most recent 30 from http://mathoverflow.net 2013-05-26T07:18:11Z http://mathoverflow.net/feeds/user/5080 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20112/interesting-results-in-algebraic-geometry-accessible-to-3rd-year-undergraduates ifk 2010-04-01T23:12:58Z 2010-10-24T23:46:49Z

<p>On <a href="http://mathoverflow.net/questions/20077/how-would-you-encourage-graduate-students-to-learn-algebraic-geometry-and-or-comp" rel="nofollow">another thread</a> I asked how I could encourage my final year undergraduate colleagues to take an algebraic geometry or complex analysis courses during their graduate studies.</p> <p>Willie Wong proposed me following idea - to show them some interesting results in this fields with relatively simply proofs and some consequences in other fields. Thus by 'interesting' result in algebraic geometry I here mean the result which may convince 3rd year undergraduate student to study algebraic geometry.</p> <p>In fact I'm supposed to give some talk during the seminar dedicated to final year undergraduates, and I can propose my own topic, so I thought that it could work. But my problem is that I'm just wanna-be student of algebraic geometry and I don't have enough insight and knowledge to find a topic which 'could work'.</p> <p>Also I doubt whether it is possible to present some intriguing ideas of algebraic geometry to audience without any preparation in this field.</p> <p>So in short my first question is as above:</p> <blockquote> <p>Is it possible to present some intriguing ideas of algebraic geometry to audience without any preparation in this field?</p> </blockquote> <p>To be more precise - the talk should take a one or two meetings, 90 minutes each one. The audience will be, as I said 3rd year undergraduate students, all of them after two semester course in algebra, some of them after one or two semesters in commutative algebra. All taking the course in one complex variable, and after one semester introductory course in differential geometry. Some of them may be interested in number theory.</p> <p>The second, related question is as follows:</p> <blockquote> <p>If You think that the answer to the previuos question is positive, please try to give an example of idea/theorem/result which would be accessible in such time for such audience, and which You find interesting enough to make them consider possibility of studying algebraic geometry. Try to think about the results which shows connections of AG with some other fields of mathematics.</p> </blockquote>

http://mathoverflow.net/questions/20826/when-is-the-set-of-zero-divisors-equal-to-the-union-of-the-minimal-primes-in-a-re/20833#20833 ifk 2010-04-09T13:04:47Z 2010-04-09T13:04:47Z

<p>The answer is: <em>always</em>, and argument is pretty simple:</p> <p>Let $R$ be a reduced commutative unital ring. If $a\in R$ is a zero divisor, then $ab=0,$ for some $b\neq 0.$ Hence $b\not \in 0 = \text{nil}(R)= \bigcap \text{Spec}(R) =\bigcap \text{Specmin}(R),$ where $\text{Specmin}(R)$ states for family of all minimal prime ideals of $R.$ So there exists a minimal prime ideal $\mathfrak{p}\triangleleft R$ s.t. $b\not \in \mathfrak{p}.$ But $\mathfrak{p}$ is prime, and $ab\in \mathfrak{p},$ thus $a\in \mathfrak{p}.$ Hence the set of zero divisors is contained in union of minimal prime ideals. </p> <p>On the other hand it is well-known fact that minimal prime ideals in commutative unital rings consist of zero divisors, so the set of zero divisors in reduced commutative unital ring is exactly the union of minimal prime ideals.</p>

http://mathoverflow.net/questions/16829/what-are-your-favorite-instructional-counterexamples/20329#20329 ifk 2010-04-04T20:07:09Z 2010-04-04T20:07:09Z

<p>Here is some simple counterexample in commutative algebra, which I found really cute when I first meet it:</p> <p>Let $k$ be a field, $A = k[X_{1},X_{2},X_{3}\ldots],$ $I = (X_{1}, X_{2}^{2}, X_{3}^{3},\ldots)$ and $R = A/I.$ Then $\text{Spec}(R)$ consists of one point (because $\text{rad}(I)$ is maximal ideal of $A$); in particular $\text{Spec}(R)$ is a noetherian space, and $\dim R = 0$; although $R$ is not noetherian ring (since $\text{nil}(R)^{n}\neq 0$ for every $n$).</p>

http://mathoverflow.net/questions/20077/how-would-you-encourage-graduate-students-to-learn-algebraic-geometry-and-or-comp ifk 2010-04-01T15:05:08Z 2010-04-01T20:04:58Z

<p>Hello,</p> <p>I am the 3rd year undegraduate student of mathematics. After I obtain a bachelor degree I want to study maths at graduate level, especially algebraic geometry and complex analysis. This fields of mathematics are well-represented at my univeristy, so at the first glance this plan looks fine.</p> <p>Unfortunately almost all of my collegues are not interested in this fields (they think that AG and CA are too technically involved; most of them are interested mainly in functional analysis and topology). It will have such unpleasant effect on my studies, that most probably the most (or all) of courses in AG and CA in the upcoming year won't start at my university.</p> <p>So I'll end up learning alone, from books. It's not that it's a big problem to learn from the book, it's not a problem at all. But I think such learning have no comparison with the regular course, where I could discuss problems and see another approches of other (more gifted) students.</p> <p>To avoid such situation I may try to convince my collegues to study CA and AG, but I don't have many arguments since I'm still an ignorant in this fields. And here arise my questions:</p> <blockquote> <ol> <li>How would You encourage graduate students to learn algebraic geometry?</li> <li>How would You encourage graduate students to learn complex analysis?</li> </ol> </blockquote>

http://mathoverflow.net/questions/16829/what-are-your-favorite-instructional-counterexamples/20329#20329 ifk 2010-04-05T09:47:48Z 2010-04-05T09:47:48Z

Nice, thank You Mariano. However I don't think it's really much simpler than above.

