-
Proposition de cration dquipe
GeoSpec
LabEx PERSYVAL 29/04/2016
-
Institut Fourier
Membres du projet dquipe
-
Laboratoire Jean Kuntzman
Institut Fourier
Membres du projet dquipe
-
Laboratoire Jean Kuntzman Lama
Institut Fourier
Membres du projet dquipe
-
Laboratoire Jean Kuntzman Lama
Institut Fourier
Membres du projet dquipe
-
Laboratoire Jean Kuntzman Lama
Institut Fourier
Membres du projet dquipe
-
Laboratoire Jean Kuntzman Lama
Institut Fourier
Membres du projet dquipe
-
Contexte et objectif de laction Mathmatiques fondamentales et
outils numriques
-
Contexte et objectif de laction
Calcul formel (Maple, SAGE, MuPAD, etc.).
Mathmatiques fondamentales et outils numriques
-
Contexte et objectif de laction
Calcul formel (Maple, SAGE, MuPAD, etc.).
Mathmatiques fondamentales et outils numriques
Thorme des quatre couleurs : logiciel Coq et dmonstrateurs,
certification.
-
Contexte et objectif de laction
Calcul formel (Maple, SAGE, MuPAD, etc.).
Mathmatiques fondamentales et outils numriques
Thorme des quatre couleurs : logiciel Coq et dmonstrateurs,
certification.
Plus proche du calcul scientifique lapproche de Hales :
conjecture de Kepler et des Honeycombs. Approche par arithmtique
dintervalle et optimisation. Dmonstration de la conjecture de
Kepler !
-
Contexte et objectif de laction
Calcul formel (Maple, SAGE, MuPAD, etc.).
Mathmatiques fondamentales et outils numriques
Thorme des quatre couleurs : logiciel Coq et dmonstrateurs,
certification.
Plus proche du calcul scientifique lapproche de Hales :
conjecture de Kepler et des Honeycombs. Approche par arithmtique
dintervalle et optimisation. Dmonstration de la conjecture de
Kepler !
Un succs rcent rgional : le tore plat (PNAS, Pour La Science,
etc.).
-
Contexte et objectif de laction
Calcul formel (Maple, SAGE, MuPAD, etc.).
Mathmatiques fondamentales et outils numriques
Objectif des porteurs du projet
Thorme des quatre couleurs : logiciel Coq et dmonstrateurs,
certification.
Plus proche du calcul scientifique lapproche de Hales :
conjecture de Kepler et des Honeycombs. Approche par arithmtique
dintervalle et optimisation. Dmonstration de la conjecture de
Kepler !
Un succs rcent rgional : le tore plat (PNAS, Pour La Science,
etc.).
-
Contexte et objectif de laction
Calcul formel (Maple, SAGE, MuPAD, etc.).
Mathmatiques fondamentales et outils numriques
Valorisation du potentiel Grenoblois mathmatiques fondamentales
et calcul scientifique.
Objectif des porteurs du projet
Thorme des quatre couleurs : logiciel Coq et dmonstrateurs,
certification.
Plus proche du calcul scientifique lapproche de Hales :
conjecture de Kepler et des Honeycombs. Approche par arithmtique
dintervalle et optimisation. Dmonstration de la conjecture de
Kepler !
Un succs rcent rgional : le tore plat (PNAS, Pour La Science,
etc.).
-
Contexte et objectif de laction
Calcul formel (Maple, SAGE, MuPAD, etc.).
Mathmatiques fondamentales et outils numriques
Valorisation du potentiel Grenoblois mathmatiques fondamentales
et calcul scientifique.
Objectif des porteurs du projet
Dveloppement dune nouvelle dynamique originale au niveau
national et international autour de jeunes chercheurs brillants de
disciplines transverses. Plus que de nouveaux thormes, les
perspectives sont de proposer de nouveaux clairages sur des sujets
trs actuels.
Thorme des quatre couleurs : logiciel Coq et dmonstrateurs,
certification.
Plus proche du calcul scientifique lapproche de Hales :
conjecture de Kepler et des Honeycombs. Approche par arithmtique
dintervalle et optimisation. Dmonstration de la conjecture de
Kepler !
Un succs rcent rgional : le tore plat (PNAS, Pour La Science,
etc.).
