User kristianjs - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T06:47:46Z http://mathoverflow.net/feeds/user/19367 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/130348/elementary-extensions-and-type-spaces Elementary extensions and type spaces KristianJS 2013-05-11T15:21:24Z 2013-05-11T22:10:26Z <p>If $M$ and $N$ are two $L$-structures, and $f: M \rightarrow N$ is an elementary extension, then given any subset $A$ of $M$, $f$ induces in a natural way a morphism $S^M_n(A) \rightarrow S^N_n(f(A))$ of type spaces which is in fact a homeomorphism of Stone spaces (in fact the underlying sets are equal: see the answer below). </p> <p>What results exist about converse statements? If two type spaces are equal, when can you say it's because of the existence of an elementary embedding? What if you have a non-trivial homeomorphism, or some more general injective continuous map? Can you sometimes conclude that that it's induced by a suitable morphism between the structures?</p> http://mathoverflow.net/questions/103243/what-are-some-interesting-problems-in-the-intersection-of-algebraic-number-theory/103351#103351 Answer by KristianJS for What are some interesting problems in the intersection of Algebraic Number Theory and Algebraic Topology? KristianJS 2012-07-27T22:49:28Z 2012-07-27T22:49:28Z <p>I think the whole field of anabelian geometry fits the bill, even if it's perhaps more focused on going the other way around (i.e. applying homotopy theory to number theory). Anabelian geometry is a 'program' launched by Grothendieck in his famous Esquisse d'un Programme, and is all about translating arithmetic geometric problems to problems in homotopy theory.</p> <p>As an example of a specific instance of the anabelian philosophy, we have Grothendieck's celebrated 'section conjecture', which states (in one form) that for a 'nice' curve $X$ over a number field $F$, the rational points are in bijection with the sections of the exact sequence $$1 \rightarrow \pi_1(X_{\bar{F}}) \rightarrow \pi_1(X) \rightarrow G_F \rightarrow 1$$ where $G_F$ is the absolute Galois group of $F$ and $\pi_1$ is the algebraic (etale) fundamental group. In case the curve is over the complex numbers, the etale $\pi_1$ is the profinite completion of the regular fundamental group, so there is a very close connection to the classical stuff of Hatcher. The conjecture is still a wide open problem, but any proof would mean you could check something of number theoretic interest (existence of rational points on curves) by studying maps between certain generalized homotopy groups!</p> http://mathoverflow.net/questions/92927/why-does-the-section-conjecture-exclude-curves-of-genus-1 Why does the Section Conjecture exclude curves of genus 1? KristianJS 2012-04-02T21:21:53Z 2012-04-03T12:46:36Z <p>Let $X$ be an integral proper normal curve over a (perfect) field $F$, of genus $\geq 2$. One variant of Grothendieck's "section conjecture" states that the sections $G_F \rightarrow \pi_1(X)$ of the exact sequence $$1 \rightarrow \pi_1(X_{\bar{F}}) \rightarrow \pi_1(X) \rightarrow G_F \rightarrow 1$$ are, up to conjugation, in bijection with the $F$-rational points of $X$, where $G_F$ is the absolute Galois group of $F$ and $\pi_1$ is the algebraic fundamental group. </p> <p><b>Question: what is the reason for excluding genus 1 curves?</b></p> <p>I understand why genus 0 curves must be excluded: if $F$ has characteristic zero, it is a general fact that the 'geometric' fundamental group $\pi_1(X_{\bar{F}})$ is just the profinite completion of the regular <i>topological</i> fundamental group of $X$, seen as a curve over $\mathbb{C}$. For genus 0, the topological fundamental group is trivial, and thus the above exact sequence induces an isomorphism $\pi_1(X) \rightarrow G_F$. Hence there is always at least one section even if $X$ has no rational points whatsoever. </p> <p>However, I don't know of a good reason why genus 1 curves should be excluded here. The above argument obviously won't do since the topological fundamental group is no longer trivial for genus 1. Are there even so known counter-examples for genus 1 curves? What goes wrong?</p> <p>I know the philosophy is that one should expect 'anabelian behaviour' only when the fundamental group is 'far from being abelian', which excludes the genus 1 case. But I would be more satisfied with a more concrete, less philosophical, reason!