User eric zaslow - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T15:46:26Z http://mathoverflow.net/feeds/user/1186 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94347/eigenforms-for-laplacian-on-a-non-flat-two-torus Eigenforms for Laplacian on a non-flat two-torus Eric Zaslow 2012-04-18T02:20:17Z 2012-04-20T11:40:44Z <p>Does anyone know an explicit, exact description of the eigenforms of the Laplacian on a non-flat two-torus?</p> http://mathoverflow.net/questions/36444/definition-of-an-e-infinity-algebra Definition of an E-infinity algebra Eric Zaslow 2010-08-23T13:20:53Z 2011-05-26T21:20:19Z <p>Hello. Can anyone give me a plain-and-simple definition of an E-infinity algebra without using the words "operad," "ring spectrum," or "stable homotopy"?</p> <p>Sorry, but I honestly couldn't find it using all on-line resources at my disposal.</p> <p>Thanks!</p> http://mathoverflow.net/questions/58169/why-do-a-infty-functors-form-an-a-infty-category/60014#60014 Answer by Eric Zaslow for Why do A_\infty functors form an A_\infty category? Eric Zaslow 2011-03-29T19:24:38Z 2011-03-30T01:59:07Z <p>I can explain the pictures I usually draw to think of $A_\infty$ functors, but I don't know if they're standard. Anyway, I'll describe what is just a rubric for ingesting the long formulas, nothing more.</p> <p>Let's consider first the Yoneda embedding $Y$, which re-thinks an object $L$ in an $A_\infty$-category $A$ as an $A$-module, or functor from $A^{op}$ to chain complexes. So $Y_L(M) = hom_A(M,L).$</p> <p>I confess that when I confront these formulas/concepts, I always think in terms of the Fukaya category, which is very amenable to pictures and for which the $A_\infty$ structures are geometric.</p> <p>So I draw a curve on a piece of paper and label it $L$. (The curve is literally a Lagrangian submanifold of my ${\mathbb R}^2$ piece of paper.) When I want to think of $L$ in terms of its Yoneda image, I draw the SAME curve, but as a squiggly line.</p> <p>So what is the data that the squiggly line gives us? For each object $M$ (a regular curve on my paper), we have the intersection points, which form a graded vector space $hom_A^*(M,L).$ This vector space has the structure of a chain complex (Floer), with differential given by football-shaped bi-gons with one regular side and one squiggly side. For a pair of other objects, $M_1, M_2,$ we get a map $$\mu^2: hom_A(M_2,L)\otimes hom_A(M_1,M_2) \rightarrow hom_A(M_1,L),$$ and so on for all the structure of a module (section 1j, p. 19).</p> <p>For the Fukaya category, the equations 1.19 follow (for non-squiggly lines) from studying degenerations of 1-parameter families of holomorphic polygons. Now squigglifying those same pictures gives 1.19 for an arbitrary module, and the equations are similar for not just modules but arbitrary functor between two $A_\infty$-categories.</p> <p>What data do we have if we have <em>two</em> squiggly lines $L_1$ and $L_2$? They should intersect at a morphism between functors (and it should have a degree). This morphism of functores gives more data, using the Fukaya perspective. If we added one normal line $M$, we'd have the spaces $Y_{L_1}(M)$ and $Y_{_2}(M)$, and have a triangle which is a map between them. Higher polygons and the relations between them (by considering one-parameter families) should give you all the equations and give you a hint as to verify them. (But no promises!)</p> <p>Hope that lengthy and pretty vague description was worth our time. (Oh, geez, this was a March 11 question? Probably stale by now!)</p> http://mathoverflow.net/questions/57520/examples-in-mirror-symmetry-that-can-be-understood/58199#58199 Answer by Eric Zaslow for Examples in mirror symmetry that can be understood. Eric Zaslow 2011-03-11T21:08:08Z 2011-03-11T21:08:08Z <p>Here is my biased view of a simple example: the two-torus. Everything I know about homological mirror symmetry stems from this example.</p> <p>Because the example is one-dimensional, a symplectic form is just an area form, and Lagrangians are simply curves, and the holomorphic maps which are part of the Fukaya category are simply topological disks. (By uniformization of Riemann surfaces, there is one holomorphic map for each topological disk satisfying the appropriate boundary conditions.) Even better, you can go to the universal cover, which is $R^2,$ and just draw Lagrangians as straight lines with rational slope. The holomorphic disks which determine compositions in the category are simply triangles. </p> <p>On the mirror side, we're talking about a complex two-torus, or elliptic curve. A typical object would be a line bundle on the elliptic curve, such as the theta line bundle, whose sections are theta functions, once we lift them up to the complex plane.