International Workshop for Young Scientists
2017
Analysis and its Applications to Geometry
5 – 9 June 2017
School of Science, Tokyo Institute of Technology
Room H201, the Main Building

ProgramMonday, 5 June
10:30 - Registration / Coﬀee
11:00 - Opening• Welcome address by the Dean, Prof. Tetsuo Okada• Introduction of Tokyo Tech by Prof. Yasuhiro Ohshima• Remarks• Group Photo
12:00 - 14:00 Lunch
14:00 - 15:30 Research Presentations by the Participants (1)15:30 - 16:00 Coﬀee Break
16:00 - 17:30 Research Presentations by the Participants (2)18:00 - 20:00 Welcome Party at Dai-ichi Shokudo (Student CO-OP)

Tuesday, 6 June
9:00 - 12:00 Lectures A1 - A2 / Coﬀee Break• Lecturer: Prof. Eiji Yanagida (Tokyo Tech)• Title: Stability Analysis for Dynamical Systems
12:00 - 14:00 Lunch
14:00 - 16:30 Research Projects A3 / Coﬀee Break
16:30 - 17:30 Presentations A41

Wednesday, 7 June
9:00 - 12:00 Lectures B1 - B2 / Coﬀee Break• Lecturer: Prof. Hiroshige Shiga (Tokyo Tech)• Title: Riemann Surface and Moduli Space
12:00 - 14:00 Lunch
14:00 - 16:30 Research Projects B3 / Coﬀee Break
16:30 - 17:30 Presentations B4

Thursday, 8 June
9:00 - 12:00 Lectures C1 - C2 / Coﬀee Break• Lecturer: Prof. Kazuo Akutagawa (Tokyo Tech)• Title: Diﬀerential Geometry — Existence of Closed Geodesics on Surfaces —12:00 - 14:00 Lunch
14:00 -Intercultural Study

Friday, 9 June
9:00 - 12:00 Lectures D1 - D2 / Coﬀee Break• Lecturer: Prof. Yoshihiro Tonegawa (Tokyo Tech)• Title: How to Construct a Mean Curvature Flow with Singularities
12:00 - 14:00 Lunch
14:00 - 16:30 Research Projects D3 / Coﬀee Break
16:30 - 17:30 Presentations D4
18:00 - 20:00 Farewell Party at Issyoh (near Ookayama Station)2

Abstracts
Lectures A1 and A2 (Tuesday)
Lecturer. Professor Eiji Yanagida (Tokyo Tech)
Title. Stability Analysis for Dynamical Systems
Abstract. Stability is one of the most important concepts in the theory of dynamical systems. I will analyze the stability of equilibria in systems of ordinary
diﬀerential equations, systems of linear parabolic partial diﬀerential equations withconstant coeﬃcients, and scalar reaction-diﬀusion equations. To this purpose, themethod of Lyapunov and linearized eigenvalue problems are introduced.
References.
[1] N. Chafee, Asymptotic behavior for solutions of a one-dimensional parabolic
equation with homogeneous Neumann boundary conditions, J. Diﬀ. Eqs. 18(1975), 111–134.
[2] L. C. Evans, Partial Diﬀerential Equations, American Mathematical Society,2010.
[3] M. Hirsch, S. Smale and R. L. Devaney Diﬀerential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 2003.
[4] J. Smoller, Shock Waves and Reaction-Diﬀusion Equations, Springer, 1994.
[5] A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc.
London, B237 (1952), 37–72.

Lectures B1 and B2 (Wednesday)
Lecturer. Professor Hiroshige Shiga (Tokyo Tech)
Title. Riemann Surface and Moduli SpaceAbstract.
A one-dimensional complex manifold is called a Riemann surface.
While it is a simple notion, one may see that Riemann surfaces appear in many
fields of mathematics and also in physics. In this lecture, after presenting somefundamental facts on Riemann surfaces, we give concrete constructions of Riemannsurfaces to understand the moduli spaces which are deformation spaces of Riemannsurfaces.

3Lectures C1 and C2 (Thursday)
Lecturer. Professor Kazuo Akutagawa (Tokyo Tech)
Title. Diﬀerential Geometry — Existence of Closed Geodesics on Surfaces —Abstract. In these two lectures, we study on the existence of closed geodesics on
a closed surface, by using the direct method and the min-max method. The
min-max method will be more powerful in the future, especially for the existence
problem of Einstein metrics on closed manifolds. The plan of my lectures is thefollowing:(1) What are “geodesics”?
(2) Existence of closed geodesics on a closed surface of genus ≥ 1.
(3) Existence of closed geodesics on 2-sphere.

References.
Diﬀerential Geometry• J. A. Thorpe, Elementary Topics in Diﬀerential Geometry, Springer-Verlag(1979).
• I. Chavel, Riemannian Geometry; A Modern Introduction, 2nd ed., Cambridge Univ. Press (2006).
• P. Petersen, Riemannian Geometry, 3rd ed., GMT 171, Springer (2016).
Variational Methods• S. Nishaikawa, Variational Methods in Geometry, Transl. Math. Monographs,Amer. Math. Soc. (2002).
• M. Struwe, Variational Methods, 3rd ed., EMG 34, Springer (2000).
Closed Geodesics• W. Klingenberg, Lectures on Closed Geodesics, GMW 230, Springer-Verlag(1978).

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Lectures D1 and D2 (Friday)
Lecturer. Professor Yoshihiro Tonegawa (Tokyo Tech)
Title. How to Construct a Mean Curvature Flow with Singularities
Abstract. A time-parametrized family of surfaces is called mean curvature flow if
the velocity of motion is equal to the mean curvature at each point and time. Mean
curvature flow is a flow of surfaces which decreases the surface area and is one of
the most fundamental geometric flow problems along with the Ricci flow. In thislecture, I will describe some essence of how we construct a mean curvature flow given
an arbitrary closed set with singularities in our recent paper: Lami Kim, Yoshihiro
Tonegawa, On the mean curvature flow of grain boundaries, Annales de l’InstitutFourier (Grenoble) 66, (2017) no. 1, 43 – 142.

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