User adam gal - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T20:52:41Z http://mathoverflow.net/feeds/user/4477 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/35748/is-completeness-of-a-field-an-algebraic-property Is completeness of a field an algebraic property? Adam Gal 2010-08-16T11:18:16Z 2010-11-13T18:25:55Z <p>Pretty straitforward: If a field has a metric in which it is complete can it have a metric in which it is not complete? By metric I mean field norm of course</p> http://mathoverflow.net/questions/35468/widely-accepted-mathematical-results-that-were-later-shown-wrong/35751#35751 Answer by Adam Gal for Widely accepted mathematical results that were later shown wrong? Adam Gal 2010-08-16T11:27:45Z 2010-08-16T11:27:45Z <p>Verma proved that the multiplicities of all simple modules in a verma module are 1 or 0. When BGG tried to repeat his proof for some other case they found an error. This led to the study of multiplicities in category O etc.</p> http://mathoverflow.net/questions/9754/magic-trick-based-on-deep-mathematics/31667#31667 Answer by Adam Gal for Magic trick based on deep mathematics Adam Gal 2010-07-13T06:15:02Z 2010-07-13T06:15:02Z <p><strong>The coffee mug trick</strong></p> <p>Give a coffee mug (full if you're brave) to someone and ask them to rotate 360 degrees without spilling the (real or imaginary) coffee, so that their hand ends up in the same position.</p> <p>This is impossible, so you get to smirk while they contort themselves and become more and more baffled (this works better with more than one person since it turns into a kind of "competition")</p> <p>Finally, take the cup and show that while it's impossible to turn it once (as has been "proven"), it's possible to turn it twice (!) and end up in the same position.</p> <p>Has to do with the fundamental group of SO(3) being $\mathbb{Z}/2\mathbb{Z}$, and when we require the cup to stay upright we end with a non-trivial loop.</p> http://mathoverflow.net/questions/28686/what-would-be-good-to-know-before-starting-my-undergraduate-studies-to-become-a-g/28737#28737 Answer by Adam Gal for What would be good to know before starting my undergraduate studies to become a good mathematician? Adam Gal 2010-06-19T09:25:30Z 2010-06-19T09:25:30Z <p>You need to know that you know nothing. Seriously though, that should be the guiding thought when learning mathematics. There is always some subtlety that can be missed or a concept that can be understood the wrong way (and is by many people) when you are too sure of your knowledge.</p> http://mathoverflow.net/questions/23414/centre-of-a-lie-algebra/23480#23480 Answer by Adam Gal for Centre of a Lie algebra Adam Gal 2010-05-04T21:06:00Z 2010-05-04T21:06:00Z <p>If the algebra is reductive it has no center. The meaningful center is the center of the universal enveloping algebra. Even there, the elements are not only combinations of what you call diagonalizable elements - they are btw called a cartan subalgebra of g. There is the Harish-Chandra theorem that says that the center is isomorphic to symmetric functions on the cartan subalgebra.</p> http://mathoverflow.net/questions/23384/how-long-is-the-average-piece-of-string/23388#23388 Answer by Adam Gal for How long is the average piece of string? Adam Gal 2010-05-03T23:11:04Z 2010-05-03T23:11:04Z <p>1 meter. Now it only remains to define a meter...</p> http://mathoverflow.net/questions/21152/do-non-associative-objects-have-a-natural-notion-of-representation/21153#21153 Answer by Adam Gal for Do non-associative objects have a natural notion of representation? Adam Gal 2010-04-12T20:18:11Z 2010-04-12T20:18:11Z <p>Taking the cue from Lie algebras you could try considering something like the enveloping algebra. In the Lie case a representation of $g$ is just a usual representation of $U(g)$ so maybe here you can make the same construction.</p> http://mathoverflow.net/questions/18085/question-about-ext Question about Ext Adam Gal 2010-03-13T18:48:36Z 2010-03-15T22:54:40Z <p>I heard that $Ext(M,N)$ is naturally isomorphic to $Ext(M^*\otimes N,1)$ where 1 is the trivial representation and $M,N$ some representations of a group $G$. Can anyone explain why? Is there an explicit construction of a map from one to the other or does it just follow from some general considerations about derived functors?</p> <p>Thanks.</p> http://mathoverflow.net/questions/18280/are-low-dimensional-modular-representations-of-sl2fp-completely-reducible Are low dimensional modular representations of SL2(Fp) completely reducible? Adam Gal 2010-03-15T16:37:48Z 2010-03-15T22:44:33Z <p>More specifically, is it true that a representation of $\dim &lt; p+1$ of the algebraic group $SL_2(\mathbb{F}_p)$ is always completely reducible? (of course above this dimension there are non completely reducible examples)</p> <p>More general results that might help in this direction are also welcome.