</p> <p>The two-torus is circle-fibered over a base circle, and the elliptic curve is circle-fibered by the dual circle (i.e., $U(1)$ local systems on the original circle). This is called T-duality, and it explains how to construct the mirror equivalence going from Lagrangians to line bundles, or vice versa. For example, the Lagrangian ${y=0}$ represents a family of trivial $U(1)$ local systems, corresponding to the trivial holomorphic line bundle whose sections are just holomorphic functions. The Lagrangian ${y=nx}$ corresponds to a line bundle of degree $n$. After making these definitions, one checks that compositions match up.</p> http://mathoverflow.net/questions/51204/is-there-a-crash-course-book-on-abelian-varieties-e-g-an-introduction-for-ph/51212#51212 Answer by Eric Zaslow for Is there a "crash-course" book on Abelian varieties (e.g., an introduction for physicists)? Eric Zaslow 2011-01-05T16:08:53Z 2011-01-05T16:08:53Z <p>The book by Polishchuk should be very helpful, and close in spirit to the nonexistent crash course on Mumford's paper discussed by Charles Matthews in his reponse:</p> <p><a href="http://www.cambridge.org/gb/knowledge/isbn/item1169225/?site_locale=en_GB" rel="nofollow">http://www.cambridge.org/gb/knowledge/isbn/item1169225/?site_locale=en_GB</a></p> http://mathoverflow.net/questions/19065/when-your-paper-makes-a-borderline-case-for-a-top-journal/50309#50309 Answer by Eric Zaslow for When your paper makes a borderline case for a top journal Eric Zaslow 2010-12-24T22:04:28Z 2010-12-24T22:04:28Z <p>This requires a conditional response. First, who is submitting?</p> <ul> <li><p>Tenured professor: Not much repurcussion either way. Worth the wait if <em>you</em> think it deserves publication.</p></li> <li><p>Graduate student: Some downside to waiting a long time and not having the thing published come application time, but this can happen at a lesser journal, too (for grads, the paper is often submitted only shortly before they are graduating). And postdoc hiring is not as publications-based as tenure-track hiring. Letters of recommendation mean more.</p></li> <li><p>Postdoc. Here, submitting with a high probability of a rejection after a long wait can be a major gambit. Mitigating factor: does the postdoc have other worthy publications? If this will be the flagship result, it's hard not to think that a slightly lesser journal would have a higher expected yield (in terms of jobs). Note that it is true that some journals may reject your paper quickly -- then you can turn around and submit somewhere else -- but those papers are not really the marginally-great ones being asked about.</p></li> </ul> <p>In the end, you are left with a hard decision. I don't think there's a formula which can help. In this case, you either go for broke or you don't.</p> <ul> <li>Collaboration: decide based on the member with the most to lose.</li> </ul> http://mathoverflow.net/questions/49024/mirror-symmetry-with-algebraic-geometry/49488#49488 Answer by Eric Zaslow for mirror symmetry with algebraic geometry? Eric Zaslow 2010-12-15T06:23:15Z 2010-12-15T06:23:15Z <p>Here are a few scattered observations:</p> <ul> <li><p>Our ability to construct examples (e.g. of CY manifolds) is limited, and the tools of algebraic geometry are perfectly suited to doing so (as has been noted). </p></li> <li><p>Toric varieties are a source of many examples -- Batyrev-Borisov pairs -- and they are even "more" than algebraic, they're combinatorial. In fact, the whole business is really about integers in the end, so combinatorics reigns supreme.</p></li> <li><p>The fuzziness of $A_\infty$ structures is more suited to algebraic <em>topology</em> rather than geometry.</p></li> <li><p>Continuity of certain structures (which are created from counting problems) across walls, scores some points for analysis over combinatorics and algebra.</p></li> <li><p>Elliptic curves are only "kinda" algebraic, and the mirror phenomenon there is certainly transcendental.</p></li> <li><p>Physics indeed does not care too much about how the spaces are constructed, but (as has been noted) even the non-topological version of mirror symmetry is an equivalence of a very algebraic structure (which includes representations of superconformal algebras).</p></li> </ul> <p>I was hoping to unify these idle thoughts into a coherent response, but I don't think I can. Maybe the algebraic geometric aspects just grew faster because the mathematics is "easier" (or at least better understood by more mathematicians): witness the slow uptake of BCOV and its antiholomorphicity within mathematics.