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Seminar information archive
Seminar information archive ～07/03｜Today's seminar 07/04 | Future seminars 07/05～
2016/01/26
PDE Real Analysis Seminar
10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Salomé Oudet (University of Tokyo)
Hamilton-Jacobi equations for optimal control on 2-dimensional junction (English)
Salomé Oudet (University of Tokyo)
Hamilton-Jacobi equations for optimal control on 2-dimensional junction (English)
[ Abstract ]
We are interested in infinite horizon optimal control problems on 2-dimensional junctions (namely a union of half-planes sharing a common straight line) where different dynamics and different running costs are allowed in each half-plane. As for more classical optimal control problems, ones wishes to determine the Hamilton-Jacobi equation which characterizes the value function. However, the geometric singularities of the 2-dimensional junction and discontinuities of data do not allow us to apply the classical results of the theory of the viscosity solutions.
We will explain how to skirt these difficulties using arguments coming both from the viscosity theory and from optimal control theory. By this way we prove that the expected equation to characterize the value function is well posed. In particular we prove a comparison principle for this equation.
We are interested in infinite horizon optimal control problems on 2-dimensional junctions (namely a union of half-planes sharing a common straight line) where different dynamics and different running costs are allowed in each half-plane. As for more classical optimal control problems, ones wishes to determine the Hamilton-Jacobi equation which characterizes the value function. However, the geometric singularities of the 2-dimensional junction and discontinuities of data do not allow us to apply the classical results of the theory of the viscosity solutions.
We will explain how to skirt these difficulties using arguments coming both from the viscosity theory and from optimal control theory. By this way we prove that the expected equation to characterize the value function is well posed. In particular we prove a comparison principle for this equation.
2016/01/25
Tokyo Probability Seminar
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
Atsushi Nakayasu (Graduate School of Mathematical Sciences, The University of Tokyo)
Hamilton-Jacobi equations in metric spaces
Atsushi Nakayasu (Graduate School of Mathematical Sciences, The University of Tokyo)
Hamilton-Jacobi equations in metric spaces
Seminar on Geometric Complex Analysis
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Kunio Obitsu (Kagoshima Univ.)
(Japanese)
Kunio Obitsu (Kagoshima Univ.)
(Japanese)
2016/01/22
FMSP Lectures
15:00 -16:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Aurelien Djament (Nantes/CNRS)(by video conference system) and Christine Vespa (Strasbourg) (ENGLISH)
Functor categories and stable homology of groups (8) (ENGLISH)
[ Reference URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Djament%26Vespa.pdf
Aurelien Djament (Nantes/CNRS)(by video conference system) and Christine Vespa (Strasbourg) (ENGLISH)
Functor categories and stable homology of groups (8) (ENGLISH)
[ Reference URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Djament%26Vespa.pdf
FMSP Lectures
16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Aurelien Djament (Nantes/CNRS)(by video conference system) and Christine Vespa (Strasbourg)
Functor categories and stable homology of groups (9) (ENGLISH)
[ Reference URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Djament%26Vespa.pdf
Aurelien Djament (Nantes/CNRS)(by video conference system) and Christine Vespa (Strasbourg)
Functor categories and stable homology of groups (9) (ENGLISH)
[ Reference URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Djament%26Vespa.pdf
Operator Algebra Seminars
15:00-17:00 Room #118 (Graduate School of Math. Sci. Bldg.)
Reiji Tomatsu (Hokkaido Univ.)
Introduction to $C^*$-tensor categories
Reiji Tomatsu (Hokkaido Univ.)
Introduction to $C^*$-tensor categories
2016/01/21
FMSP Lectures
15:00-16:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Aurelien Djament (Nantes/CNRS)(by video conference system) and Christine Vespa (Strasbourg)
Functor categories and stable homology of groups (6) (ENGLISH)
[ Reference URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Djament%26Vespa.pdf
Aurelien Djament (Nantes/CNRS)(by video conference system) and Christine Vespa (Strasbourg)
Functor categories and stable homology of groups (6) (ENGLISH)
[ Reference URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Djament%26Vespa.pdf
FMSP Lectures
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Aurelien Djament (Nantes/CNRS)(by video conference system) and Christine Vespa (Strasbourg)
Functor categories and stable homology of groups (7) (ENGLISH)
[ Reference URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Djament%26Vespa.pdf
Aurelien Djament (Nantes/CNRS)(by video conference system) and Christine Vespa (Strasbourg)
Functor categories and stable homology of groups (7) (ENGLISH)
[ Reference URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Djament%26Vespa.pdf
Operator Algebra Seminars
15:00-17:00 Room #118 (Graduate School of Math. Sci. Bldg.)