-
Complexit dynamique : sections de Birkhoff
-
Complexit dynamique : sections de Birkhoff
Notion de section associe un champ de vecteur
-
Complexit dynamique : sections de Birkhoff
Notion de section associe un champ de vecteur
Comprendre la complexit dun champ de vecteur par ses
sections
-
Complexit dynamique : sections de Birkhoff
Notion de section associe un champ de vecteur
Comprendre la complexit dun champ de vecteur par ses
sections
Les gomtres travaillent dans des classes de surfaces !
-
Complexit dynamique : sections de Birkhoff
Notion de section associe un champ de vecteur
Comprendre la complexit dun champ de vecteur par ses
sections
Les gomtres travaillent dans des classes de surfaces ! Le thorme
non constructif de Thurston garantit lexistence de
reprsentations lmentaires sur des surfaces plates dans le cadre
hyperbolique. Comment les identifier ?
-
Optimisation parmi les mtriques
pp v 2 Tx
S, limn!1
1
nlog ||D
x
fn(v)|| = log()
-
Optimisation parmi les mtriques
Paramtrisation des mtriques : problmes avec contraintes de type
SDP locales. Rsultats de non convergence pour P1 ! Plusieurs
chelles sont ncessaires.
pp v 2 Tx
S, limn!1
1
nlog ||D
x
fn(v)|| = log()
-
Optimisation parmi les mtriques
Paramtrisation des mtriques : problmes avec contraintes de type
SDP locales. Rsultats de non convergence pour P1 ! Plusieurs
chelles sont ncessaires.
Benchmarks : recherche dexemples explicites de Thurston. Autres
paramtrisation du problme ?
pp v 2 Tx
S, limn!1
1
nlog ||D
x
fn(v)|| = log()
-
Optimisation parmi les mtriques
Paramtrisation des mtriques : problmes avec contraintes de type
SDP locales. Rsultats de non convergence pour P1 ! Plusieurs
chelles sont ncessaires.
Benchmarks : recherche dexemples explicites de Thurston. Autres
paramtrisation du problme ?
pp v 2 Tx
S, limn!1
1
nlog ||D
x
fn(v)|| = log()
Linarisation, projection //, algorithme non lisse large
scale.
-
Complexit godsique et spectrale :
-
Complexit godsique et spectrale : Quest ce quune godsique sur
une surface plate ?
-
Complexit godsique et spectrale :
Un problme gomtrique
QuestionTrouver la godsique ferme la plus courte sur une surface
plate S.
La surface est reprsente par un sous-ensemble ouvert de R2
pourlequel des morceaux de bord sont identifies (exemple: le tore
plat).Une godsique est une trajectoire dune particule dans un
champvectorielle V : S R2 constante sur S.
Sx0
V
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
Quest ce quune godsique sur une surface plate ?
-
Complexit godsique et spectrale :
Un problme gomtrique
QuestionTrouver la godsique ferme la plus courte sur une surface
plate S.
La surface est reprsente par un sous-ensemble ouvert de R2
pourlequel des morceaux de bord sont identifies (exemple: le tore
plat).Une godsique est une trajectoire dune particule dans un
champvectorielle V : S R2 constante sur S.
Sx0
V
Emmanuel Russ, Bozhidar Velichkov Problmes spectrauxTrouver la
godsique la plus courte ferme sur une surface plate donne
Quest ce quune godsique sur une surface plate ?
-
Complexit godsique et spectrale :
Un problme gomtrique
QuestionTrouver la godsique ferme la plus courte sur une surface
plate S.
La surface est reprsente par un sous-ensemble ouvert de R2
pourlequel des morceaux de bord sont identifies (exemple: le tore
plat).Une godsique est une trajectoire dune particule dans un
champvectorielle V : S R2 constante sur S.
Sx0
V
Emmanuel Russ, Bozhidar Velichkov Problmes spectrauxTrouver la
godsique la plus courte ferme sur une surface plate donne
Quest ce quune godsique sur une surface plate ?
Complexit combinatoire : tous les points, toutes les
directions
-
Complexit godsique et spectrale :
Un problme gomtrique
QuestionTrouver la godsique ferme la plus courte sur une surface
plate S.
La surface est reprsente par un sous-ensemble ouvert de R2
pourlequel des morceaux de bord sont identifies (exemple: le tore
plat).Une godsique est une trajectoire dune particule dans un
champvectorielle V : S R2 constante sur S.