</p> <p>To respond personally: these days, I try to transfer the algebraic and symplectic structures to combinatorics so that I can hold them in my hand and try to understand them better.</p> http://mathoverflow.net/questions/48671/examples-of-non-rigorous-but-efficient-mathematical-methods-in-physics/48704#48704 Answer by Eric Zaslow for Examples of non-rigorous but efficient mathematical methods in physics Eric Zaslow 2010-12-09T03:28:18Z 2010-12-09T03:28:18Z <p>Finally, a Math Overflow question that addresses my specialty: non-rigor!</p> <p>Here are a few examples of non-rigor as applied to evidence for dualities:</p> <ol> <li><p>Heterotic-Type II. In earlier times, the best evidence for heterotic-Type-II duality was a) counting the number of supersymmetries of the theory, and (b) comparing the moduli spaces.</p></li> <li><p>AdS-CFT. For AdS-CFT the earliest and best comparisons were counting the so-called anomalous dimensions of various operators. To date, I think the tests are far from rigorized (and yes, this would be a great problem to make mathematically precise).</p></li> <li><p>Mirror Symmetry, early days. Recall that mirror symmetry in CY moduli space came from constructing a chart of the Euler characteristics of CY complete intersections and noticing the symmetry of the chart about zero. Other non-rigorous arguments involve counting the dimensions (just the dimensions) of the moduli of purportedly mirror objects. Then there's the old compute-on-flat-space-and-let-supersymmetry-take-care-of-the-rest trick.</p></li> <li><p>Low energy effective field theory. The "fact" that string theory reduces to an oft-identifiable QFT in a low energy limit is a huge source of argumentation/inspiration in string theory. Accounting for (effective) black holes helped lead to M-theory in one context, and to the microscopic description of black-hole entropy in another. One can also argue for dualities by identifying equivalent field contents in two different models. This brings up another point.</p></li> <li><p>Invariance of BPS states under perturbation. It is great to take a quantity that does not vary and evaluate it in a limit where it is easy to compute. This argument appears again and again in physics -- and also in math, of course (e.g. in the heat-kernel proof of the index theorem). BPS numbers are just that. (Of course, they do vary, and the continuity of the relevant <em>physical</em> parameters [numbers are not necessarily physical quantities] is what underlies interesting explanations of wall-crossing.)</p></li> </ol> <p>I'm probably including too many that don't fit and excluding a lot that do. Very non-rigorous of me!</p> http://mathoverflow.net/questions/47852/examples-of-mixed-hodge-structures Examples of Mixed Hodge Structures Eric Zaslow 2010-12-01T01:59:00Z 2010-12-03T14:54:26Z <p>Does anyone know a user-friendly, example-laden introduction to mixed Hodge structures? I get from Wikipedia how to calculate for a punctured and pinched curve (http://en.wikipedia.org/wiki/Hodge_structure#Mixed_Hodge_structures), but I want more! I want tables and numbers and everything explicit and spoonfed. Thanks!</p> http://mathoverflow.net/questions/47770/string-theory-computation-for-math-undergrad-audience/47790#47790 Answer by Eric Zaslow for String theory "computation" for math undergrad audience Eric Zaslow 2010-11-30T13:17:29Z 2010-11-30T13:17:29Z <p>Two counting problems -- from my own very biased and personal viewpoint -- that can perhaps be motivated:</p> <ol> <li>Counting triangles on the torus = theta function relation for elliptic curve. (I tried to squeeze this into a public lecture one time.)</li> <li>Counting symmetric polynomials of degree k in 24 variables = partition function of chiral bosonic string => counting curves on K3 by heterotic duality: 24, 24 + 24*25/2 = 324, etc.</li> </ol> <p>But these can't beat calculating an actual partition function (as in Richard Eager's answer), unless you're trying to emphasize mathiness.</p> http://mathoverflow.net/questions/47466/how-do-i-compute-the-compact-cohomology-of-a-hypersurface How do I compute the compact cohomology of a hypersurface? Eric Zaslow 2010-11-26T21:48:20Z 2010-11-27T23:18:52Z <p>How do I compute the compact cohomology of a hypersurface? For example, let $f$ be a Newton polynomial of a polytope in $\mathbb{R}^n$ and let $X = (f=0)$ inside $(\mathbb{C}^*)^n$ (maybe there is some dependency on the coefficients of $f\;$?). Can you tell me anything about $H^*_c(X)$? Perhaps I should know better, but I don't. Thanks!