Reiji Tomatsu (Hokkaido Univ.)
Introduction to $C^*$-tensor categories
Reiji Tomatsu (Hokkaido Univ.)
Introduction to $C^*$-tensor categories
2016/01/20
FMSP Lectures
16:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Aurelien Djament (Nantes/CNRS)(by video conference system) and Christine Vespa (Strasbourg)
Functor categories and stable homology of groups (5) (ENGLISH)
[ Reference URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Djament%26Vespa.pdf
Aurelien Djament (Nantes/CNRS)(by video conference system) and Christine Vespa (Strasbourg)
Functor categories and stable homology of groups (5) (ENGLISH)
[ Reference URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Djament%26Vespa.pdf
Operator Algebra Seminars
15:00-17:00 Room #118 (Graduate School of Math. Sci. Bldg.)
Reiji Tomatsu (Hokkaido Univ.)
Introduction to $C^*$-tensor categories
Reiji Tomatsu (Hokkaido Univ.)
Introduction to $C^*$-tensor categories
Seminar on Probability and Statistics
13:00-17:00 Room #123 (Graduate School of Math. Sci. Bldg.)
Enzo Orsingher (Sapienza University of Rome)
Fractional calculus and some applications to stochastic processes
Enzo Orsingher (Sapienza University of Rome)
Fractional calculus and some applications to stochastic processes
[ Abstract ]
1) Riemann-Liouville fractional integrals and derivatives
2) integrals of derivatives and derivatives of integrals
3) Dzerbayshan-Caputo fractional derivatives
4) Marchaud derivative
5) Riesz potential and fractional derivatives
6) Hadamard derivatives and also Erdelyi-Kober derivatives
7) Laplace transforms of Riemann.Liouville and Dzerbayshan-Caputo fractional derivatives
8) Fractional diffusion equations and related special functions (Mittag-Leffler and Wright functions)
9) Fractional telegraph equations (space-time fractional equations and also their mutidimensional versions)
10) Time-fractional telegraph Poisson process
11) Space fractional Poisson process
13) Other fractional point processes (birth and death processes)
14) We shall present the relationship between solutions of wave and Euler-Poisson-Darboux equations through the Erdelyi-Kober integrals.
In these lessons we will introduce the main ideas of the classical fractional calculus. The results and theorems will be presented with all details and calculations. We shall study some fundamental fractional equations and their interplay with stochastic processes. Some details on the iterated Brownian motion will also be given.
1) Riemann-Liouville fractional integrals and derivatives
2) integrals of derivatives and derivatives of integrals
3) Dzerbayshan-Caputo fractional derivatives
4) Marchaud derivative
5) Riesz potential and fractional derivatives
6) Hadamard derivatives and also Erdelyi-Kober derivatives
7) Laplace transforms of Riemann.Liouville and Dzerbayshan-Caputo fractional derivatives
8) Fractional diffusion equations and related special functions (Mittag-Leffler and Wright functions)
9) Fractional telegraph equations (space-time fractional equations and also their mutidimensional versions)
10) Time-fractional telegraph Poisson process
11) Space fractional Poisson process
13) Other fractional point processes (birth and death processes)
14) We shall present the relationship between solutions of wave and Euler-Poisson-Darboux equations through the Erdelyi-Kober integrals.
In these lessons we will introduce the main ideas of the classical fractional calculus. The results and theorems will be presented with all details and calculations. We shall study some fundamental fractional equations and their interplay with stochastic processes. Some details on the iterated Brownian motion will also be given.