Sx0
V
Emmanuel Russ, Bozhidar Velichkov Problmes spectrauxTrouver la
godsique la plus courte ferme sur une surface plate donne
Quest ce quune godsique sur une surface plate ?
Complexit combinatoire : tous les points, toutes les directions
Contexte non plat, problme de lapproximation godsique
-
Complexit godsique et spectrale :
Un problme gomtrique
QuestionTrouver la godsique ferme la plus courte sur une surface
plate S.
La surface est reprsente par un sous-ensemble ouvert de R2
pourlequel des morceaux de bord sont identifies (exemple: le tore
plat).Une godsique est une trajectoire dune particule dans un
champvectorielle V : S R2 constante sur S.
Sx0
V
Emmanuel Russ, Bozhidar Velichkov Problmes spectrauxTrouver la
godsique la plus courte ferme sur une surface plate donne
Quest ce quune godsique sur une surface plate ?
Complexit combinatoire : tous les points, toutes les directions
Contexte non plat, problme de lapproximation godsique La complexit
explose avec la dimension
-
Un problme gomtrique
S
V
fl0 = x0
x
= fl(t, x)
fl0 = fl(t, x)
Les godsiques sont solutions de lquation de transport
t
fl(t, x) + V x
fl(t, x) = 0, fl(0, ) = x0 . (1)
"La godsique la plus courte corresponde un champ vectoriel Vet
une donne initiale x0 S pour lesquelles la solution de (??) etla
moins diuse."
QuestionDfinir la diusion pour un systme dynamique: topological
mixing,
mixing, ergodicity.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
Une approche de nature EDP
@
t
(t, x) + V rx
(t, x) = 0, (0, ) = x0
-
Un problme gomtrique
S
V
fl0 = x0
x
= fl(t, x)
fl0 = fl(t, x)
Les godsiques sont solutions de lquation de transport
t
fl(t, x) + V x
fl(t, x) = 0, fl(0, ) = x0 . (1)
"La godsique la plus courte corresponde un champ vectoriel Vet
une donne initiale x0 S pour lesquelles la solution de (??) etla
moins diuse."
QuestionDfinir la diusion pour un systme dynamique: topological
mixing,
mixing, ergodicity.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
Une approche de nature EDP Un problme gomtrique
S
V
fl0 = x0
x
= fl(t, x)
fl0 = fl(t, x)
Les godsiques sont solutions de lquation de transport
t
fl(t, x) + V x
fl(t, x) = 0, fl(0, ) = x0 . (1)
"La godsique la plus courte corresponde un champ vectoriel Vet
une donne initiale x0 S pour lesquelles la solution de (??) etla
moins diuse."
QuestionDfinir la diusion pour un systme dynamique: topological
mixing,
mixing, ergodicity.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
@
t
(t, x) + V rx
(t, x) = 0, (0, ) = x0
-
Un problme gomtrique
S
V
fl0 = x0
x
= fl(t, x)
fl0 = fl(t, x)
Les godsiques sont solutions de lquation de transport
t
fl(t, x) + V x
fl(t, x) = 0, fl(0, ) = x0 . (1)
"La godsique la plus courte corresponde un champ vectoriel Vet
une donne initiale x0 S pour lesquelles la solution de (??) etla
moins diuse."
QuestionDfinir la diusion pour un systme dynamique: topological
mixing,
mixing, ergodicity.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
Une approche de nature EDP Un problme gomtrique
S
V
fl0 = x0
x
= fl(t, x)
fl0 = fl(t, x)
Les godsiques sont solutions de lquation de transport
t
fl(t, x) + V x
fl(t, x) = 0, fl(0, ) = x0 . (1)
"La godsique la plus courte corresponde un champ vectoriel Vet
une donne initiale x0 S pour lesquelles la solution de (??) etla
moins diuse."
QuestionDfinir la diusion pour un systme dynamique: topological
mixing,
mixing, ergodicity.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
@
t
(t, x) + V rx
(t, x) = 0, (0, ) = x0
@t(t, x) = (t, x) V r(t, x), (0, x) = '(x).On considre lquation
de la chaleur avec un terme de drift
-
Un problme gomtrique
S
V
fl0 = x0
x
= fl(t, x)
fl0 = fl(t, x)
Les godsiques sont solutions de lquation de transport
t
fl(t, x) + V x
fl(t, x) = 0, fl(0, ) = x0 . (1)
"La godsique la plus courte corresponde un champ vectoriel Vet
une donne initiale x0 S pour lesquelles la solution de (??) etla
moins diuse."