</p> http://mathoverflow.net/questions/47082/exterior-derivative-on-almost-complex-manifolds/47092#47092 Answer by Eric Zaslow for Exterior derivative on almost complex manifolds Eric Zaslow 2010-11-23T13:27:20Z 2010-11-23T13:27:20Z <p>In writing $\omega$ you used a symbol $dz$ which doesn't make sense unless there is a holomorphic coordinate. Your $dz$ should really be an element of a frame of (1,0) 1-forms, which need not be closed (as you have assumed).</p> http://mathoverflow.net/questions/46524/non-kahler-calabi-yau/46543#46543 Answer by Eric Zaslow for Non-Kahler "Calabi-Yau"? Eric Zaslow 2010-11-18T21:54:54Z 2010-11-18T21:54:54Z <p>This is covered in Andrei Halanay's answer, but it's worth mentioning the simplest examples, which are primary Kodaira surfaces. For the simplest of these:</p> <p>Take C^2 and quotient by the group generated by these a_k:</p> <p>a_1 : z -> z + 1</p> <p>a_2 : z -> z + i</p> <p>a_3 : w -> w + z + 1</p> <p>a_4 : w -> w - iz + i</p> <p>(I think this is it.)</p> <p>The quotient group is nonabelian. Here z is the fiber and w the base.</p> http://mathoverflow.net/questions/46418/relation-between-toric-geometry-and-log-geometry/46454#46454 Answer by Eric Zaslow for relation between toric geometry and log geometry Eric Zaslow 2010-11-18T03:43:08Z 2010-11-18T03:43:08Z <p>Sloppy stab at a reformulation:</p> <p>I think we are assuming P is commutative? (\sigma^\vee \cap M certainly is.) If so, then isn't P spanned by all linear combinations of the generators? But then there are relations, too, etc. So the question might be, does P embed as a submonoid of a free abelian group over Z? (I don't know.) If so, convexity of the corresponding domain is clear, and finite generatedness means it's cut out by a finite number of conditions. Your lattice is the Z-span of P and your cone is then the (dual of the) R-convex hull.</p> http://mathoverflow.net/questions/45802/undergraduate-math-research/46040#46040 Answer by Eric Zaslow for Undergraduate math research Eric Zaslow 2010-11-14T13:39:36Z 2010-11-14T13:39:36Z <p>Most major topics have been covered in discussion, so just two remarks/experiences:</p> <ol> <li><p>While director of graduate studies at Northwestern (2007-2010), I led a committee which valued undergraduate preparedness over research experience. So at least as far as Northwestern was concerned during that time frame, research (especially research for which an undergrad may not be fully prepared) did not help as much as you might have thought.</p></li> <li><p>In trying to use RTG funds toward undergraduates, rather than try to simulate a research environment, I and my co-PI's created an "undergraduate conference" to try to offer a supplement to standard undergraduate curricula, without yet getting on toward research. Here is the link: <a href="http://www.math.northwestern.edu/summerconference/" rel="nofollow">http://www.math.northwestern.edu/summerconference/</a> Maybe we'll do it again next year?</p></li> </ol> http://mathoverflow.net/questions/17325/why-is-cotangent-more-canonical-than-tangent Why is cotangent more canonical than tangent? Eric Zaslow 2010-03-06T22:40:59Z 2010-10-24T19:28:48Z <p>You don't need a metric to define the differential of a function, and the cotangent bundle carries a canonical one-form.</p> <p>But you do need a metric to define the gradient, and the tangent bundle does not have a canonical vector field.</p> <p>These are not difficult truths, but still... why the preference toward "co"?</p> http://mathoverflow.net/questions/38651/compute-the-kahler-moduli-of-an-elliptic-curve/38723#38723 Answer by Eric Zaslow for compute the Kähler moduli of an elliptic curve Eric Zaslow 2010-09-14T18:34:07Z 2010-09-14T18:43:13Z <p>The curve you wrote in equations lies in C^2, while the "elliptic curve" of your text is presumably a compact projective variety -- meaning you imagine making your equations homogeneous (or even quasi-homogeneous) and considering the closure of the set of points described by your equation in a (quasi-)projective plane. Not every "homogenization" will lead to an elliptic curve (Calabi-Yau) upon compactification, so you have to do this correctly (as noted by Kevin Buzzard above).</p> <p>Having said that, the answer is that every projective variety is also Kahler: just restrict e.g. the Fubini-Study(-like) Kahler form. In plain English, since a Kahler form on a complex curve is just a volume form, the volume of the compact curve inside projective space gives you your answer.