GeoSpec LabEx PERSYVAL June 14th 2017

Transcription

1 GeoSpec LabEx PERSYVAL June 14th 2017

2 Members of the project Institut Fourier

3 Members of the project Institut Fourier Laboratoire Jean Kuntzman

4 Members of the project Institut Fourier Laboratoire Jean Kuntzman Lama

8 Fundamental Mathematics and Numerical Tools

9 Fundamental Mathematics and Numerical Tools Formal computation (Maple, SAGE, MuPAD, etc.).

10 Fundamental Mathematics and Numerical Tools Formal computation (Maple, SAGE, MuPAD, etc.). Four color Theorem: Coq software and «Proof Assistant», certification.

11 Fundamental Mathematics and Numerical Tools Formal computation (Maple, SAGE, MuPAD, etc.). Four color Theorem: Coq software and «Proof Assistant», certification. Hales approach, scientific computing: Kepler s and the Honeycomb conjecture. Interval arithmetic and Optimization. Proof of Kepler s conjecture!

12 Fundamental Mathematics and Numerical Tools Formal computation (Maple, SAGE, MuPAD, etc.). Four color Theorem: Coq software and «Proof Assistant», certification. Hales approach, scientific computing: Kepler s and the Honeycomb conjecture. Interval arithmetic and Optimization. Proof of Kepler s conjecture! A local success story: the Flat Torus (PNAS, Pour La Science, etc.).

13 Fundamental Mathematics and Numerical Tools Formal computation (Maple, SAGE, MuPAD, etc.). Four color Theorem: Coq software and «Proof Assistant», certification. Hales approach, scientific computing: Kepler s and the Honeycomb conjecture. Interval arithmetic and Optimization. Proof of Kepler s conjecture! A local success story: the Flat Torus (PNAS, Pour La Science, etc.). Objectives of the project

14 Fundamental Mathematics and Numerical Tools Formal computation (Maple, SAGE, MuPAD, etc.). Four color Theorem: Coq software and «Proof Assistant», certification. Hales approach, scientific computing: Kepler s and the Honeycomb conjecture. Interval arithmetic and Optimization. Proof of Kepler s conjecture! A local success story: the Flat Torus (PNAS, Pour La Science, etc.). Objectives of the project Take benefit of Grenoble s potential which has a strong expertise both in pure mathematics and computer science.

15 Fundamental Mathematics and Numerical Tools Formal computation (Maple, SAGE, MuPAD, etc.). Four color Theorem: Coq software and «Proof Assistant», certification. Hales approach, scientific computing: Kepler s and the Honeycomb conjecture. Interval arithmetic and Optimization. Proof of Kepler s conjecture! A local success story: the Flat Torus (PNAS, Pour La Science, etc.). Objectives of the project Take benefit of Grenoble s potential which has a strong expertise both in pure mathematics and computer science. We plan to develop a new synergy at a national and international level around young researchers of several disciplines. More than new theorems, we would like to introduce new highlights of difficult and important mathematical questions.

16 Dynamic Complerxity : Birkhoff s sections

17 Dynamic Complerxity : Birkhoff s sections Section associated to a vector field

18 Dynamic Complerxity : Birkhoff s sections Section associated to a vector field Understand the complexity of a vector field by a section

19 Dynamic Complerxity : Birkhoff s sections Section associated to a vector field Understand the complexity of a vector field by a section Specialists of differential geometry work with classes of surfaces

20 Dynamic Complerxity : Birkhoff s sections Section associated to a vector field Understand the complexity of a vector field by a section Specialists of differential geometry work with classes of surfaces The non constructive Thurston theorem ensures the existence of simple representations for flat surfaces in the Hyperbolic context. How to identify such representations?

21 Optimization among metrics pp v 2 T x S, lim n!1 1 n log D xf n (v) = log( )

22 Optimization among metrics pp v 2 T x S, lim n!1 1 n log D xf n (v) = log( ) Parametrization of metrics: local constraints of SDP type. Non convergence for P1 representations! Different scales are required to obtain convergence.

23 Optimization among metrics pp v 2 T x S, lim n!1 1 n log D xf n (v) = log( ) Parametrization of metrics: local constraints of SDP type. Non convergence for P1 representations! Different scales are required to obtain convergence. Benchmarks: Explicit identification of Thurston invariant. Other parametrization of the problem?

24 Optimization among metrics pp v 2 T x S, lim n!1 1 n log D xf n (v) = log( ) Parametrization of metrics: local constraints of SDP type. Non convergence for P1 representations! Different scales are required to obtain convergence. Benchmarks: Explicit identification of Thurston invariant. Other parametrization of the problem? Linearization, projection //, non smooth and large scale algorithm.

25 Geodesic and Spectral complexity:

26 Geodesic and Spectral complexity: What is a geodesic on a flat surface?

27 Geodesic and Spectral complexity: What is a geodesic on a flat æ surface? V S x 0

28 Geodesic and Spectral complexity: What is a geodesic on a flat æ surface? V S x 0 Identify the non trivial shortest geodesic?