QuestionDfinir la diusion pour un systme dynamique: topological
mixing,
mixing, ergodicity.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
Une approche de nature EDP Un problme gomtrique
S
V
fl0 = x0
x
= fl(t, x)
fl0 = fl(t, x)
Les godsiques sont solutions de lquation de transport
t
fl(t, x) + V x
fl(t, x) = 0, fl(0, ) = x0 . (1)
"La godsique la plus courte corresponde un champ vectoriel Vet
une donne initiale x0 S pour lesquelles la solution de (??) etla
moins diuse."
QuestionDfinir la diusion pour un systme dynamique: topological
mixing,
mixing, ergodicity.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
quations de diusion sur une surface plate
On considre lquation de la chaleur avec un terme de drift
t
fl(t, x) = fl(t, x) V fl(t, x), fl(0, x) = (x). (2)
S
V
fl(t, x)
La vitesse de diusion de fl est dtermine par la valeur
propre
u + V u = u,
S
u = 0,
S
u
2 = 1. (3)
La fonction la moins diuse est la fonction propre u.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
@
t
(t, x) + V rx
(t, x) = 0, (0, ) = x0
@t(t, x) = (t, x) V r(t, x), (0, x) = '(x).On considre lquation
de la chaleur avec un terme de drift
-
Un problme gomtrique
S
V
fl0 = x0
x
= fl(t, x)
fl0 = fl(t, x)
Les godsiques sont solutions de lquation de transport
t
fl(t, x) + V x
fl(t, x) = 0, fl(0, ) = x0 . (1)
"La godsique la plus courte corresponde un champ vectoriel Vet
une donne initiale x0 S pour lesquelles la solution de (??) etla
moins diuse."
QuestionDfinir la diusion pour un systme dynamique: topological
mixing,
mixing, ergodicity.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
Une approche de nature EDP Un problme gomtrique
S
V
fl0 = x0
x
= fl(t, x)
fl0 = fl(t, x)
Les godsiques sont solutions de lquation de transport
t
fl(t, x) + V x
fl(t, x) = 0, fl(0, ) = x0 . (1)
"La godsique la plus courte corresponde un champ vectoriel Vet
une donne initiale x0 S pour lesquelles la solution de (??) etla
moins diuse."
QuestionDfinir la diusion pour un systme dynamique: topological
mixing,
mixing, ergodicity.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
quations de diusion sur une surface plate
On considre lquation de la chaleur avec un terme de drift
t
fl(t, x) = fl(t, x) V fl(t, x), fl(0, x) = (x). (2)
S
V
fl(t, x)
La vitesse de diusion de fl est dtermine par la valeur
propre
u + V u = u,
S
u = 0,
S
u
2 = 1. (3)
La fonction la moins diuse est la fonction propre u.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
@
t
(t, x) + V rx
(t, x) = 0, (0, ) = x0
@t(t, x) = (t, x) V r(t, x), (0, x) = '(x).On considre lquation
de la chaleur avec un terme de drift
La vitesse de diusion de est determinee par la valeur propre
u + V ru = u,Z
Su = 0,
Z
Su2 = 1.
La fonction la moins diusee est la fonction propre u.
-
Un problme gomtrique
S
V
fl0 = x0
x
= fl(t, x)
fl0 = fl(t, x)
Les godsiques sont solutions de lquation de transport
t
fl(t, x) + V x
fl(t, x) = 0, fl(0, ) = x0 . (1)
"La godsique la plus courte corresponde un champ vectoriel Vet
une donne initiale x0 S pour lesquelles la solution de (??) etla
moins diuse."
QuestionDfinir la diusion pour un systme dynamique: topological
mixing,
mixing, ergodicity.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
Un problme gomtrique
S
V
fl0 = x0
x
= fl(t, x)
fl0 = fl(t, x)
Les godsiques sont solutions de lquation de transport
t
fl(t, x) + V x
fl(t, x) = 0, fl(0, ) = x0 . (1)
"La godsique la plus courte corresponde un champ vectoriel Vet
une donne initiale x0 S pour lesquelles la solution de (??) etla
moins diuse."
