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    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeMar 22nd 2010
    • (edited Mar 24th 2010)

    I wrote a stub geometrical optics with redirect geometric optics (maybe it was better other way around, I don't have the feeling which is more used). And created stub optics. Geometric optics is of course in my present program of semiclassical approximation, equivariant localization, wall crossing, (edit typo:) Stokes phenomenon and related notions. Note that in physicscontents we still do have it listed but not written entry deformation quantization. So I just starting a stub.

    • CommentRowNumber2.
    • CommentAuthorIan_Durham
    • CommentTimeMar 23rd 2010
    In English I've more often heard it as "geometrical optics" which I think is technically more grammatically correct.
    • CommentRowNumber3.
    • CommentAuthorIan_Durham
    • CommentTimeMar 23rd 2010
    I like that you've added this, by the way. It seems as if category theory is well-suited to describe it.
    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeMar 24th 2010

    Important piece about wall crossing and slopes. Slopes appear in various fields of mathematics. Now I emphasise as a physicist semi-classical approximation as the Stokes phenomenon is seen there the best. However, this is just about the phenomenon related to the solutions of certain equation. This can be formalized taking certain category of monodromy data or say certain category of D-modules (or perverse sheaves). In any case we have some category of sheaf like objects which is additive and we have a stability slope filtration on this one! Therefore this is in complete analogy with stability on objects of various categories which come when we create some moduli space. Therefore the stability like in the study of say Gromov Witten or Donaldson Thomas invariants and the Stokes phenomenon for the solutions of ODEs are of the related nature. Similar in other situatiojns like Hodge theory etc. I wish there would be some detailed comparison in the literature. I should create an entry on slopes ! There is an article of Yves Andre with manzy cases but not treating the basic one of asymptotic analysis and the one of Fuchsian equations...

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 24th 2010
    • (edited Mar 24th 2010)

    Thanks, Zoran, seems like a big subject that you are attacking here. It looks to me that if i wanted to understand slope filtrations right now, i would have to do a fair bit of reading and thinking.

    Can we connect this somehow to something that we already have more material on?

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeMar 24th 2010
    • (edited Mar 24th 2010)

    There are many entry points. This whole subject is very close to what the lab is centered on, especially the business of monodromies, Tannakain reconstruction, connections and quantization.

    Some of the entry points are e.g.

    • to understand the cohomological origin of Maslov index, among the references is a paper of Weinstein on "Maslov gerbe". This should fit with the things we were doing much.

    • to understand the business of the monodromy of differential equations, e.g. the differential Galois group. In general monodromy is very central to much of this subject. The relation between the monodromies and equations is the famous Riemann-Hilbert correspondence which is certain equivalence of categories of pretty geometric origin. Tannakian reasoning is pretty central to these issues.

    • equivariant localization -- this appears in the asymptotic expansion of some Feynman integrals, i.e. those with enough special properties, look at the reference of Szabo at semiclassical approximation. This is closely related to Cartan calculus, Chevalley-Eilenberg complex, Weil algebra. Some of the key words are Duistermaat localization formula etc. Witten index which is an element in elliptic cohomology also was first heuristically introduced via these methods.

    • the nc Hodge structures are defined using meromorphic connections and one of the applications is cyclic cohomology of stable Calabi Yau categories which are sometimes equipped with such structures...

    I could list many more...

    I can not resist one in fact: the moduli spaces. Almost every important moduli space in mathematical physics is constructed with consideration of the stability ("moduli space of stable curves") so the slopes used for stability must be familiar to anybody doing Gromov Witten or anything like that...

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