Facts from algebraic geometry that are useful to non-algebraic geometers - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T23:28:24Z http://mathoverflow.net/feeds/question/36471 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/36471/facts-from-algebraic-geometry-that-are-useful-to-non-algebraic-geometers Facts from algebraic geometry that are useful to non-algebraic geometers Charles Staats 2010-08-23T17:49:00Z 2010-11-02T03:24:32Z <p>A professor of mine (a geometric topologist, I believe) once criticized the core graduate curriculum at my institution because it teaches all sorts of esoteric algebra, but does not include basic information about Galois theory and algebraic geometry, which, according to him, are important even for non-algebraists.</p> <blockquote> <p>What are some useful facts from algebraic geometry that are useful for non-algebraic geometers? Ideally, the statements at least should be accessible without knowing much algebraic geometry. </p> </blockquote> <p>Edit: Please do not post results that are only relevant to people who already know massive amounts of algebraic geometry anyway. In particular: Be very cautious about posting statements whose only applications are in number theory.</p> <p>Example: Here is a basic statement that I have seen applied outside algebraic geometry, if not necessarily outside of algebra:</p> <p>Let $U \subset \mathbb{C}^n$. If there is some nonzero polynomial satisfied by every point of $\mathbb{C}^n \smallsetminus U$, then $U$ is dense in $\mathbb{C}^n$ (with the usual topology), and in fact contains a dense open subset of $\mathbb{C}^n$.</p> <p>[Sketch of proof: Given any point $p \in \mathbb{C}^n$, find a complex line $L$ passing through $p$ that intersects $U$. Then $L \cap (\mathbb{C}^n \smallsetminus U)$ is algebraic, hence contains only finitely many points of $L$, and so $p$ is a limit point of $U$.] </p> http://mathoverflow.net/questions/36471/facts-from-algebraic-geometry-that-are-useful-to-non-algebraic-geometers/36472#36472 Answer by Ricky for Facts from algebraic geometry that are useful to non-algebraic geometers Ricky 2010-08-23T17:55:33Z 2010-08-23T17:55:33Z <p>Any compact Riemann surface is projective and algebraic.</p> <p>Riemann surfaces are studied in analysis an differential geometry, and of course is easier to work with polynomial equations. This statement is useful also for studying non compact Riemann surfaces.</p> http://mathoverflow.net/questions/36471/facts-from-algebraic-geometry-that-are-useful-to-non-algebraic-geometers/36474#36474 Answer by Steve Huntsman for Facts from algebraic geometry that are useful to non-algebraic geometers Steve Huntsman 2010-08-23T18:19:15Z 2010-08-23T18:19:15Z <p>The basic theory of curves is essential to modern communication theory, particularly in the construction of <a href="http://en.wikipedia.org/wiki/Algebraic_geometric_code" rel="nofollow">error-correcting codes</a> and <a href="http://en.wikipedia.org/wiki/Elliptic_curve_cryptography" rel="nofollow">elliptic-curve cryptosystems</a>.</p> http://mathoverflow.net/questions/36471/facts-from-algebraic-geometry-that-are-useful-to-non-algebraic-geometers/36476#36476 Answer by Helge for Facts from algebraic geometry that are useful to non-algebraic geometers Helge 2010-08-23T18:29:31Z 2010-08-23T18:29:31Z <p>Hyperelliptic curves play a basic role in the construction of solutions to completely integrable systems (soliton equations), e.g. KdV. But it should be noted that these solutions are of non-soliton type.</p> <p>The basic idea is, that such a system can be written as a Lax pair: $$\dot{L} = [P_j, L]$$ for some $j$. Different $j$ correspond to different members of the hierachy. Now to construct such an algebro-geometric solution. Consider some $\ell > j$, and look for an $L$ such that $$[L, P_{\ell}] = 0$$ then some general theory implies that this solution satisfies a polynomial equation, and can thus be written in terms of data on a curve.</p> http://mathoverflow.net/questions/36471/facts-from-algebraic-geometry-that-are-useful-to-non-algebraic-geometers/36477#36477 Answer by David Speyer for Facts from algebraic geometry that are useful to non-algebraic geometers David Speyer 2010-08-23T18:35:08Z 2010-08-23T18:35:08Z <p>If $p_1$, $p_2$, ..., $p_m$ are polynomials in $n$ variables, with $m>n$, then there is a polynomial $q$ such that $q(p_1, p_2, \ldots, p_m)$ is identically zero.