2016/01/19
FMSP Lectures
13:30 -14:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Aurelien Djament (Nantes/CNRS)(by video conference system) and Christine Vespa (Strasbourg)
Functor categories and stable homology of groups (3) (ENGLISH)
[ Reference URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Djament%26Vespa.pdf
Aurelien Djament (Nantes/CNRS)(by video conference system) and Christine Vespa (Strasbourg)
Functor categories and stable homology of groups (3) (ENGLISH)
[ Reference URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Djament%26Vespa.pdf
FMSP Lectures
16:30 -18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Aurelien Djament (Nantes/CNRS)(by video conference system) and Christine Vespa (Strasbourg)
Functor categories and stable homology of groups (4) (ENGLISH)
[ Reference URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Djament%26Vespa.pdf
Aurelien Djament (Nantes/CNRS)(by video conference system) and Christine Vespa (Strasbourg)
Functor categories and stable homology of groups (4) (ENGLISH)
[ Reference URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Djament%26Vespa.pdf
Operator Algebra Seminars
15:00-17:00 Room #118 (Graduate School of Math. Sci. Bldg.)
Reiji Tomatsu (Hokkaido Univ.)
Introduction to $C^*$-tensor categories
Reiji Tomatsu (Hokkaido Univ.)
Introduction to $C^*$-tensor categories
Tuesday Seminar on Topology
15:00-16:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Hikaru Yamamoto (The University of Tokyo)
Ricci-mean curvature flows in gradient shrinking Ricci solitons (JAPANESE)
Hikaru Yamamoto (The University of Tokyo)
Ricci-mean curvature flows in gradient shrinking Ricci solitons (JAPANESE)
[ Abstract ]
A Ricci-mean curvature flow is a coupled parabolic PDE system of a mean
curvature flow and a Ricci flow.
In this talk, we consider a Ricci-mean curvature flow in a gradient
shrinking Ricci soliton, and give a generalization of a well-known result
of Huisken which states that if a mean curvature flow in a Euclidean space
develops a singularity of type I, then its parabolic rescaling near the singular
point converges to a self-shrinker.
A Ricci-mean curvature flow is a coupled parabolic PDE system of a mean
curvature flow and a Ricci flow.
In this talk, we consider a Ricci-mean curvature flow in a gradient
shrinking Ricci soliton, and give a generalization of a well-known result
of Huisken which states that if a mean curvature flow in a Euclidean space
develops a singularity of type I, then its parabolic rescaling near the singular
point converges to a self-shrinker.
PDE Real Analysis Seminar
10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Hao Wu (Fudan University)
Well-posedness and stability of the full Ericksen-Leslie system for incompressible nematic liquid crystal flows
Hao Wu (Fudan University)
Well-posedness and stability of the full Ericksen-Leslie system for incompressible nematic liquid crystal flows
[ Abstract ]
In this talk, the general Ericksen-Leslie (E-L) system modelling the incompressible nematic liquid crystal flow will be discussed.
We shall prove the well-posedness and long-time behavior of the E-L system under proper assumptions on the viscous Leslie coefficients.
In particular, we shall discuss the connection between Parodi's relation and stability of the E-L system.
In this talk, the general Ericksen-Leslie (E-L) system modelling the incompressible nematic liquid crystal flow will be discussed.
We shall prove the well-posedness and long-time behavior of the E-L system under proper assumptions on the viscous Leslie coefficients.
In particular, we shall discuss the connection between Parodi's relation and stability of the E-L system.
2016/01/18
Seminar on Geometric Complex Analysis
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Hiroshige Shiga (Tokyo Institute of Technology)
Holomorphic motions and the monodromy (Japanese)
Hiroshige Shiga (Tokyo Institute of Technology)
Holomorphic motions and the monodromy (Japanese)
[ Abstract ]
Holomorphic motions, which was introduced by Mane, Sad and Sullivan, is a useful tool for Teichmuller theory as well as for complex dynamics. In particular, Slodkowski’s theorem makes a significant contribution to them. The theorem says that every holomorphic motion of a closed set on the Riemann sphere parametrized by the unit disk is extended to a holomorphic motion of the whole Riemann sphere parametrized by the unit disk. In this talk, we consider a generalization of the theorem. If time permits, we will discuss applications of our results.