</p> <p>Thanks</p> http://mathoverflow.net/questions/56677/what-notions-are-used-but-not-clearly-defined-in-modern-mathematics/56717#56717 Comment by Adam Gal Adam Gal 2011-10-07T13:37:12Z 2011-10-07T13:37:12Z &quot;made without choices&quot; is the usual &quot;definition&quot; The problem is that in many cases (maybe in all cases?) you just don't notice that you have made arbitrary choices along the way. Of course you can always say it's the canonical object given these choices, but if these choices are not &quot;natural&quot; (another one for the list btw) then the notion will not be useful because there would be no intrinsic reason for other objects to have this set of properties. http://mathoverflow.net/questions/50360/jacobson-radical-intersection-of-all-maximal-two-sided-ideals Comment by Adam Gal Adam Gal 2010-12-25T18:58:00Z 2010-12-25T18:58:00Z I wont venture so far as to say its bs in general, but it certainly doesnt apply to the k algebra case, since a simple algebra always splits over a finite extension of the base field http://mathoverflow.net/questions/35468/widely-accepted-mathematical-results-that-were-later-shown-wrong/35751#35751 Comment by Adam Gal Adam Gal 2010-08-28T13:09:47Z 2010-08-28T13:09:47Z I heard it from Bernstein in a lecture. Maybe he didn't mean exactly what I wrote but he definitely said that Verma had an error of this type. Maybe Verma claimed that what you mentioned implies the multiplicity freeness, I'm not sure. http://mathoverflow.net/questions/35748/is-completeness-of-a-field-an-algebraic-property/35750#35750 Comment by Adam Gal Adam Gal 2010-08-16T16:57:23Z 2010-08-16T16:57:23Z Thanks alot! Of course I was asking about Qp to begin with... http://mathoverflow.net/questions/35468/widely-accepted-mathematical-results-that-were-later-shown-wrong/35686#35686 Comment by Adam Gal Adam Gal 2010-08-16T11:29:40Z 2010-08-16T11:29:40Z It probably wasn't a real error, just a drool stain :-D http://mathoverflow.net/questions/9754/magic-trick-based-on-deep-mathematics/19919#19919 Comment by Adam Gal Adam Gal 2010-07-13T06:37:38Z 2010-07-13T06:37:38Z The one with the plasticine is more of an exercise for mathematicians to figure out what is wrong with the method. http://mathoverflow.net/questions/30081/origin-of-symbol-l-for-a-prime-different-from-a-fixed-prime/30090#30090 Comment by Adam Gal Adam Gal 2010-06-30T19:34:41Z 2010-06-30T19:34:41Z l and p are the first and last letters in the rythm segment l-m-n-o-p in the abc song. http://mathoverflow.net/questions/23414/centre-of-a-lie-algebra/23480#23480 Comment by Adam Gal Adam Gal 2010-05-04T21:26:45Z 2010-05-04T21:26:45Z You're right, I was thinking of semi-simple. http://mathoverflow.net/questions/23384/how-long-is-the-average-piece-of-string/23388#23388 Comment by Adam Gal Adam Gal 2010-05-04T13:02:07Z 2010-05-04T13:02:07Z A. it was a joke B. I'm pretty sure the original asker is a troll... http://mathoverflow.net/questions/23384/how-long-is-the-average-piece-of-string/23388#23388 Comment by Adam Gal Adam Gal 2010-05-03T23:30:04Z 2010-05-03T23:30:04Z and a second is? :-O http://mathoverflow.net/questions/7389/what-are-the-most-overloaded-words-in-mathematics/8499#8499 Comment by Adam Gal Adam Gal 2010-04-21T10:54:12Z 2010-04-21T10:54:12Z Good is used in the same way... http://mathoverflow.net/questions/20872/when-can-we-cancel-vector-bundles-from-tensor-products/21253#21253 Comment by Adam Gal Adam Gal 2010-04-13T20:24:47Z 2010-04-13T20:24:47Z I'm sure he meant direct sum, as evidenced by his orthogonal complement notation, where such examples come from. http://mathoverflow.net/questions/21152/do-non-associative-objects-have-a-natural-notion-of-representation/21153#21153 Comment by Adam Gal Adam Gal 2010-04-12T20:23:51Z 2010-04-12T20:23:51Z My guess would be then that representations in a vector space in such a case would be meaningless, and you would need to search for some other structure. http://mathoverflow.net/questions/20946/criteria-to-determine-whether-a-real-coefficient-polynomial-has-real-root Comment by Adam Gal Adam Gal 2010-04-10T23:50:16Z 2010-04-10T23:50:16Z Just a thought: This set is probably not a Zariski closed set of the polynomials since it is a union of infinitely many codim 1 sets which intersect in codim 2 or more. So probably there is no algebraic condition for belonging to it. http://mathoverflow.net/questions/20471/why-are-free-groups-residually-finite/20485#20485 Comment by Adam Gal Adam Gal 2010-04-06T09:42:02Z 2010-04-06T09:42:02Z You are assuming all the generators in the word are different, but this is not really the case in a free group. What about the word ababababab?