</p> http://mathoverflow.net/questions/36471/facts-from-algebraic-geometry-that-are-useful-to-non-algebraic-geometers/36484#36484 Answer by Jim Humphreys for Facts from algebraic geometry that are useful to non-algebraic geometers Jim Humphreys 2010-08-23T19:27:21Z 2010-08-23T19:27:21Z <p>In the classical theory of semisimple Lie algebras over the complex numbers (and elsewhere in Lie theory), it's convenient to apply easy Zariski density arguments for some underlying affine spaces. For instance, a natural proof of Harish-Chandra's basic theorem on the structure and characters of the center of the universal enveloping algebra involves restriction of polynomial functions from the Lie algebra to a Cartan subalgebra. Here the density of "regular" elements makes it possible to focus just on their behavior. Similarly, some classical conjugacy theorems for the Lie algebras relative to the adjoint group action are most easily studied in geometric terms. The point is that polynomials play a prominent role, making even the most elementary parts of algebraic geometry helpful.</p> http://mathoverflow.net/questions/36471/facts-from-algebraic-geometry-that-are-useful-to-non-algebraic-geometers/36495#36495 Answer by Piero D'Ancona for Facts from algebraic geometry that are useful to non-algebraic geometers Piero D'Ancona 2010-08-23T20:47:14Z 2010-08-23T20:47:14Z <p>The Tarski-Seidenberg theorem states that if you project a semialgebraic set (i.e. given by polynomial in/equalities) from $\mathbb{R}^{n+1}$ to $\mathbb{R}^{n}$ you obtain another semialgebraic set. This was used by Lars Hormander (maybe even earlier by Lars Garding) in a few spectacular applications to characterize the solvability and regularity properties of constant coefficient PDEs. I understand that the TS theorem has applications to logic, model theory, functional analysis, and other fields, but I have no direct knowledge of them so maybe some expert might be willing to comment.</p> http://mathoverflow.net/questions/36471/facts-from-algebraic-geometry-that-are-useful-to-non-algebraic-geometers/36499#36499 Answer by algori for Facts from algebraic geometry that are useful to non-algebraic geometers algori 2010-08-23T21:23:55Z 2010-08-24T20:37:17Z <p>I would mention B&eacute;zout's theorem. Forgetting the complicated general definition of the intersection index one of the consequences is: whenever two curves in $\mathbf{P}^2(K), K$ an algebraically closed field have no common component, the number of the intersection points is finite and is always at most the product of the degrees. Moreover, it is the product of the degrees provided no intersection point is a singular point and all intersections are transversal. To state this over $\mathbf{C}$ we essentially only need multivariable calculus. But a proof requires a bit of algebraic geometry.</p> <p>Let me also mention two algebraic geometry books that present precisely the kind of material people in other areas are likely to use. One is "Algebraic geometry, a first course" by Joe Harris; the other is "Undergraduate algebraic geometry" by Miles Reid. If memory serves, it says somewhere in the latter book that it covers (together with Atiyah-MacDonald) all algebraic geometry questions that the author was ever asked by his colleagues who specialize in other areas.</p> http://mathoverflow.net/questions/36471/facts-from-algebraic-geometry-that-are-useful-to-non-algebraic-geometers/36510#36510 Answer by ABayer for Facts from algebraic geometry that are useful to non-algebraic geometers ABayer 2010-08-24T01:09:59Z 2010-08-24T01:09:59Z <p>I would vote for Chevalley's theorem as <em>the</em> most basic fact in algebraic geometry:</p> <blockquote> <p>The image of a constructible map is constructible.</p> </blockquote> <p>More down to earth, its most basic case (which, I think, already captures the essential content), is the following: the image of a polynomial map $\mathbb{C}^n \to \mathbb{C}^m$, $z_1, \dots, z_n \mapsto f_1(\underline{z}), \dots, f_m(\underline{z})$ can always be described by a set of polynomial equations $g_1= \dots = g_k = 0$, combined with a set of polynomial ''unequations'' (*) $h_1 \neq 0, \dots, h_l \neq 0$.