</p> http://mathoverflow.net/questions/36596/refereeing-a-paper/38069#38069 Answer by Eric Zaslow for Refereeing a Paper Eric Zaslow 2010-09-08T16:07:00Z 2010-09-08T16:07:00Z <p>I've had long discussions about whether the referee report should be author-dependent. If mathematician X and mathematician Y submit the same paper, should the report be the same?</p> http://mathoverflow.net/questions/37384/mirror-of-flop/37403#37403 Answer by Eric Zaslow for Mirror of Flop? Eric Zaslow 2010-09-01T17:07:26Z 2010-09-01T17:07:26Z <p>I assume the question regards the coherent sheaves on these two CY's. These CY's should be regarded as the "same" complex manifold with two different choices of complexified symplectic forms ("Kahler form," in physics terminology).</p> <p>The mirrors are a "single" symplectic manifold with two different complex structures on it. There is a curve of complex structures relating the two.</p> <p>That's about it. The tricky part is to "parallel transport" the category of coherent sheaves along this curve, using a "flat family of categories" defined by stability conditions. Doing so should provide a preferred isomorphism of the categories. Examples have been studied, but general statements (like the ones I have glibly been making) are not proven.</p> http://mathoverflow.net/questions/30000/do-you-understand-syz-conjecture/32197#32197 Answer by Eric Zaslow for Do you understand SYZ conjecture Eric Zaslow 2010-07-16T16:00:36Z 2010-07-16T16:00:36Z <p>Hi-</p> <p>Just saw this thread. Maybe I should comment. The conjecture can be viewed from the perspective of various categories: geometric, symplectic, topological. Since the argument is physical, it was written in the most structured (geometric) context -- but it has realizations in the other categories too.</p> <p>Geometric: this is the most difficult and vague, mathematically, since the geometric counterpart of even a conformal field theory is approximate in nature. For example, a SUSY sigma model with target a compact complex manifold X is believed to lie in the universality class of a conformal field theory when X is CY, but the CY metric does not give a conformal field theory on the nose -- only to one loop. Likewise, the arguments about creating a boundary conformal field theory using minimal (CFT) + Lagrangian (SUSY) are only valid to one loop, as well. To understand how the corrections are organized, we should compare to (closed) GW theory, where "corrections" to the classical cohomology ring come from worldsheet instantons -- holomorphic maps contributing to the computation by a weighting equal to the exponentiated action (symplectic area). The "count" of such maps is equivalent by supersymmetry to an algebraic problem. No known quantity (either spacetime metric or Kahler potential or aspect of the complex structure) is so protected in the open case, with boundary. That's why the precise form of the instanton corrections is unknown, and why traction in the geometric lines has been made in cases "without corrections" (see the work of Leung, e.g.). Nevertheless, the corrections should take the form of <em>some</em> instanton sum, with known weights. The sums seem to correspond to flow trees of Kontsevich-Soibelman/ Moore-Nietzke-Gaiotto/Gross-Siebert, but I'm already running out of time.</p> <p>Topological: Mark Gross has proven that the dual torus fibration compactifies to produce the mirror manifold.</p> <p>Symplectic: Wei Dong Ruan has several preprints which address dual Lagrangian torus fibrations, which come to the same conclusion as Gross (above). I don't know much more than that.</p> <p>Also-</p> <p>Auroux's treatment discusses the dual Lagrangian torus fibration (even dual slag, properly understood) for toric Fano manifolds, and produces the mirror Landau-Ginzburg theory (with superpotential) from this.</p> <p>With Fang-Liu-Treumann, we have used T-dual fibrations for the same fibration to map holomorphic sheaves to Lagrangian submanifolds, proving an equivariant version of homological mirror symmetry for toric varieties. (There are many other papers with similar results by Seidel, Abouzaid, Ueda, Yamazaki, Bondal, Auroux, Katzarkov, Orlov -- sorry for the biased view!)</p> <p>Reversing the roles of A- and B-models, Chan-Leung relate quantum cohomology of a toric Fano to the Jacobian ring of the mirror superpotential via T-duality.</p> <p>Help or hindrance?</p> http://mathoverflow.net/questions/3591/mathematicians-who-were-late-learners-list/22446#22446 Answer by Eric Zaslow for Mathematicians who were late learners?