http://mathoverflow.net/questions/16829/what-are-your-favorite-instructional-counterexamples/20329#20329 ifk 2010-04-04T21:35:51Z 2010-04-04T21:35:51Z

Nevertheless, I think that it's not obvious that there exist commutative rings with only one prime (not only with one maximal) ideal that are not noetherian.

http://mathoverflow.net/questions/20112/interesting-results-in-algebraic-geometry-accessible-to-3rd-year-undergraduates/20115#20115 ifk 2010-04-02T21:52:28Z 2010-04-02T21:52:28Z

@hilbertthm90: In &quot;13 Lectures on Fermat Last Theorem&quot; by Paolo Ribenboim (pp. 265-266) there is a proof (using Picard Theorem) of following result: If $f,g$ are entire functions, $p\neq 0$ is a polynomial, $\deg p \le n-3$ (he remarks that the result is still valid for $n-2$ here) satisfying equation $(f(z))^n+(g(z))^n=p(z)$ then $f,g,p$ are constant. An immediate consequence is that if $f(z),g(z),h(z)$ are entire functions, $h$ is nonvanishing and $(f(z))^n+(g(z))^n=(h(z))^n$ for $n\ge 3$ then $f=ah,g=bh$ for some $a,b\in \mathbb{C}$ such that $a^n+b^n=1$

http://mathoverflow.net/questions/20123/problems-concerning-r-and-rx/20127#20127 ifk 2010-04-02T10:12:28Z 2010-04-02T10:12:28Z

Note that one can give a very similar proof of fact that if integral domain is an artinian ring then it is a field (this also follows from consideration involving dimension).

http://mathoverflow.net/questions/20112/interesting-results-in-algebraic-geometry-accessible-to-3rd-year-undergraduates/20115#20115 ifk 2010-04-02T09:08:51Z 2010-04-02T09:08:51Z

Perhaps the missing assumption is that $a,b,c$ are coprime. There is nice, quite elementary result, known as Mason's theorem, which implies that the equation $a^n+b^n=c^n$ (or even a more general one) doesn't have any solutions in coprime polynomials over arbitrary algebraic closed fields of characteristic zero (hence over arbitrary field of characteristic zero). You can see the details in the 11th paragraph of the Polynomial Excursions by Edward Barbeau.

http://mathoverflow.net/questions/20077/how-would-you-encourage-graduate-students-to-learn-algebraic-geometry-and-or-comp/20102#20102 ifk 2010-04-01T22:45:07Z 2010-04-01T22:45:07Z

Yes, of course. This idea comes to my mind too late this year (also I'm not sure if I have a formal ability to leave on the first semester of graduate studies), but I will serously consider such possibility as soon as it will be available.

http://mathoverflow.net/questions/20077/how-would-you-encourage-graduate-students-to-learn-algebraic-geometry-and-or-comp ifk 2010-04-01T21:51:40Z 2010-04-01T21:51:40Z

@Willie: To be honest I've got some alternative, i.e. in Poland there is for example Warsaw University at which mathematics are on similar level as at Jagiellonian University. I think I'll take enterance exams there anyway, but I've heard that my present university is better place to study CA, and this is the reason why I want to stay.

http://mathoverflow.net/questions/20077/how-would-you-encourage-graduate-students-to-learn-algebraic-geometry-and-or-comp/20102#20102 ifk 2010-04-01T21:39:13Z 2010-04-01T21:39:13Z

As to the another interests of fellow students - one of them is interested in number theory and I think it should help. Another one said me that he want to have good general knowledge about mathematics, so there is a hope. Another one study phisics and mathematics simultanously. Two others study theoretical computer science and mathematics. Nevertheless I need six students to run the course

http://mathoverflow.net/questions/20077/how-would-you-encourage-graduate-students-to-learn-algebraic-geometry-and-or-comp/20102#20102 ifk 2010-04-01T21:25:42Z 2010-04-01T21:25:42Z

Thanks for Your advice. I'm certain that I should leave the country sooner or later (maybe sooner) and I will, but I'm still not ready for such decision for many reasons (the main are money and language; also I'm afraid that it's a little too late for such decision as leaving in next semester).

http://mathoverflow.net/questions/20077/how-would-you-encourage-graduate-students-to-learn-algebraic-geometry-and-or-comp ifk 2010-04-01T15:41:12Z 2010-04-01T15:41:12Z

@Pete: I think about such options as leaving to another university or even studying in two places, but in my country the complex analysis is especially strong at my current university.

http://mathoverflow.net/questions/20077/how-would-you-encourage-graduate-students-to-learn-algebraic-geometry-and-or-comp ifk 2010-04-01T15:37:56Z 2010-04-01T15:37:56Z

Of course I'm not asking You about options avalaible at my university, I know what they are. In particular I'm aware of availability of independent study, but as I said I think that at this level regular course may be more profitable.

http://mathoverflow.net/questions/20077/how-would-you-encourage-graduate-students-to-learn-algebraic-geometry-and-or-comp ifk 2010-04-01T15:37:42Z 2010-04-01T15:37:42Z

As to background course - I'm after 2-semester course in algebra which covered some Galois theory, 2-semester course in commutative-algebra, and now I'm taking course in algebraic number theory. I'm taking also a course in 1 complex variable. Now it's a little too late to take some course as undergrad, since it's my final semester of my undergraduate studies. The current graduates also (with some few exceptions) are not interested in GA.

http://mathoverflow.net/questions/20077/how-would-you-encourage-graduate-students-to-learn-algebraic-geometry-and-or-comp ifk 2010-04-01T15:36:58Z 2010-04-01T15:36:58Z

I was talking about my collegues who are now as I 3rd year undegraduate students, and in upcoming year will be 1st year graduates, not about the ones who are already graduates.