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Shohei Honda
Spectral convergence under bounded Ricci curvature
29 October 2015 (R), 13:30~, Large lecture room 2F GSIS -
Colette Ann\'e (Universit\' e de Nantes, France)
Signature and the gap in the spectrum of the Hodge Laplacian
8 September 2015 (T), 14:00~, Large lecture room 2F GSIS -
Motohiro Sobashima (Tokyo University of Science)
Generation of analytic semigroups on $L^p$ by scale-invariant elliptic operators
28 July 2015 (T), 10:00~, Small lecture room 6F GSIS -
Noboru Okazawa (Tokyo University of Science)
Spectral theory of linear m-sectorial operators in Banach and Hilbert spaces -historical review-
27 July 2015 (M), 16:00~, Lecture room 2F GSIS -
Koichi Taniguchi (Chuo Univ)
L^p-mapping properties for Schrodinger operators in open
sets of R^d
6 June 2015 (S), 10:00~, Small lecture room 6F GSIS - Tokio Matsuyama (Chuo Univ)
Gevrey class solutions to the Kirchhoff equation
5 June 2015 (F), 13:00~, Small lecture room 6F GSIS -
Matthias Keller (Univ. Jena)
Intrinsic metric on graphs
7 May 2015 (R), 13:00~15:30, Small lecture room 6F GSIS -
Vincenzo Ferone (Univ. di Napoli)
Isoperimetric inequalities and symmetrization methods: an approach to PDE's and to functional inequalities
13 April 2015 (M), 13:30~15:00, Middle lecture room 2F GSIS
14 April 2015 (T), 15:00~18:00, Middle lecture room 2F GSIS
15 April 2015 (W), 15:00~18:00, Middle lecture room 2F GSIS - Comparison results for nonlinear elliptic and parabolic partial differential equations.
- Existence and regularity for solutions to nonlinear elliptic and parabolic partial differential equations in the case of non-regular data.
- Isoperimetric inequalities concerning functionals of the Calculus of Variations and investigation about the best constants in functional inequalities.
- Optimization problems for functionals defined on classes of equimeasurable functions.
- Other symmetrization methods.
- Andrea Colesanti (Univ. di Firenze)
From the Brunn-Minkowski inequality to elliptic PDE's
9 March 2015 (M), 13:30~16:45, Conference room 2F GSIS
10 March 2015 (T), 13:30~16:45, Conference room 2F GSIS
11 March 2015 (W), 10:30~12:00, 13:30~15:00, Conference room 2F GSIS - Kazuo Akutagawa (TIT)
Harmonic maps between asymptotically hyperbolic manifolds
27 January 2015 (T), 15:30~16:30, Conference room 2F GSIS - Kazuo Akutagawa (TIT)
The Yamabe invariant and singular Einstein metrics
26 January 2015 (M) , 13:30~17:00, Small lecture room 6F GSIS
Abstract:
Abstract: Gilles Carron proved that if the complete Riemannian manifold $M$ can be partitioned by a closed oriented hypersurface with non zero signature, then the spectrum of the Hodge Laplacian on $M$ is the all half line $[0, + \infty [.$ We present a result in the reciprocal way: If the closed Riemannian manifold $M$ can be partitioned by a closed oriented hypersurface with zero cohomological group in the middle degree, then one can construct on the corresponding $\Z$-covering a metric such that the spectrum of the Hodge Laplacian has as many gaps as we want.
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Paper: http://arxiv.org/abs/1405.5657v1
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http://arxiv.org/abs/1508.05305
Abstract:There are various results in Riemannian manifolds which are consequences of the close relationship between the Laplace- Beltrami operator and the Riemannian distance. For graphs many of these results fail to be true if one considers the combinatorial graph distance. Now by using the concept of intrinsic metrics re- cently introduced in the context of general regular Dirichlet forms by Frank/Lenz/Wingert analogous results can be proven in the context of weighted graphs. This solves various problems that have been open for several years and decades.
Abstract: The series of lectures will be devoted to give an introduction to the "so-called" symmetrization methods which have been successfully used in the study of solutions to elliptic and parabolic problems and in the study of functional inequalities. The starting point will be the classical isoperimetric inequality and Schwarz symmetrization. In particular, their role in the quoted contexts will be highlighted discussing some model results that nowadays can be considered classical. Typical questions which will be addressed are the following ones.
Abstract: The main scope of this series of lectures is to describe in details the Brunn-Minkowski inequality, in the realm of Convex Geometry, and its links to the Calculus of Variations and elliptic PDE's. The first part will be dedicated to background material in Convex Geometry. In particular we will introduce the space of convex bodies (compact, convex subsets of the n-dimensional Euclidean space), along with basic tools like the Minkowski addition, the Hausdorff metric and the support function. We will then pass to the Brunn-Minkowski inequality, presenting its proof through the Prekopa-Leindler inequality, and describing how it is connected with other fundamental inequalities in analysis such the isoperimetric and the Poincare inequality. In the third part we will speak about functionals defined on the class of convex bodies that verify an inequality of Brunn-Minkowski type. This will create the main link with elliptic PDE's. Indeed many classical functionals, like the principal frequency of a domain, the Newtonian capacity and the torsional rigidity, verify an inequality of this type. We will present these examples in some details, explaining some of the techniques used to prove the corresponding Brunn-Minkowski inequality.
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