[geometry-ml:03616] 研究集会GAGT2018:第2報

[芥川 和雄]

幾何学分科会の皆様，

Workshop 「Geometric Analysis in Geometry and Topology 2018」（第2報）
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中央大学・理工学部
芥川和雄

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{\bf Date：}March 11th --\ 14th, 2019\\

{\bf Place：}Chuo University (Korakuen Campus) 6th building, 61125\\

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{\bf Invited speakers} \\

$\bullet$\,{\bf Eric Bahuaud (Seattle University)}\\

$\bullet$\,{\bf Renato Bettiol (City University of New York)}\\

$\bullet$\,{\bf Boris Botvinnik (University of Oregon)}\\

$\bullet$\,{\bf Kei Irie (University of Tokyo)}\\

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\underline{\bf Schedule}\\

・・・・・・・・・・・ 10:00--11:00\qquad 11:30--12:30\qquad 14:00--15:00\qquad 15:30--16:30\\

{\bf March\,11}\quad B.\,Botvinnik-1\b\ B.\,Botvinnik-2\ \ R.\,Bettiol-1\quad \ \ R.\,Bettiol-2 \\

{\bf March\,12}\quad E.\,Bahuaud-1\quad E.\,Bahuaud-2\quad B.\,Botvinnik-3\ \ B.\,Botvinnik-4\\

{\bf March\,13}\quad K.\,Irie-1\qquad \quad \ \ K.\,Irie-2 \qquad \quad {\bf Short Excursion} \\

{\bf March\,14}\quad R.\,Bettiol-3\qquad R.\,Bettiol-4\qquad E.\,Bahuaud-3\quad E.\,Bahuaud-4 \\

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{\bf Organizers}\\

$\bullet$\,Shu Nakamura (University of Tokyo)\\

$\bullet$\,Mikio Furuta (University of Tokyo)\\

$\bullet$\,Shinichiroh Matsuo (Nagoya University)\\

$\bullet$\,Tsuyoshi Kato (Kyoto University)\\

$\bullet$\,Yoshihiko Matsumoto (Osaka University)\\

$\bullet$\,Nobuhiko Otoba (University of Regensburg)\\

$\bullet$\,Rafe Mazzeo (Stanford University, Foreign adviser)\\

$\bullet$\,Kazuo Akutagawa (Chuo University)\\

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$\boxed{\bf Titles\ \&\ Abstracts\ of\ Lectures}$\\

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$\bullet$\ \ {\bf Eric Bahuaud (Seattle University, USA)}\\

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\underline{\bf Lectures 1--2\,:\,Poincar\'e-Einstein metrics with prescribed conformal infinity}\\

\underline{\bf Abstract}\ \ The seminal work of Fefferman and Graham on conformal invariants introduced

an important existence problem: given a compact conformal manifold $(X, [g])$, is it possible to find a manifold with boundary $M$ with boundary$(M) = X$

and a conformally compact Einstein metric $g$ on the interior of $M$ that has $[g]$ as conformal infinity ?

In these talks I will introduce this problem and the explain the Graham-Lee theorem

that proves the existence of such metrics nearby the hyperbolic metric on the ball.

I will then discuss recent work with Lee on low regularity Poincar\'e-Einstein metrics.\\

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\underline{\bf Lectures 3--4\,:\,Ricci flow of asymptotically hyperbolic metrics}\\

\underline{\bf Abstract}\ \ In these two talks I'll survey results concerning the normalized Ricci flow evolving from a conformally compact asymptotically hyperbolic metric.

I'll discuss joint work with Mazzeo and Woolgar on the behavior of the renormalized volume along the flow of both asymptotically Poincar\'e-Einstein metrics

and metrics with an even expansion.