29 Geodesic and Spectral complexity: What is a geodesic on a flat æ surface? V S x 0 Identify the non trivial shortest geodesic? Combinatorial complexity: all the points/directions, Galois s team

30 Geodesic and Spectral complexity: What is a geodesic on a flat æ surface? V S x 0 Identify the non trivial shortest geodesic? Combinatorial complexity: all the points/directions, Galois s team Non Flat setting, geodesic approximation

31 Geodesic and Spectral complexity: What is a geodesic on a flat æ surface? V S x 0 Identify the non trivial shortest geodesic? Combinatorial complexity: all the points/directions, Galois s team Non Flat setting, geodesic approximation Curse of dimension

32 A PDE approach V fl 0 = x0 x = fl(t, x) t (t, x)+v r x (t, x) =0, (0, ) = x0

33 A PDE approach V fl 0 = fl 0 Ï= x0 S x fl(t, = fl(t, x) t (t, x)+v r x (t, x) =0, (0, ) = x0

34 A PDE approach V fl 0 = fl 0 Ï= x0 S x fl(t, = fl(t, x) t (t, x)+v r x (t, x) =0, (0, ) = x0 We consider the heat equation with a drive t (t, x) = (t, x) V r (t, x), (0,x)='(x).

35 A PDE approach Ò V Ï fl 0 = fl 0 = x0 S x flfl(t, = Á fl(t, x) x) e de di usion de est déterminée par la valeur t (t, x)+v r x (t, x) =0, We consider the heat equation with a drive term (0, ) = t (t, x) = (t, x) V r (t, x), (0,x)='(x).

36 A PDE approach Ò V Ï fl 0 = fl 0 = x0 S x flfl(t, = Á fl(t, x) x) e de di usion de est déterminée par la valeur t (t, x)+v r x (t, x) =0, We consider the heat equation with a drive term (0, ) = t (t, x) = (t, x) V r (t, x), (0,x)='(x). The speed of di usion of is represented by the eigenvalue Z Z u + V ru = u, u =0, u 2 =1. The less di used function is the eigenfunction u. S S

37 A PDE approach Ò V Ï fl 0 = fl 0 = x0 S x flfl(t, = Á fl(t, x) x) e de di usion de est déterminée par la valeur pro

38 A PDE approach Ò V Ï fl 0 = fl 0 = x0 S x flfl(t, = Á fl(t, x) x) e de di usion de est déterminée par la valeur pro For a fixed, détermine the vector field V which maximizes ( +V r). Study the properties of the optimal fiel V and of the corresponding eigenfunction u.

39 A PDE approach Ò V Ï fl 0 = fl 0 = x0 S x flfl(t, = Á fl(t, x) x) e de di usion de est déterminée par la valeur pro For a fixed, détermine the vector field V which maximizes ( +V r). Study the properties of the optimal fiel V and of the corresponding eigenfunction u. Non Variational PDE, Non smooth Optimization when the multiplicity is >0. Bundle s method.

40 A PDE approach Ò V Ï fl 0 = fl 0 = x0 S x flfl(t, = Á fl(t, x) x) e de di usion de est déterminée par la valeur pro For a fixed, détermine the vector field V which maximizes ( +V r). Study the properties of the optimal fiel V and of the corresponding eigenfunction u. Non Variational PDE, Non smooth Optimization when the multiplicity is >0. Bundle s method. Non convex problem: relaxation by considering small parameters?

41 Metric complexity:

42 Metric complexity: Flat Torus and distances: Nash s Isometric problem

45 Metric complexity: Flat Torus and distances: Nash s Isometric problem Canonical embeding, an alternative to Gromov s approach

46 Metric complexity: Flat Torus and distances: Nash s Isometric problem Canonical embeding, an alternative to Gromov s approach Singular objects have to be parametrized

47 Metric complexity: Flat Torus and distances: Nash s Isometric problem Canonical embeding, an alternative to Gromov s approach Singular objects have to be parametrized Generalization to genus > 1

48 Metric complexity: Flat Torus and distances: Nash s Isometric problem Canonical embeding, an alternative to Gromov s approach Singular objects have to be parametrized Generalization to genus > 1 Fractal behavior has to be expected, quantitative estimate

49 A Spectral approach

50 A Spectral approach Nadirashvili s Optimality Theorem max S =, genre(s)=1 (S) = (T equi)

51 A Spectral approach Nadirashvili s Optimality Theorem max S =, genre(s)=1 (S) = (T equi) Eigenvalue Problem for singular Surfaces. Spectral parametric tools. Mesh displacement is not tractable!

52 A Spectral approach Nadirashvili s Optimality Theorem max S =, genre(s)=1 (S) = (T equi) Eigenvalue Problem for singular Surfaces. Spectral parametric tools. Mesh displacement is not tractable! How to parametrize surfaces of genus >1 in a compatible way of scientific computing. Conformal classes and representative members.

53 A Spectral approach Nadirashvili s Optimality Theorem max S =, genre(s)=1 (S) = (T equi) Eigenvalue Problem for singular Surfaces. Spectral parametric tools. Mesh displacement is not tractable! How to parametrize surfaces of genus >1 in a compatible way of scientific computing. Conformal classes and representative members. Geodesic distance, qualitative evaluation of the regularity of the embedding.

57 Some realizations

58 Some realizations Two formal students started a Phd with Pure and Applied mathematicians as supervisors

59 Some realizations Two formal students started a Phd with Pure and Applied mathematicians as supervisors Lecture group every week: around 15 researchers from different areas

60 Some realizations Two formal students started a Phd with Pure and Applied mathematicians as supervisors Lecture group every week: around 15 researchers from different areas At least two workshops organized by the members of the project

61 Some realizations Two formal students started a Phd with Pure and Applied mathematicians as supervisors Lecture group every week: around 15 researchers from different areas At least two workshops organized by the members of the project Energetic formulation of the Systole s Problem

62 Some realizations Two formal students started a Phd with Pure and Applied mathematicians as supervisors Lecture group every week: around 15 researchers from different areas At least two workshops organized by the members of the project Energetic formulation of the Systole s Problem Recursive process to parametrize singular objects