| 科目名 | 応用解析学特別講義Ⅲ・Ⅳ |
| 日程 | 7月30日（月）～8月3日（金） （談話会：8月1日（水） 16:30～17:30） |
| 講演者（所属） | 小薗 英雄（早稲田大学） |
| タイトル | New function space in general unbounded domains and its applications to the Navier-Stokes equations |
| 場所 | 数学講究室（3040） |
| 講義内容 | We consider problems on the mathematical fluid mechanics in general unbounded domains $\Omega$ with non-compact boundaries.
In such domains $\Omega$, it is known that the usual Helmholtz decompositions in $L^r(\Omega)$, $1 < r < \infty$ does not hold except
$r = 2$, and hence we need to introduce another function space $\tilde L^r(\Omega)$ defined by $\tilde L^r(\Omega) = L^r(\Omega) + L^2(\Omega)$
for $1< r\le 2$ and $\tilde L^r(\Omega) = L^r(\Omega) \cap L^2(\Omega)$ for $2\le r < \infty$, respectively. This new function space $\tilde L^r(\Omega)$
plays a substitutionary role for $L^r(\Omega)$ so that the Helmholtz decomposition holds. Defining the Stokes operator $A$ in $\tilde L^r(\Omega)$,
we can develop well-known techniques like analyticity of the semi-group $\{e^{-tA}\}_{t > 0}$ and the maximal regularity theorem on $\partial_t + A$
as well as the characterization of the domains of the fractional powers $A^{\alpha}$, $0 < \alpha < 1 $. As applications of the theory on $\tilde L^r(\Omega)$,
we are able to treat fundamental problems on uniqueness, regularity and decay properties of weak solutions of the Navier-Stokes equations in $\Omega$.
Contents: 1. Introduction; Leray's structure theorem 2. Helmholtz decomposition in $\tilde L^r(\Omega)$ 3. Stokes operator in $\tilde L^r_{\sigma}(\Omega)$ 4. Applications to the Navier-Stokes equations; strong energy inequality, uniqueness criterion |
| 科目名 | 数理解析学特別講義Ⅲ・Ⅳ |
| 日程 | 11月12日（月）～11月16日（金） （談話会：11月14日（水） 16:30～17:30） |
| 講演者（所属） | 小森 洋平（早稲田大学） |
| タイトル | 入門複素解析幾何 |
| 場所 | 数学講究室（共通研究棟 3階 301） |
| 講義内容 | 複素解析に関する集中講義ということで、今回は複素関数を使って幾何をする、複素解析幾何という分野の入門的な話をします。 前半は三角関数など周期関数の一般化である、２重周期関数（楕円関数）の話から始めます。 楕円関数が住む自然な場所として複素 トーラスが登場します。 複素トーラスを空間のなかで実現しようとすると、代数曲線としての表示が得られます（楕円曲線）。 逆に 代数曲線としての表示から複素トーラスの姿も再構成できます（楕円積分）。 後半は楕円曲線を個別でなく族で考えて、族の中での個々の楕円曲線の変化の様子について調べます（モジュライ問題）。 特に今回は 複素数パラメータについて正則に変化する族（正則族）について考え、族の「端」で楕円曲線が壊れる様子を調べます（退化とコンパクト化の問題）。 |
| 科目名 | 数理科学D |
| 日程 | 7月9日（月）～7月13日（金） （談話会：7月11日（水） 16:30～17:30） |
| 講演者（所属） | 原田 芽ぐみ（McMaster大学） |
| タイトル | An introduction to integrable systems, toric degenerations, and Okounkov bodies |
| 場所 | 数学講究室（3040） |
| 講義内容 | 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 paritcular 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. |