</p> http://mathoverflow.net/questions/81209/galois-theory-for-polynomials-in-several-variables/82621#82621 Answer by KristianJS for Galois theory for polynomials in several variables KristianJS 2011-12-04T13:37:46Z 2011-12-04T13:37:46Z <p>(This should really be a comment I think, but I'm not highly rated enough to leave one, so please bear with me)</p> <p>A Galois Theoretic condition for a polynomial in two variables to be solvable by radicals is found in the following paper: <a href="http://arxiv.org/abs/math/0305226" rel="nofollow">http://arxiv.org/abs/math/0305226</a>. It seems to indicate that something similar can be done for higher variables. Perhaps I'll ask Jochen next time I see him about this.</p> http://mathoverflow.net/questions/130348/elementary-extensions-and-type-spaces/130361#130361 Comment by KristianJS KristianJS 2013-05-12T13:50:28Z 2013-05-12T13:50:28Z I would say in that example that you know something very interesting about the relationship between M and N actually. However, I appreciate the point that type spaces are perhaps quite bad at telling structures apart. Perhaps in certain specific cases though you can say something. I'll leave this question open for now. http://mathoverflow.net/questions/130348/elementary-extensions-and-type-spaces/130361#130361 Comment by KristianJS KristianJS 2013-05-11T22:11:22Z 2013-05-11T22:11:22Z Thanks for pointing this out. However, it does not answer the question. I have edited it to be more general to take into account non-trivial maps as well. http://mathoverflow.net/questions/108860/anabelian-geometry-study-materials/108910#108910 Comment by KristianJS KristianJS 2012-10-05T13:01:13Z 2012-10-05T13:01:13Z I'm sure this book will be the one to get, once it comes out. If you start with Szamuely as an introduction, you could then move on to this afterwards. http://mathoverflow.net/questions/108842/the-significance-of-modularity-for-all-galois-representations Comment by KristianJS KristianJS 2012-10-04T23:37:54Z 2012-10-04T23:37:54Z Just to nitpick, in your reference to Minhyong's talk, he's not saying that the Frey curve is an example of the sort of canonical association of points with the path space. He's just pointing out how associating a point with something with more structure is often useful, and that the association of a point with the etale path-space is an entirely systematic way of doing this for any scheme. http://mathoverflow.net/questions/107379/model-theoretic-content-of-mochizukis-teichmuller-theory-papers Comment by KristianJS KristianJS 2012-09-17T14:04:03Z 2012-09-17T14:04:03Z Oh, I should add that I'm pretty sure he's doing the above not just for schemes, but for even more abstract stuff (what he calls 'species' I think?) http://mathoverflow.net/questions/107379/model-theoretic-content-of-mochizukis-teichmuller-theory-papers Comment by KristianJS KristianJS 2012-09-17T13:59:18Z 2012-09-17T13:59:18Z they should, theoretically, be described by a formula in the language of set theory. The philosophy of model theory tells us it should be profitable to study this formula in lots of different models of ZFC (or, as I believe Mochizuki does, in ZFC + Grothendieck's Universe Axiom). It seems to me that the novelty is actually believing you can extract something even remotely concrete out of this! Now if only I actually understood anything about how he does it, then I could maybe submit an answer and not a comment.... http://mathoverflow.net/questions/107379/model-theoretic-content-of-mochizukis-teichmuller-theory-papers Comment by KristianJS KristianJS 2012-09-17T13:58:57Z 2012-09-17T13:58:57Z It is my impression that model theorists have not really been much interested in applying the basic philosophy of model theory (i.e. that it is profitable to study a set of formulas by looking at a whole bunch of models as opposed to just a 'canonical' one) to abstract schemes. This kinda makes sense, since in one way the point of schemes is that they let you do geometry without actually having to worry about specific equations, and so they don't a priori seem to lend themselves well to model theory, which absolutely requires explicit formulas. Being mathematical objects however...(cont.) http://mathoverflow.net/questions/93163/solve-3x-sin-x-cosx-3-equation Comment by KristianJS KristianJS 2012-04-04T20:15:45Z 2012-04-04T20:15:45Z Try <a href="http://math.stackexchange.com/" rel="nofollow">math.stackexchange.com</a> instead. http://mathoverflow.net/questions/92927/why-does-the-section-conjecture-exclude-curves-of-genus-1/92992#92992 Comment by KristianJS KristianJS 2012-04-03T19:25:46Z 2012-04-03T19:25:46Z Thanks, this is exactly the kind of answer I was hoping for!