-list Eric Zaslow 2010-04-24T18:03:15Z 2010-04-24T18:03:15Z <p>Unfortunately, all these exceptions appear to be reaches, thus proving the rule.</p> http://mathoverflow.net/questions/18631/mirror-of-local-calabi-yau/19320#19320 Answer by Eric Zaslow for Mirror of local Calabi-Yau Eric Zaslow 2010-03-25T15:45:29Z 2010-03-25T15:45:29Z <p>I think this is a stubborn case which does not fit into the general picture. For example, if you use the standard toric procedure to try to construct a differential equation for the log periods of the mirror, then try to find the GW invariants from the solutions (in this case, to recover the famous 1/d^3 formula), then it won't quite work. There are various adjustments you can make, based on knowing the answer ahead of time, and I once saw a paper (sorry, I forget where or by whom) which tried to make sense of all this, but I don't think there is a good general picture of this case. I would check Klemm's written record for some guidance.</p> http://mathoverflow.net/questions/8692/higher-genus-closed-string-b-model/17319#17319 Answer by Eric Zaslow for Higher genus closed string B-model Eric Zaslow 2010-03-06T21:42:29Z 2010-03-06T21:42:29Z <p>One thing missing from this discussion is the even-more-mysterious holomorphic <em>ambiguity</em> (not "anomaly"). BCOV is not deterministic, and should probably be thought of as part of a general schema for a B-model-type topological theory. What I mean is that there are other solutions to the BCOV equation which do not yield GW invariants. The reason is that BCOV determines the partition function only up to a holomorphic function at each genus. The choice of this function -- analogous to initial conditions -- is usually set by matching to GW invariants or by imposing some structure at the singularities of CY moduli space. </p> <p>I don't think there is a comprehensive understanding of how this ambiguity is resolved, though there are cases where it is fixed by other symmetries or properties of the full partition function.</p> http://mathoverflow.net/questions/17238/which-part-of-physical-b-model-is-not-rigorous/17283#17283 Answer by Eric Zaslow for Which part of physical B model is not rigorous? Eric Zaslow 2010-03-06T13:55:16Z 2010-03-06T13:55:16Z <p>To define (as Kevin Lin does above) the B-model purely as the derived category of coherent sheaves is fine and rigorous, but it ignores the higher-genus aspects of mirror symmetry -- which was the original question. As I wrote above, Kevin Costello gives a rigorous description of the higher-genus amplitudes, but it is still conjectural whether this agrees with the physics. The issue is that higher-genus string amplitudes depend on an integration over the moduli space of Riemann surfaces (or a space of maps from them, depending on the model), and this demands compactification. The full, non-topological theory is of course an ordinary two-dimensional quantum field theory, with all the usual difficulties in making the path integral rigorous.</p> http://mathoverflow.net/questions/17238/which-part-of-physical-b-model-is-not-rigorous/17245#17245 Answer by Eric Zaslow for Which part of physical B model is not rigorous? Eric Zaslow 2010-03-05T23:47:56Z 2010-03-05T23:47:56Z <p>Kevin Costello's mathematical definition of the B-model (math/0509264) is rigorous. It's an open problem whether this definition agrees with the BCOV construction, as far as I know.</p> http://mathoverflow.net/questions/7823/negative-gromov-witten-invariants/7873#7873 Answer by Eric Zaslow for Negative Gromov-Witten invariants Eric Zaslow 2009-12-05T16:14:30Z 2009-12-05T16:14:30Z <p>Let me use an example. If I recall correctly (I am too lazy to look), the GW invariant for "local P^2" (i.e. the canonical bundle O(-3) over P^2) in genus zero and degree 1 is 3, and in degree 2 is -45/8. Now -45/8 breaks down, after trying to account for multiple covers, to 3/2^3 - 6, giving an effective number of degree 2 curves as -6. What up? This number is how many degree 2 curves the P^2 should "account for," when it pops into existence within a family of Calabi-Yau's. (Here of course we have a family of maps into the P^2.) Then you should ask, "Does this mean that for a compact CY to have just rigid genus zero curves and an embedded P^2 (and no other surface) and lie in a family that includes a CY with only rigid genus zero curves and no other surface, that it must have at least 6 degree 2 rational curves somewhere else?" Presumably.