QuestionDfinir la diusion pour un systme dynamique: topological
mixing,
mixing, ergodicity.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
quations de diusion sur une surface plate
On considre lquation de la chaleur avec un terme de drift
t
fl(t, x) = fl(t, x) V fl(t, x), fl(0, x) = (x). (2)
S
V
fl(t, x)
La vitesse de diusion de fl est dtermine par la valeur
propre
u + V u = u,
S
u = 0,
S
u
2 = 1. (3)
La fonction la moins diuse est la fonction propre u.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
Une approche de nature EDP
-
Un problme gomtrique
S
V
fl0 = x0
x
= fl(t, x)
fl0 = fl(t, x)
Les godsiques sont solutions de lquation de transport
t
fl(t, x) + V x
fl(t, x) = 0, fl(0, ) = x0 . (1)
"La godsique la plus courte corresponde un champ vectoriel Vet
une donne initiale x0 S pour lesquelles la solution de (??) etla
moins diuse."
QuestionDfinir la diusion pour un systme dynamique: topological
mixing,
mixing, ergodicity.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
Un problme gomtrique
S
V
fl0 = x0
x
= fl(t, x)
fl0 = fl(t, x)
Les godsiques sont solutions de lquation de transport
t
fl(t, x) + V x
fl(t, x) = 0, fl(0, ) = x0 . (1)
"La godsique la plus courte corresponde un champ vectoriel Vet
une donne initiale x0 S pour lesquelles la solution de (??) etla
moins diuse."
QuestionDfinir la diusion pour un systme dynamique: topological
mixing,
mixing, ergodicity.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
quations de diusion sur une surface plate
On considre lquation de la chaleur avec un terme de drift
t
fl(t, x) = fl(t, x) V fl(t, x), fl(0, x) = (x). (2)
S
V
fl(t, x)
La vitesse de diusion de fl est dtermine par la valeur
propre
u + V u = u,
S
u = 0,
S
u
2 = 1. (3)
La fonction la moins diuse est la fonction propre u.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
Pour fixee, determinee le champ vectoriel V qui maximise (+V
r).
Etudier les proprietes du champ optimal V et de la fonction
propre ucorrespondante.
Une approche de nature EDP
-
Un problme gomtrique
S
V
fl0 = x0
x
= fl(t, x)
fl0 = fl(t, x)
Les godsiques sont solutions de lquation de transport
t
fl(t, x) + V x
fl(t, x) = 0, fl(0, ) = x0 . (1)
"La godsique la plus courte corresponde un champ vectoriel Vet
une donne initiale x0 S pour lesquelles la solution de (??) etla
moins diuse."
QuestionDfinir la diusion pour un systme dynamique: topological
mixing,
mixing, ergodicity.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
Un problme gomtrique
S
V
fl0 = x0
x
= fl(t, x)
fl0 = fl(t, x)
Les godsiques sont solutions de lquation de transport
t
fl(t, x) + V x
fl(t, x) = 0, fl(0, ) = x0 . (1)
"La godsique la plus courte corresponde un champ vectoriel Vet
une donne initiale x0 S pour lesquelles la solution de (??) etla
moins diuse."
QuestionDfinir la diusion pour un systme dynamique: topological
mixing,
mixing, ergodicity.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
quations de diusion sur une surface plate
On considre lquation de la chaleur avec un terme de drift
t
fl(t, x) = fl(t, x) V fl(t, x), fl(0, x) = (x). (2)
S
V
fl(t, x)
La vitesse de diusion de fl est dtermine par la valeur
propre
u + V u = u,
S
u = 0,
S
u
2 = 1. (3)
La fonction la moins diuse est la fonction propre u.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
Pour fixee, determinee le champ vectoriel V qui maximise (+V
r).
Etudier les proprietes du champ optimal V et de la fonction
propre ucorrespondante.
EDP non variationelle, optimisation non lisse probable en cas de
multiplicit >0. Mthodes bundle.
Une approche de nature EDP
-
Un problme gomtrique
S
V
fl0 = x0
x
= fl(t, x)
fl0 = fl(t, x)
Les godsiques sont solutions de lquation de transport
t
fl(t, x) + V x
fl(t, x) = 0, fl(0, ) = x0 . (1)
"La godsique la plus courte corresponde un champ vectoriel Vet
une donne initiale x0 S pour lesquelles la solution de (??) etla
moins diuse."