I'll then discuss joint work with Woolgar on the long-time existence of the flow for rotationally symmetric asymptotically hyperbolic initial data,

and ongoing work with Guenther and Isenberg on the stability of these flows.\\

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$\bullet$\ \ {\bf Renato Bettiol (City University of New York, USA)}\\

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\underline{\bf Lectures 1-- 4\,:\,Applications of Bifurcation Theory to Geometric Analysis}\\

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\underline{\bf Lecture 1\,:\,Introduction to Bifurcation Theory}\\

\underline{\bf Abstract}\ \ Bifurcation Theory originates from problems in Applied Sciences and Engineering (such as the buckling of columns under compressive stress),

and was developed by mathematicians into a powerful toolkit that uses the instability of solutions to certain problems to prove the existence of other solutions nearby.

These "bifurcating" solutions are often less symmetric and harder to find directly, but can provide very interesting examples.

In this first lecture, I will give a broad overview of bifurcation methods and some classical applications.\\

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\underline{\bf Lecture 2\,:\,Bifurcating Constant Mean Curvature Hypersurfaces}\\

\underline{\bf Abstract}\ \ In this second lecture, I will survey on applications of Bifurcation Theory to establish the existence of many hypersurfaces with Constant Mean Curvature.

In particular, I will describe a construction of Delaunay-type hypersurfaces in cohomogeneity one manifolds, which generalizes classical Delaunay surfaces.\\

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\underline{\bf Lecture 3\,:\,Bifurcating solutions to the classical Yamabe problem}\\

\underline{\bf Abstract}\ \ The third lecture will focus on proving the existence of infinitely many solutions to the (classical) Yamabe problem on certain closed manifolds,

that is, Riemannian metrics with constant scalar curvature in prescribed conformal classes. Key examples are products and total spaces of Riemannian submersions.\\

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\underline{\bf Lecture 4: Bifurcating solutions to generalizations of the Yamabe problem}\\

\underline{\bf Abstract}\ \

In this last lecture, I will describe how similar bifurcation techniques yield infinitely many solutions to certain generalizations of the Yamabe problem to noncompact manifolds,

as well as to higher order problems in conformal geometry, such as the fourth-order Q-curvature problem.\\

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$\bullet$\ \ {\bf Boris Botvinnik (University of Oregon, USA)}\\

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\underline{\bf Main topic\,:\,Spaces of psc-metrics and moduli spaces of psc-metrics}\\

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\underline{\bf Lectures 1--3\,:\,The space of positive scalar curvature\,:\,constructions,} \\

\underline{\bf results and conjectures}\\

\underline{\bf Abstract}\ \ \underline{Lecture 1}\ : Geometric constructions and results concerning the relevant

spaces of metrics, conformal classes and corresponding moduli spaces.

Surgery results, homotopy invariance of the space of psc-metrics under surgery.\\

\underline{Lecture 2}\ : Dirac operator and psc-curvature, Gromov-Lawson-Stolz results;

concordance/isotopy, index theory for families of metrics. Hitchin's results.

Index theory and moduli spaces of metrics.\\

\underline{Lecture 3}\ : Spaces of smooth manifolds and of manifolds with psc-metrics.

Review of the results on the moduli spaces following Madse-Weiss, Galatious, Randal-Williams.\\

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\underline{\bf Lecture 4\,:\,Positive scalar curvature metrics on manifolds with fibered}\\

\underline{\bf singularities}\\

\underline{\bf Abstract}\ \ Index-difference map and review of the results following Botvinnik--Ebert--Randal-Williams

on the homotopy type of the space of psc-metrics.\\

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$\bullet$\ \ {\bf Kei Irie (University of Tokyo, Japan)}\\

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\underline{\bf Lectures 1--2\,:\,Symplectic homology}\\

\underline{\bf Abstract}\ \

Symplectic homology was introduced by Floer-Hofer and Viterbo in early

nineties, combining two important ideas in symplectic geometry:

symplectic capacity and Floer homology. It is now one of standard tools

in symplectic topology and its applications to Hamiltonian dynamics.

The goal of my talks will be to outline the theory with a few sample

applications, such as a proof of Weinstein conjecture on periodic orbits

of Hamiltonian dynamics on symplectic vector spaces, and a proof of

Gromov non-squeezing theorem.

If time permits, I will discuss a few topics from recent research. \\

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