</p> http://mathoverflow.net/questions/101314/are-donaldson-thomas-invariants-a-model-or-b-model/103701#103701 Comment by Eric Zaslow Eric Zaslow 2012-08-15T12:31:43Z 2012-08-15T12:31:43Z Fun! DT is B-model and GW is A-model. Nothing spooky there (yet). [Richard: GW numbers are locally invariant under symplectic deformations, just like your DT numbers. (I say &quot;locally&quot; because we could cross Kahler cone walls.) The &quot;dependence&quot; is in the parameter governing the generating function.] The DT partition function encodes nonperturbative (in the string coupling, g) contributions from the B-branes, while the GW partition function encodes perturbative (in g) contributions from A-branes. The relation between the perturb/nonperturb expansions in Type IIB is called S-duality. http://mathoverflow.net/questions/103120/does-the-moduli-space-of-smooth-curves-of-genus-g-contain-an-elliptic-curve/103140#103140 Comment by Eric Zaslow Eric Zaslow 2012-07-26T05:42:09Z 2012-07-26T05:42:09Z Jim, if I understood their talks correctly, your old result was just used by Donagi and Witten to show that the moduli space of super-Riemann surfaces does not split -- meaning superstring perturbation theory itself needs to be &quot;revisited.&quot; http://mathoverflow.net/questions/99352/important-open-questions-in-the-field-of-tropical-geometry Comment by Eric Zaslow Eric Zaslow 2012-06-13T14:27:48Z 2012-06-13T14:27:48Z Perhaps the best way to demonstrate why I voted for this question is to reveal my own ignorance and confusion about this subject. Many younger researchers may carry the same uncertainties. How much of the package'' of algebraic geometry carries over to the tropical geometry? When working with the tropical semigroup ring, what topological spaces can you construct from tropical varieties and how exactly do they compare to varieties over C? What general results have been proven in tropical Gromov-Witten theory? If these questions are not open, there must be related ones that are, right? http://mathoverflow.net/questions/31789/statistical-physics-of-string-theory/31791#31791 Comment by Eric Zaslow Eric Zaslow 2012-05-20T01:46:29Z 2012-05-20T01:46:29Z I'm confused by this dispute. If I understand, Clay Cordova says &quot;gravitons are always there,&quot; so gravity does not emerge through some averaging process, while jeremy says that the thermodynamics of classical gravitational systems is described by microstates, so represents an aggregate notion. It seems both parties are conscientious and knowledgeable and both are correct by their own definition of what it means to be &quot;statistical&quot; (a weighted word). The problem lies in the vagueness of the question, not in the answers. http://mathoverflow.net/questions/94347/eigenforms-for-laplacian-on-a-non-flat-two-torus/94627#94627 Comment by Eric Zaslow Eric Zaslow 2012-04-20T15:44:42Z 2012-04-20T15:44:42Z Thanks, Robert! That's exactly where I had arrived, having considered a torus with $S^1$ isometry. For particular $F$ you can make it look a lot like, but not precisely equal to, the &quot;quantum pendulum.&quot; That system is described by the well-studied Matthieu equation, but again it's not precisely the correct equation (or at least I can't make it be). http://mathoverflow.net/questions/94347/eigenforms-for-laplacian-on-a-non-flat-two-torus Comment by Eric Zaslow Eric Zaslow 2012-04-18T11:43:59Z 2012-04-18T11:43:59Z Ooops! Sorry, Robert et al! Yes, the question I wrote is trivial. I meant to say the eigenforms of the Hodge Laplacian! Stupid me. Will update. http://mathoverflow.net/questions/85856/places-to-learn-about-landau-ginzburg-models Comment by Eric Zaslow Eric Zaslow 2012-02-18T06:58:06Z 2012-02-18T06:58:06Z If you know that in physics the energy is a sum of kinetic energy and potential energy, then the appearance of a potential in a physical theory should not be too surprising. The potential here is constructed from a holomorphic function, of which there is none for a compact manifold, but possibly many if the target space is noncompact. http://mathoverflow.net/questions/84954/can-you-make-the-cotangent-bundle-to-a-complex-manifold Comment by Eric Zaslow Eric Zaslow 2012-01-06T05:14:07Z 2012-01-06T05:14:07Z The Nijenhuis tensor lives upstairs and doesn't have so many symmetries, while the curvature tensor lives downstairs (with half the number of vectors) with more symmetries -- so the relation is not direct, and involves the splitting of T(TM) into TM+TM. Maybe the exact relation is in Kobayashi-Nomizu? http://mathoverflow.net/questions/84954/can-you-make-the-cotangent-bundle-to-a-complex-manifold Comment by Eric Zaslow Eric Zaslow 2012-01-05T15:06:38Z 2012-01-05T15:06:38Z The metric-induced almost complex structure is not integrable