QuestionDfinir la diusion pour un systme dynamique: topological
mixing,
mixing, ergodicity.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
Un problme gomtrique
S
V
fl0 = x0
x
= fl(t, x)
fl0 = fl(t, x)
Les godsiques sont solutions de lquation de transport
t
fl(t, x) + V x
fl(t, x) = 0, fl(0, ) = x0 . (1)
"La godsique la plus courte corresponde un champ vectoriel Vet
une donne initiale x0 S pour lesquelles la solution de (??) etla
moins diuse."
QuestionDfinir la diusion pour un systme dynamique: topological
mixing,
mixing, ergodicity.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
quations de diusion sur une surface plate
On considre lquation de la chaleur avec un terme de drift
t
fl(t, x) = fl(t, x) V fl(t, x), fl(0, x) = (x). (2)
S
V
fl(t, x)
La vitesse de diusion de fl est dtermine par la valeur
propre
u + V u = u,
S
u = 0,
S
u
2 = 1. (3)
La fonction la moins diuse est la fonction propre u.
Emmanuel Russ, Bozhidar Velichkov Problmes spectraux
Pour fixee, determinee le champ vectoriel V qui maximise (+V
r).
Etudier les proprietes du champ optimal V et de la fonction
propre ucorrespondante.
Problme non convexe : relaxation par petit paramtre ?
EDP non variationelle, optimisation non lisse probable en cas de
multiplicit >0. Mthodes bundle.
Une approche de nature EDP
-
Complexit mtrique :
-
Complexit mtrique : Tore plat et distance : problme du
plongement isomtrique de Nash
-
Complexit mtrique : Tore plat et distance : problme du
plongement isomtrique de Nash
1
The flat torus
I Realization of the square torus in 3D
Distances are changed during the transformation.
) The torus of revolution is not an isometric embedding of the
square flat torus
-
Complexit mtrique : Tore plat et distance : problme du
plongement isomtrique de Nash
1
The flat torus
I Realization of the square torus in 3D
Distances are changed during the transformation.
) The torus of revolution is not an isometric embedding of the
square flat torus
science received a particular attention in the last decades (for
instance when it comes to shapematching). Many practical questions
have been addressed related to the comprehension of thegeometry of
an object and its spectral data. Whereas progress has been made
regarding for in-stance the parametrization of meshed surface,
manifold with a large genus (that is with manyholes) are still very
challenging to manipulate or cluster. We expect that progresses in
the math-ematical understanding of the link between the geometry
and analytic or dynamical propertiesmay also lead to significant
progresses in these fields.
2 ObjectivesThe general goal of this project is to combine the
knowledge of theoretical and applied mathe-maticians in order to
study optimal manifolds with respect to complex criteria, under
complexgeometrical constraints. The main difficulty to achieve this
purpose lies in that the comprehen-sion of extremal manifolds
requires fine mathematical parametrization. Related tools have
beendeveloped in the areas of topology and dynamical systems but
are still beyond the reach of ap-plied mathematicians or computer
scientists. This project plans to benefit from the expertise oftwo
teams in Grenoble University in the related fields of theoretical
and applied mathematics toproduce original and promising studies in
this direction: we indeed believe that the researchersof the
Fourier Institute, of the LAMA and of the Jean Kuntzmann Laboratory
involved in thisproject have the exact complementary skills which
will make it possible to develop new ap-proaches in the numerical
approximation of optimal or critical manifolds and the analysis
ofshape optimization problems under complex geometrical
constraints.
We focus our project on three problematics in which both
theoretical and applied mathemati-cians are involved. By this
proposal, our team targets progress on understanding deeper
thefollowing three different fields of metric structures, dynamical
systems and spectral theory.
3 Metric complexity [D.Bucur - G. Besson - E. Oudet - B.
Thibert]Metrics are the relevant mathematical tools to generalize
the notion of length, shortest path andcurvatures for general
surfaces. In the last decades, the theoretical point of view on
these classicalgeometrical notions generated a wide range of
applications in many fields like data analysis inhigh dimension,
mesh matching, optimal transportation, etc.