Holomorphic motions, which was introduced by Mane, Sad and Sullivan, is a useful tool for Teichmuller theory as well as for complex dynamics. In particular, Slodkowski’s theorem makes a significant contribution to them. The theorem says that every holomorphic motion of a closed set on the Riemann sphere parametrized by the unit disk is extended to a holomorphic motion of the whole Riemann sphere parametrized by the unit disk. In this talk, we consider a generalization of the theorem. If time permits, we will discuss applications of our results.
FMSP Lectures
15:00-16:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Aurelien Djament (Nantes/CNRS)(by video conference system) and Christine Vespa (Strasbourg)
Functor categories and stable homology of groups (1) (ENGLISH)
[ Reference URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Djament%26Vespa.pdf
Aurelien Djament (Nantes/CNRS)(by video conference system) and Christine Vespa (Strasbourg)
Functor categories and stable homology of groups (1) (ENGLISH)
[ Reference URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Djament%26Vespa.pdf
FMSP Lectures
16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Aurelien Djament (Nantes/CNRS)(by video conference system) and Christine Vespa (Strasbourg)
Functor categories and stable homology of groups (2) (ENGLISH)
[ Reference URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Djament%26Vespa.pdf
Aurelien Djament (Nantes/CNRS)(by video conference system) and Christine Vespa (Strasbourg)
Functor categories and stable homology of groups (2) (ENGLISH)
[ Reference URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Djament%26Vespa.pdf
Operator Algebra Seminars
15:00-17:00 Room #118 (Graduate School of Math. Sci. Bldg.)
Reiji Tomatsu (Hokkaido Univ.)
Introduction to $C^*$-tensor categories (日本語)
Reiji Tomatsu (Hokkaido Univ.)
Introduction to $C^*$-tensor categories (日本語)
Seminar on Probability and Statistics
13:00-17:00 Room #123 (Graduate School of Math. Sci. Bldg.)
Enzo Orsingher (Sapienza University of Rome)
Fractional calculus and some applications to stochastic processes
Enzo Orsingher (Sapienza University of Rome)
Fractional calculus and some applications to stochastic processes
[ Abstract ]
1) Riemann-Liouville fractional integrals and derivatives
2) integrals of derivatives and derivatives of integrals
3) Dzerbayshan-Caputo fractional derivatives
4) Marchaud derivative
5) Riesz potential and fractional derivatives
6) Hadamard derivatives and also Erdelyi-Kober derivatives
7) Laplace transforms of Riemann.Liouville and Dzerbayshan-Caputo fractional derivatives
8) Fractional diffusion equations and related special functions (Mittag-Leffler and Wright functions)
9) Fractional telegraph equations (space-time fractional equations and also their mutidimensional versions)
10) Time-fractional telegraph Poisson process
11) Space fractional Poisson process
13) Other fractional point processes (birth and death processes)
14) We shall present the relationship between solutions of wave and Euler-Poisson-Darboux equations through the Erdelyi-Kober integrals.
In these lessons we will introduce the main ideas of the classical fractional calculus. The results and theorems will be presented with all details and calculations. We shall study some fundamental fractional equations and their interplay with stochastic processes. Some details on the iterated Brownian motion will also be given.
1) Riemann-Liouville fractional integrals and derivatives
2) integrals of derivatives and derivatives of integrals
3) Dzerbayshan-Caputo fractional derivatives
4) Marchaud derivative
5) Riesz potential and fractional derivatives
6) Hadamard derivatives and also Erdelyi-Kober derivatives
7) Laplace transforms of Riemann.Liouville and Dzerbayshan-Caputo fractional derivatives
8) Fractional diffusion equations and related special functions (Mittag-Leffler and Wright functions)
9) Fractional telegraph equations (space-time fractional equations and also their mutidimensional versions)
10) Time-fractional telegraph Poisson process
11) Space fractional Poisson process
13) Other fractional point processes (birth and death processes)
14) We shall present the relationship between solutions of wave and Euler-Poisson-Darboux equations through the Erdelyi-Kober integrals.
In these lessons we will introduce the main ideas of the classical fractional calculus. The results and theorems will be presented with all details and calculations. We shall study some fundamental fractional equations and their interplay with stochastic processes. Some details on the iterated Brownian motion will also be given.