</p> <p>David's post is a special case (if $m > n$, then the image can't be dense, hence $k > 0$). Tarski-Seidenberg is basically a version of Chevalley's theorem in ''semialgebraic real geometry''. More generally, I would argue it is the reason why engineers buy Cox, Little, O'Shea ("Using algebraic geometry"): in the right coordinates, you can parametrize the possible configurations of a robotic arm by polynomials. Then Chevalley says the possible configuration can also be described by equations.</p> <p>(*) Really it seems that "inequalities" would be the right word her...might be a little late to change terminology though...</p> http://mathoverflow.net/questions/36471/facts-from-algebraic-geometry-that-are-useful-to-non-algebraic-geometers/36566#36566 Answer by Jim Humphreys for Facts from algebraic geometry that are useful to non-algebraic geometers Jim Humphreys 2010-08-24T16:31:20Z 2010-08-24T16:31:20Z <p>This is another answer, in a different direction (about which I personally know little and can't offer any value judgments). While applied mathematicians are unlikely to be interested in the more abstract parts of modern algebraic geometry, some fairly sophisticated ideas have found their way into the literature of "systems theory". Chris Byrnes, one of our enterprising UMass Ph.D. students in the 1970s, went in that direction after learning from John Fogarty and others about moduli spaces. Chris spent time later with Roger Brockett's group at Harvard, then had an active career in university teaching and administration. One of his early papers gives an indication of how geometric ideas interacted with more applied problems:</p> <p>Christopher I. Byrnes, On the control of certain deterministic, infinite-dimensional systems by algebro-geometric techniques. Amer. J. Math. 100 (1978), no. 6, 1333–1381.</p> http://mathoverflow.net/questions/36471/facts-from-algebraic-geometry-that-are-useful-to-non-algebraic-geometers/36573#36573 Answer by Mitch Harris for Facts from algebraic geometry that are useful to non-algebraic geometers Mitch Harris 2010-08-24T17:53:37Z 2010-08-24T17:53:37Z <p>Grobner basis calculation is very practical for engineering, but also theoretically, in combinatorics (e.g. to prove colorability of given graph classes) and theoretical computer science (e.g. for polynomial interpretations to prove termination of programs).</p> http://mathoverflow.net/questions/36471/facts-from-algebraic-geometry-that-are-useful-to-non-algebraic-geometers/36586#36586 Answer by Artie Prendergast-Smith for Facts from algebraic geometry that are useful to non-algebraic geometers Artie Prendergast-Smith 2010-08-24T19:02:54Z 2010-08-24T19:02:54Z <p>This falls outside the scope of "basic facts", but it seems interesting enough to mention here: apparently the seminar in our AG group in Hannover this week will be about applying resolution of singularities (more specifically, the concept of log canonical thresholds) to a problem of Bayesian statistics. Maybe this is a standard technique, but it certainly surprised me.</p> <p>Abstract: <a href="http://www.iag.uni-hannover.de/de/oberseminar/abstracts/abstract.php?in=lin_de.html" rel="nofollow">http://www.iag.uni-hannover.de/de/oberseminar/abstracts/abstract.php?in=lin_de.html</a></p> http://mathoverflow.net/questions/36471/facts-from-algebraic-geometry-that-are-useful-to-non-algebraic-geometers/38983#38983 Answer by Denis Serre for Facts from algebraic geometry that are useful to non-algebraic geometers Denis Serre 2010-09-16T15:35:48Z 2010-09-22T14:16:14Z <p>Just have a look at the XIXth century. Say that you look for a primitive of an algebraic expression. The general question is whether this primitive can be written in terms of elementary functions (rational fraction and logarithms). The algebraic expression is usually associated to some algebraic curve. The answer is yes iff the curve admits a rational parametrization. When it is non-singular, this is equivalent to having genus $0$.</p> <p>For instance, if $R$ is rational, then $$\int R\left(x,\sqrt{x^2+ax+b}\right)dx$$ can be expressed in terms of elementary functions. On the contrary, $$\int \sqrt{x^3+ax+b}\,dx$$ cannot, unless the polynomial $x^3+ax+b$ has a double root.</p> <p>A more advanced situation is that of hyperbolic linear Partial Differential Equations. The differential operator defines a symbol, which is a polynomial in several variables. The properties of its zero set, an algebraic variety, are crucial in many aspects, for instance in determining whether Huyghens principle holds (theory of lacunas). In the Russian school, prominent researchers in PDE were also active in algebraic geometry (Petrovski, Oleinik).