We develop in the following paragraphs the part of our project
related to the understandingof abstract metrics. More precisely, we
focus here on the identification of specific geometricalobjects
which help to describe and quantify metric structures.
Figure 1: Two approximate isometric embeddings of a flat torus
(the left illustration comes from[29] and the central picture has
been obtained by from [4]).
3
-
Complexit mtrique : Tore plat et distance : problme du
plongement isomtrique de Nash
Plongement canonique, alternative lapproche de Gromov
1
The flat torus
I Realization of the square torus in 3D
Distances are changed during the transformation.
) The torus of revolution is not an isometric embedding of the
square flat torus
science received a particular attention in the last decades (for
instance when it comes to shapematching). Many practical questions
have been addressed related to the comprehension of thegeometry of
an object and its spectral data. Whereas progress has been made
regarding for in-stance the parametrization of meshed surface,
manifold with a large genus (that is with manyholes) are still very
challenging to manipulate or cluster. We expect that progresses in
the math-ematical understanding of the link between the geometry
and analytic or dynamical propertiesmay also lead to significant
progresses in these fields.
2 ObjectivesThe general goal of this project is to combine the
knowledge of theoretical and applied mathe-maticians in order to
study optimal manifolds with respect to complex criteria, under
complexgeometrical constraints. The main difficulty to achieve this
purpose lies in that the comprehen-sion of extremal manifolds
requires fine mathematical parametrization. Related tools have
beendeveloped in the areas of topology and dynamical systems but
are still beyond the reach of ap-plied mathematicians or computer
scientists. This project plans to benefit from the expertise oftwo
teams in Grenoble University in the related fields of theoretical
and applied mathematics toproduce original and promising studies in
this direction: we indeed believe that the researchersof the
Fourier Institute, of the LAMA and of the Jean Kuntzmann Laboratory
involved in thisproject have the exact complementary skills which
will make it possible to develop new ap-proaches in the numerical
approximation of optimal or critical manifolds and the analysis
ofshape optimization problems under complex geometrical
constraints.
We focus our project on three problematics in which both
theoretical and applied mathemati-cians are involved. By this
proposal, our team targets progress on understanding deeper
thefollowing three different fields of metric structures, dynamical
systems and spectral theory.
3 Metric complexity [D.Bucur - G. Besson - E. Oudet - B.
Thibert]Metrics are the relevant mathematical tools to generalize
the notion of length, shortest path andcurvatures for general
surfaces. In the last decades, the theoretical point of view on
these classicalgeometrical notions generated a wide range of
applications in many fields like data analysis inhigh dimension,
mesh matching, optimal transportation, etc.
We develop in the following paragraphs the part of our project
related to the understandingof abstract metrics. More precisely, we
focus here on the identification of specific geometricalobjects
which help to describe and quantify metric structures.
Figure 1: Two approximate isometric embeddings of a flat torus
(the left illustration comes from[29] and the central picture has
been obtained by from [4]).
3
-
Complexit mtrique : Tore plat et distance : problme du
plongement isomtrique de Nash
Plongement canonique, alternative lapproche de Gromov
Les objets manipuler sont de rgularit restreinte
1
The flat torus
I Realization of the square torus in 3D
Distances are changed during the transformation.
) The torus of revolution is not an isometric embedding of the
square flat torus
science received a particular attention in the last decades (for
instance when it comes to shapematching). Many practical questions
have been addressed related to the comprehension of thegeometry of
an object and its spectral data. Whereas progress has been made
regarding for in-stance the parametrization of meshed surface,
manifold with a large genus (that is with manyholes) are still very
challenging to manipulate or cluster. We expect that progresses in
the math-ematical understanding of the link between the geometry
and analytic or dynamical propertiesmay also lead to significant
progresses in these fields.
2 ObjectivesThe general goal of this project is to combine the
knowledge of theoretical and applied mathe-maticians in order to
study optimal manifolds with respect to complex criteria, under
complexgeometrical constraints. The main difficulty to achieve this
purpose lies in that the comprehen-sion of extremal manifolds
requires fine mathematical parametrization. Related tools have
beendeveloped in the areas of topology and dynamical systems but
are still beyond the reach of ap-plied mathematicians or computer
scientists. This project plans to benefit from the expertise oftwo
teams in Grenoble University in the related fields of theoretical
and applied mathematics toproduce original and promising studies in
this direction: we indeed believe that the researchersof the
Fourier Institute, of the LAMA and of the Jean Kuntzmann Laboratory
involved in thisproject have the exact complementary skills which
will make it possible to develop new ap-proaches in the numerical
approximation of optimal or critical manifolds and the analysis
ofshape optimization problems under complex geometrical
constraints.
