WORKSHOPGEOMETRIC ANALYSIS IN GEOMETRY AND TOPOLOGY 2018

Date：March 11th – 14th, 2019
Place：Chuo University (Korakuen Campus) 6th building, 61125
Invited speakers• Eric Bahuaud (Seattle University)• Renato Bettiol (City University of New York)• Boris Botvinnik (University of Oregon)• Kei Irie (University of Tokyo)
Schedule
・・・・・・・・・・・ 10:00–11:00
11:30–12:30
14:00–15:00
15:30–16:30
March 11 B. Botvinnik-1 B. Botvinnik-2 R. Bettiol-1
R. Bettiol-2¯
March 12 E. Bahuaud-1 E. Bahuaud-2 B. Botvinnik-3 B. Botvinnik-4
March 13 K. Irie-1
K. Irie-2
Short Excursion
March 14 R. Bettiol-3
R. Bettiol-4
E. Bahuaud-3 E. Bahuaud-4
Organizers• Shu Nakamura (University of Tokyo)• Mikio Furuta (University of Tokyo)• Shinichiroh Matsuo (Nagoya University)• Tsuyoshi Kato (Kyoto University)• Yoshihiko Matsumoto (Osaka University)• Nobuhiko Otoba (University of Regensburg)• Rafe Mazzeo (Stanford University, Foreign adviser)• Kazuo Akutagawa (Chuo University)———————————————–

Titles & Abstracts of Lectures• Eric Bahuaud (Seattle University, USA)
Lectures 1–2 : Poincaré-Einstein metrics with prescribed conformal infinity
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] asconformal infinity ? In these talks I will introduce this problem and the explain theGraham-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
Poincaré-Einstein metrics.
Lectures 3–4 : Ricci flow of asymptotically hyperbolic metrics
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 Poincaré-Einstein metrics and metrics
with an even expansion. I’ll then discuss joint work with Woolgar on the long-timeexistence of the flow for rotationally symmetric asymptotically hyperbolic initial
data, and ongoing work with Guenther and Isenberg on the stability of these flows.
• Renato Bettiol (City University of New York, USA)
Lectures 1– 4 : Applications of Bifurcation Theory to Geometric Analysis
Lecture 1 : Introduction to Bifurcation Theory
Abstract Bifurcation Theory originates from problems in Applied Sciences andEngineering (such as the buckling of columns under compressive stress), and wasdeveloped 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.
Lecture 2 : Bifurcating Constant Mean Curvature Hypersurfaces
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.
Lecture 3 : Bifurcating solutions to the classical Yamabe problem
Abstract The third lecture will focus on proving the existence of infinitely manysolutions 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.
Lecture 4: Bifurcating solutions to generalizations of the Yamabe problem
Abstract In this last lecture, I will describe how similar bifurcation techniques
yield infinitely many solutions to certain generalizations of the Yamabe problem tononcompact manifolds, as well as to higher order problems in conformal geometry,such as the fourth-order Q-curvature problem.

• Boris Botvinnik (University of Oregon, USA)
Main topic : Spaces of psc-metrics and moduli spaces of psc-metrics
Lectures 1–3 : The space of positive scalar curvature : constructions,results and conjectures
Abstract 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.
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.
Lecture 3 : Spaces of smooth manifolds and of manifolds with psc-metrics. Reviewof the results on the moduli spaces following Madse-Weiss, Galatious, RandalWilliams.
Lecture 4 : Positive scalar curvature metrics on manifolds with fiberedsingularities
Abstract Index-diﬀerence map and review of the results following Botvinnik–
Ebert–Randal-Williams on the homotopy type of the space of psc-metrics.
• Kei Irie (University of Tokyo, Japan)
Lectures 1–2 : Symplectic homology
Abstract Symplectic homology was introduced by Floer-Hofer and Viterbo inearly 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 beto outline the theory with a few sample applications, such as a proof of Weinsteinconjecture 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.