</p> <p>A definitely more advanced situation is the use of algebraic geometry in the analysis of linear initial-boundary value problems. Let $L$ be a differential operator, for which the Cauchy problem is well-posed. A necessary condition for an IBVP to be well-posed in ${\mathcal C}^\infty$ is the so-called <em>Lopatinskii Condition</em>, which is algebraic and parametrized by frequencies (along boundary and time). If one replaces ${\mathcal C}^\infty$ by a Sobolev space $H^s$, then the Lopatinskii condition has to be satisfied <em>uniformly</em>. In several interesting cases, LC or ULC condition turns out to be sufficient for well-posedness, but this requires the construction of a so-called <em>dissipative symmetrizer</em>, which relies upon algebraic geometry. For hyperbolic operators, see the work of H.-O. Kreiss (ULC) and the books by R. Sakamoto (LC) or by S. Benzoni-Gavage and myself (ULC).</p> http://mathoverflow.net/questions/36471/facts-from-algebraic-geometry-that-are-useful-to-non-algebraic-geometers/38987#38987 Answer by algori for Facts from algebraic geometry that are useful to non-algebraic geometers algori 2010-09-16T16:10:38Z 2010-09-16T16:23:20Z <p>Since no one has done it so far let me also mention the Riemann-Roch theorem for complex analytic or algebraic curves. (True, if one is mainly interested in the analytic case, then the algebraic version will not be of much use since to apply it one needs to show first that any Riemann surface is algebraic, which is most easily done by taking a projective embedding, which requires in turn the analytic Riemann-Roch theorem.) One needs only 1-variable complex analysis to define smooth compact complex curves and rational functions and divisors on them. Now if the degree of a divisor $D$ is $>2g-2$ where $g$ is the genus, then $\dim\mathcal{L}(D)=d-g+1$ where $d=deg(D)$. One does not even need the canonical divisor to state this.</p> <p>As a consequence one gets many results on meromorphic functions on Riemann surfaces. For instance</p> <ol> <li><p>there exist nonconstant meromorphic functions.</p></li> <li><p>the Mittag-Leffler problem (find a meromorphic differential form with given poles and given principal parts at the poles) has a solution iff the sum of the residues is 0. There is a similar statement for meromorphic functions that can be proved in a similar way.</p></li> <li><p>any algebraic curve (or Riemann surface) is projective and moreover can be embedded in $\mathbf{P}^3$.</p></li> </ol> http://mathoverflow.net/questions/36471/facts-from-algebraic-geometry-that-are-useful-to-non-algebraic-geometers/44454#44454 Answer by roy smith for Facts from algebraic geometry that are useful to non-algebraic geometers roy smith 2010-11-01T15:51:22Z 2010-11-02T03:24:32Z <p>Not a fact but a philosophy: to me the most important way of thinking in algebraic geometry that would be useful in many areas is that of a moduli space. I.e. the idea that the set of isomorphism classes of certain objects should be viewed with structure of that same type, and its properties studied as a tool for understanding the original types of objects. This I believe is basic to the work of Chris Byrnes alluded to above. This philosophy is perhaps not due to or original with algebraic geometry, but is practiced systematically there. It may derive from algebraic topology, (classification of vector bundles, E.H. Brown's representability of cohomology,....), like many other things in AG.</p> <p>It might be of interest e.g. to some high school students to know that Euclid proved the set of congruence classes of circles is an open half line, and that the set of all triangles modulo similarity is parametrized by the interior of an isosceles right triangle, modulo the reflection in the altitude on the hypotenuse (via the unordered coordinates AA given by the two largest angles), hence also the interior of an isosceles right triangle, but with the interior of one edge added in. Then the set of congruence classes of triangles can be seen as the product of this triangular region with an infinite open half line, i.e. an infinite parallelpiped, (via the ASA theorem). They might then compare this with the realization of this same space by the coordinates SSS.</p>