switching proof is not only mathematically correct but also goes to the essence of the matter. Hence it can also easily answer other related questions such as how many head-bumps occurred before all the ants fell of the rod. http://mathoverflow.net/questions/18100/theorems-with-unexpected-conclusions/18110#18110 Comment by Anonymous Anonymous 2010-03-15T06:09:30Z 2010-03-15T06:09:30Z @KConrad: Hmm, the fact that there is a polynomial whose values at the integers are primes only, even negative ones, still surprises me. Is there a reason that such a polynomial isn't very surprising? Also, why are irreducible polynomials more impressive than reducible ones? If a polynomial is reducible, it just seems much harder for its values to be primes... http://mathoverflow.net/questions/7957/books-well-motivated-with-explicit-examples/7984#7984 Comment by Anonymous Anonymous 2010-03-14T16:03:28Z 2010-03-14T16:03:28Z I agree with Andrew L here. I too was disappointed at how little stuff was covered in this book. http://mathoverflow.net/questions/18100/theorems-with-unexpected-conclusions/18110#18110 Comment by Anonymous Anonymous 2010-03-14T15:37:34Z 2010-03-14T15:37:34Z The existence of a polynomial whose positive integer values are all primes is indeed surprising. I can't help to ask: what important consequences follow from that existence? http://mathoverflow.net/questions/18100/theorems-with-unexpected-conclusions/18122#18122 Comment by Anonymous Anonymous 2010-03-14T15:33:57Z 2010-03-14T15:33:57Z @Kevin: yes, I knew that and thanks for making it clear. I was just surprised to hear that Siegel had (as it turns out) another proof of the result. http://mathoverflow.net/questions/18100/theorems-with-unexpected-conclusions/18122#18122 Comment by Anonymous Anonymous 2010-03-14T06:01:05Z 2010-03-14T06:01:05Z Thanks. The papers listed in the answers to that question is the one I remember seeing. So this means that the Theorem above is a true theorem. http://mathoverflow.net/questions/18100/theorems-with-unexpected-conclusions/18122#18122 Comment by Anonymous Anonymous 2010-03-14T04:37:19Z 2010-03-14T04:37:19Z Is it true that the only Fibonacci numbers that are cubes are $0, \pm 1, \pm 8$? I seem to recall a recent paper proving this... http://mathoverflow.net/questions/17778/books-you-would-like-to-see-translated-into-english/17883#17883 Comment by Anonymous Anonymous 2010-03-11T18:23:45Z 2010-03-11T18:23:45Z That's great to hear.