[geometry-ml:02178] 原田芽ぐみ氏の集中講義 (12/3-5)

[Mikiya Masuda]

皆様

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原田芽ぐみ氏（McMaster大学）に，大阪市大数学研究所が推進している
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　　頭脳循環を加速する戦略的国際研究ネットワー ク推進プログラム
「対称性，トポロジーとモジュライの数理，数 学研究所の国際研究 ネットワー
ク展開」
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の活 動の一つとして，以下の要領で集中講義をして頂くことにしました．
興味ある方はご参加下さい．

枡田幹 也（大阪市大）

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場 所：　大阪市大数学教室　数学講究室（共通研究棟３０１）
　アクセス方法は以下をご覧ください．　
　http://www.osaka-cu.ac.jp/ja/about/university/access
日 時：　12/3(水）： 13:30--16:30（途中休憩あり）
　　 　　12/4（木）： 10:00--11:45, 13:30--16:30（途中休憩あり）
　　 　　12/5（金）： 10:00--11:45, 13:30--16:30（途中休憩あり）

タイ トル： Newton-Okounkov bodies, Bott-Samelson varieties, and
Schubert calculus

アブ ストラクト：
The theory of Newton-Okounkov bodies is a far-reaching generalization
of the theory of toric varieties. Given a complex variety X and some
extra data (e.g. a very ample line bundle L on X and a valuation v on
its homogeneous coordinate ring), the Newton-Okounkov body of (X, L,
v) is a convex body of "maximal dimension", i.e., the (complex)
dimension of X. In many interesting cases, it is in fact a rational
polytope. In the case when X is a toric variety and the accompanying
data (L,v) is chosen to be torus-invariant, the Newton-Okounkov body
is the usual Newton polytope from toric geometry. In the case when X =
G/B is a (full) flag variety G/B of a complex semisimple algebraic
group G, Kaveh recently showed that an appropriate choice of (L,v)
yields the so-called "string polytopes" of Berenstein-Littelmann,
which are fundamental objects of study in representation theory. In
particular, the Gel'fand-Zetlin polytopes associated to
representations of GL(n,\C) are Newton-Okounkov bodies of the standard
flag variety Flags(\C^n) of nested subspaces in \C^n. It is widely
believed that the theory of Newton-Okounkov bodies will provide a
fruitful new approach to the study of Schubert calculus, through a
careful study of the Newton-Okounkov bodies of flag varieties and of
associated Bott-Samelson varieties. In particular, it can be _hoped_
that a "Schubert calculus" can be developed where "intersecting
Schubert varieties" translates to "intersecting (unions of) faces of a
single polytope", in the same spirit as the work of
Kiritchenko-Smirnov-Timorin for the GL(n,\C) (i.e. Gelfand-Zetlin)
case and the recent work of Kiritchenko on divided-difference
operators on polytopes and a "geometric mitosis" on polytopes.

This lecture series will be aimed at a broad audience and will attempt
to explain the basic background behind the "hope" mentioned above. In
particular, the lectures are not intended to explain a particular
significant theorem. Instead, I hope to, first, explain the necessary
context and, second, convey some feeling for the reasons why I believe
this subject is so interesting and promising. In particular, the
lectures will be full of questions to which I do not know the answer.
The audience is encouraged to actively participate in the discussion.
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