[geometry-ml:01492] 集中講義 7/9-13 （原田芽ぐみ）

[Mikiya Masuda]

皆様

7月17日（火）−27日（金）の期間，MSJ-SI 2012 Schubert Calculus の集会が
大阪市大で開催されますが，それに先だって，以下の要領で集中講義を開催しま
す．ウェブサイト
http://www.sci.osaka-cu.ac.jp/math/index.html
にも案内があります．重複して受け取られた方，ご容赦ください．

枡田幹也（大阪市立大学）

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講師：　原田芽ぐみ（McMaster大学）
題目：　An introduction to integrable systems, toric degenerations, and
Okounkov bodies
場所：　大阪市立大学大学院理学研究科　数学講究室（3040）
日時：　7月 9日（月） 16:30-18:30
　　　　　 10日（火） 16:30-18:30
　　　　 11日（水） 10:00-12:00 （談話会：16:30-17:30)
　　　　 12日（木） 16:30-18:30
　　　　 13日（金） 10:00-12:00

アブストラクト
　The theory of Okounkov bodies is a vast generalization of the theory
of toric varieties, recently (roughly 2008/2009) developed independently
by Kaveh-Khovanskii and Lazarsfeld-Mustata. Okounkov bodies are an
exciting new tool which connects convex geometry, in particular the
combinatorics of polytopes, to many other areas of geometry and
topology. In an interesting development, Dave Anderson observed in his
2010 research preprint that the construction of Okounkov bodies also
allows one to construct a toric degeneration from a wide class of
projective varieties X to a (possibly non-normal) toric variety X_0.
Anderson's construction vastly generalizes many toric degenerations
already in the literature (e.g. Alexeev-Brion, Kogan-Miller) and
suggests exciting new applications of toric-geometric techniques to many
other areas of mathematics, including geometric representation theory,
algebraic geometry, Schubert calculus, and symplectic topology, as well
as many others. Further developing this point of view and using Dave
Anderson's toric degeneration as a key ingredient, in joint work with
Kaveh, I have recently constructed integrable systems --- in the sense
of symplectic geometry -- on a wide class of complex projective
algebraic varieties. Our construction recovers, for example, the famous
Gel'fand-Cetlin integrable system on the complete flag variety
GL(n,\C)/B observed by Guillemin and Sternberg, but our methods are much
more general. Moreover, the `moment map image' of the our integrable
system is precisely the Okounkov body associated to X (and a choice of
valuation on its homogeneous coordinate ring), so there is an intimate
relation between the geometry of this system and the combinatorics of
the Okounkov body. Our construction significantly contributes to the set
of known examples of integrable systems in the literature, and
represents a corresponding significant expansion of the possible
applications of these systems, and their associated combinatorics, to
other research areas.

In this series of lectures, which I hope will involve a lot of audience
participation and active discussion, I plan to give a gentle
introduction to every word in the title. In particular I plan to give
plenty of motivation and concrete examples to illustrate the general
philosophy of this new and rapidly developing theory. By the end of the
lecture series I hope to have given a reasonable sketch of all of the
main ingredients in the construction of the integrable system mentioned
above.
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