FMSP Lectures
14:00-15:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Samuli Siltanen (University of Helsinki)
Blind deconvolution for human speech signals (ENGLISH)
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Siltanen.pdf
Samuli Siltanen (University of Helsinki)
Blind deconvolution for human speech signals (ENGLISH)
[ Abstract ]
The structure of vowel sounds in human speech can be divided into two independent components. One of them is the “excitation signal,” which is a kind of buzzing sound created by the vocal folds flapping against each other. The other is the “filtering effect” caused by resonances in the vocal tract, or the confined space formed by the mouth and throat. The Glottal Inverse Filtering (GIF) problem is to (algorithmically) divide a microphone recording of a vowel sound into its two components. This “blind deconvolution” type task is an ill-posed inverse problem. Good-quality GIF filtering is essential for computer-generated speech needed for example by disabled people (think Stephen Hawking). Also, GIF affects the quality of synthetic speech in automatic information announcements and car navigation systems. Accurate estimation of the voice source from recorded speech is known to be difficult with current glottal inverse filtering (GIF) techniques, especially in the case of high-pitch speech of female or child subjects. In order to tackle this problem, the present study uses two different solution methods for GIF: Bayesian inversion and alternating minimization. The first method takes advantage of the Markov chain Monte Carlo (MCMC) modeling in defining the parameters of the vocal tract inverse filter. The filtering results are found to be superior to those achieved by the standard iterative adaptive inverse filtering (IAIF), but the computation is much slower than IAIF. Alternating minimization cuts down the computation time while retaining most of the quality improvement.
[ Reference URL ]The structure of vowel sounds in human speech can be divided into two independent components. One of them is the “excitation signal,” which is a kind of buzzing sound created by the vocal folds flapping against each other. The other is the “filtering effect” caused by resonances in the vocal tract, or the confined space formed by the mouth and throat. The Glottal Inverse Filtering (GIF) problem is to (algorithmically) divide a microphone recording of a vowel sound into its two components. This “blind deconvolution” type task is an ill-posed inverse problem. Good-quality GIF filtering is essential for computer-generated speech needed for example by disabled people (think Stephen Hawking). Also, GIF affects the quality of synthetic speech in automatic information announcements and car navigation systems. Accurate estimation of the voice source from recorded speech is known to be difficult with current glottal inverse filtering (GIF) techniques, especially in the case of high-pitch speech of female or child subjects. In order to tackle this problem, the present study uses two different solution methods for GIF: Bayesian inversion and alternating minimization. The first method takes advantage of the Markov chain Monte Carlo (MCMC) modeling in defining the parameters of the vocal tract inverse filter. The filtering results are found to be superior to those achieved by the standard iterative adaptive inverse filtering (IAIF), but the computation is much slower than IAIF. Alternating minimization cuts down the computation time while retaining most of the quality improvement.
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Siltanen.pdf
FMSP Lectures
14:45-15:25 Room #126 (Graduate School of Math. Sci. Bldg.)
Tapio Helin (University of Helsinki)
Inverse scattering from random potential (ENGLISH)
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Helin.pdf
Tapio Helin (University of Helsinki)
Inverse scattering from random potential (ENGLISH)
[ Abstract ]
We consider an inverse scattering problem with a random potential. We assume that our far-field data at multiple angles and all frequencies are generated by a single realization of the potential. From the frequency-correlated data our aim is to demonstrate that one can recover statistical properties of the potential. More precisely, the potential is assumed to be Gaussian with a covariance operator that can be modelled by a classical pseudodifferential operator. Our main result is to show that the principal symbol of this
covariance operator can be determined uniquely. What is important, our method does not require any approximation and we can analyse also the multiple scattering. This is joint work with Matti Lassas and Pedro Caro.
[ Reference URL ]We consider an inverse scattering problem with a random potential. We assume that our far-field data at multiple angles and all frequencies are generated by a single realization of the potential. From the frequency-correlated data our aim is to demonstrate that one can recover statistical properties of the potential. More precisely, the potential is assumed to be Gaussian with a covariance operator that can be modelled by a classical pseudodifferential operator. Our main result is to show that the principal symbol of this
covariance operator can be determined uniquely. What is important, our method does not require any approximation and we can analyse also the multiple scattering. This is joint work with Matti Lassas and Pedro Caro.