unless M is flat. (The Nijenhuis tensor is essentially the curvature.) Now interpreting Tim Perutz's excellent response to the question you cite: if M is compact, then there is an integrable complex structure on T*M by Eliashberg's result. It is not Kahler, in general. A Kahler metric exists on T*M = TM in a neighborhood of the zero section (&quot;Grauert tube&quot;). http://mathoverflow.net/questions/57520/examples-in-mirror-symmetry-that-can-be-understood/58199#58199 Comment by Eric Zaslow Eric Zaslow 2011-03-16T12:27:11Z 2011-03-16T12:27:11Z The paper with Polishchuk works through the example in detail, as opposed to just the sketch above. You seem to be still somewhat dissatisfied. Why don't you say precisely which aspect of mirror symmetry you are looking to uncover in your example? (Three references for torus fibrations are Arinkin-Polishchuk, Leung-Yau-Z, and Mark Gross's &quot;Topological Mirror Symmetry.&quot;) http://mathoverflow.net/questions/57520/examples-in-mirror-symmetry-that-can-be-understood/58199#58199 Comment by Eric Zaslow Eric Zaslow 2011-03-14T17:31:08Z 2011-03-14T17:31:08Z Well, I can offer you my own article with Polishchuk: <a href="http://arxiv.org/abs/math/9801119" rel="nofollow">arxiv.org/abs/math/9801119</a> Below, AByer mentions that HMS implies that we get an isomorphism of our category for each loop in moduli space. These isomorphisms (&quot;autoequivalences&quot;) alone can lead to the mirror map, as demonstracted by a calculation for this example in <a href="http://arxiv.org/abs/math/0506359" rel="nofollow">arxiv.org/abs/math/0506359</a>. The T-duality aspect can be pushed in many other directions and examples, too. (Sorry for the lazy self-promotion! There are, of course, many other articles by other authors, some of which are included in the answers below.) http://mathoverflow.net/questions/51656/mathematics-of-gun-control Comment by Eric Zaslow Eric Zaslow 2011-01-11T14:13:24Z 2011-01-11T14:13:24Z @Charles: Thanks, I'll try again. @Steven Gubkin: That was a Ph.D. question in econonmics, as far as I understand, and a scholarly paper (with the opposite conclusion), so it would probably be a question under the tag &quot;reference request.&quot; Here is a link: <a href="http://en.wikipedia.org/wiki/The_Impact_of_Legalized_Abortion_on_Crime" rel="nofollow">en.wikipedia.org/wiki/&hellip;</a> http://mathoverflow.net/questions/51656/mathematics-of-gun-control Comment by Eric Zaslow Eric Zaslow 2011-01-11T09:43:58Z 2011-01-11T09:43:58Z @Andy Putman: Thanks for the reference! I don't think I am permitted to contribute to meta, so I cannot broach the issue of whether any question with math content (not nec. mine) but also with political ramifications is ever permissible on MO. Admittedly unnerving, this meta issue should (maybe already has been) be addressed, as MO is the only major free forum for math-related exchange. I &quot;get&quot; slippery-slope reluctance (lots of down votes and negative remarks!), but not all slopes are equally slippery. Up-voters may feel the issue compelling enough for the math community to weigh in on. http://mathoverflow.net/questions/51656/mathematics-of-gun-control Comment by Eric Zaslow Eric Zaslow 2011-01-11T03:04:06Z 2011-01-11T03:04:06Z @Pete L. Clark: Following your suggestion, I asked a professor of statistics. This is actually a problem of the application of statistics to law, a legitimate and specialized branch of the discipline. I doubt that a non-statistician could do this well, even given the data (but this is conjectural; certainly <i>I</i> cannot) -- and which data would be the best to collect? Now there is no Statistics Overflow (as far as I know), but there is a statistics tag on MO. The academic bona fides of the question are no worse than many soft questions, at the least. (AFAIK, issue's not in mission of UCS.) http://mathoverflow.net/questions/51656/mathematics-of-gun-control Comment by Eric Zaslow Eric Zaslow 2011-01-10T16:44:36Z 2011-01-10T16:44:36Z @JSE: Thank you for the explanation. I was hoping for a statistically meaningful and rigorous way of analyzing data. I don't statistics well enough to attach probabilities/error-bounds to any conclusions. What can one say after collecting lots of pieces of evidence and crossing against local/national laws? (I won't post any actual data, so as to keep this meta-conversation on-topic.) Is this settled matter? We have data, but not a controlled experiment. Obviously, the topic is incendiary. We could replace &quot;gun-control&quot; with &quot;math-homework-enforcement&quot; but we wouldn't have good data.