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Helin.pdf
FMSP Lectures
15:25-16:05 Room #126 (Graduate School of Math. Sci. Bldg.)
Matti Lassas (University of Helsinki)
Geometric Whitney problem: Reconstruction of a manifold from a point cloud (ENGLISH)
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Lassas.pdf
Matti Lassas (University of Helsinki)
Geometric Whitney problem: Reconstruction of a manifold from a point cloud (ENGLISH)
[ Abstract ]
We study the geometric Whitney problem on how a Riemannian manifold $(M,g)$ can be constructed to approximate a metric space $(X,d_X)$. This problem is closely related to manifold interpolation (or manifold learning) where a smooth $n$-dimensional surface $S¥subset {¥mathbb R}^m$, $m>n$ needs to be constructed to approximate a point cloud in ${¥mathbb R}^m$. These questions are encountered in differential geometry, machine learning, and in many inverse problems encountered in applications. The determination of a Riemannian manifold includes the construction of its topology, differentiable structure, and metric.
We give constructive solutions to the above problems. Moreover, we characterize the metric spaces that can be approximated, by Riemannian manifolds with bounded geometry: We give sufficient conditions to ensure that a metric space can be approximated, in the Gromov-Hausdorff or quasi-isometric sense, by a Riemannian manifold of a fixed dimension and with bounded diameter, sectional curvature, and injectivity radius. Also, we show that similar conditions, with modified values of parameters, are necessary.
Moreover, we characterise the subsets of Euclidean spaces that can be approximated in the Hausdorff metric by submanifolds of a fixed dimension and with bounded principal curvatures and normal injectivity radius.
The above interpolation problems are also studied for unbounded metric sets and manifolds. The results for Riemannian manifolds are based on a generalisation of the Whitney embedding construction where approximative coordinate charts are embedded in ${¥mathbb R}^m$ and interpolated to a smooth surface. We also give algorithms that solve the problems for finite data.
The results are done in collaboration with C. Fefferman, S. Ivanov, Y. Kurylev, and H. Narayanan.
References:
[1] C. Fefferman, S. Ivanov, Y. Kurylev, M. Lassas, H. Narayanan: Reconstruction and interpolation of manifolds I: The geometric Whitney problem. ArXiv:1508.00674
[ Reference URL ]We study the geometric Whitney problem on how a Riemannian manifold $(M,g)$ can be constructed to approximate a metric space $(X,d_X)$. This problem is closely related to manifold interpolation (or manifold learning) where a smooth $n$-dimensional surface $S¥subset {¥mathbb R}^m$, $m>n$ needs to be constructed to approximate a point cloud in ${¥mathbb R}^m$. These questions are encountered in differential geometry, machine learning, and in many inverse problems encountered in applications. The determination of a Riemannian manifold includes the construction of its topology, differentiable structure, and metric.
We give constructive solutions to the above problems. Moreover, we characterize the metric spaces that can be approximated, by Riemannian manifolds with bounded geometry: We give sufficient conditions to ensure that a metric space can be approximated, in the Gromov-Hausdorff or quasi-isometric sense, by a Riemannian manifold of a fixed dimension and with bounded diameter, sectional curvature, and injectivity radius. Also, we show that similar conditions, with modified values of parameters, are necessary.
Moreover, we characterise the subsets of Euclidean spaces that can be approximated in the Hausdorff metric by submanifolds of a fixed dimension and with bounded principal curvatures and normal injectivity radius.
The above interpolation problems are also studied for unbounded metric sets and manifolds. The results for Riemannian manifolds are based on a generalisation of the Whitney embedding construction where approximative coordinate charts are embedded in ${¥mathbb R}^m$ and interpolated to a smooth surface. We also give algorithms that solve the problems for finite data.
The results are done in collaboration with C. Fefferman, S. Ivanov, Y. Kurylev, and H. Narayanan.
References:
[1] C. Fefferman, S. Ivanov, Y. Kurylev, M. Lassas, H. Narayanan: Reconstruction and interpolation of manifolds I: The geometric Whitney problem. ArXiv:1508.00674
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Lassas.pdf
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