We focus our project on three problematics in which both
theoretical and applied mathemati-cians are involved. By this
proposal, our team targets progress on understanding deeper
thefollowing three different fields of metric structures, dynamical
systems and spectral theory.
3 Metric complexity [D.Bucur - G. Besson - E. Oudet - B.
Thibert]Metrics are the relevant mathematical tools to generalize
the notion of length, shortest path andcurvatures for general
surfaces. In the last decades, the theoretical point of view on
these classicalgeometrical notions generated a wide range of
applications in many fields like data analysis inhigh dimension,
mesh matching, optimal transportation, etc.
We develop in the following paragraphs the part of our project
related to the understandingof abstract metrics. More precisely, we
focus here on the identification of specific geometricalobjects
which help to describe and quantify metric structures.
Figure 1: Two approximate isometric embeddings of a flat torus
(the left illustration comes from[29] and the central picture has
been obtained by from [4]).
3
-
Complexit mtrique : Tore plat et distance : problme du
plongement isomtrique de Nash
Plongement canonique, alternative lapproche de Gromov
Les objets manipuler sont de rgularit restreinte Gnralisation un
genre plus grand que 1
1
The flat torus
I Realization of the square torus in 3D
Distances are changed during the transformation.
) The torus of revolution is not an isometric embedding of the
square flat torus
science received a particular attention in the last decades (for
instance when it comes to shapematching). Many practical questions
have been addressed related to the comprehension of thegeometry of
an object and its spectral data. Whereas progress has been made
regarding for in-stance the parametrization of meshed surface,
manifold with a large genus (that is with manyholes) are still very
challenging to manipulate or cluster. We expect that progresses in
the math-ematical understanding of the link between the geometry
and analytic or dynamical propertiesmay also lead to significant
progresses in these fields.
2 ObjectivesThe general goal of this project is to combine the
knowledge of theoretical and applied mathe-maticians in order to
study optimal manifolds with respect to complex criteria, under
complexgeometrical constraints. The main difficulty to achieve this
purpose lies in that the comprehen-sion of extremal manifolds
requires fine mathematical parametrization. Related tools have
beendeveloped in the areas of topology and dynamical systems but
are still beyond the reach of ap-plied mathematicians or computer
scientists. This project plans to benefit from the expertise oftwo
teams in Grenoble University in the related fields of theoretical
and applied mathematics toproduce original and promising studies in
this direction: we indeed believe that the researchersof the
Fourier Institute, of the LAMA and of the Jean Kuntzmann Laboratory
involved in thisproject have the exact complementary skills which
will make it possible to develop new ap-proaches in the numerical
approximation of optimal or critical manifolds and the analysis
ofshape optimization problems under complex geometrical
constraints.
We focus our project on three problematics in which both
theoretical and applied mathemati-cians are involved. By this
proposal, our team targets progress on understanding deeper
thefollowing three different fields of metric structures, dynamical
systems and spectral theory.
3 Metric complexity [D.Bucur - G. Besson - E. Oudet - B.
Thibert]Metrics are the relevant mathematical tools to generalize
the notion of length, shortest path andcurvatures for general
surfaces. In the last decades, the theoretical point of view on
these classicalgeometrical notions generated a wide range of
applications in many fields like data analysis inhigh dimension,
mesh matching, optimal transportation, etc.
We develop in the following paragraphs the part of our project
related to the understandingof abstract metrics. More precisely, we
focus here on the identification of specific geometricalobjects
which help to describe and quantify metric structures.
Figure 1: Two approximate isometric embeddings of a flat torus
(the left illustration comes from[29] and the central picture has
been obtained by from [4]).
3
-
Complexit mtrique : Tore plat et distance : problme du
plongement isomtrique de Nash
Plongement canonique, alternative lapproche de Gromov
Les objets manipuler sont de rgularit restreinte
Comportement fractal attendu, estimation quantitative
Gnralisation un genre plus grand que 11
The flat torus
I Realization of the square torus in 3D
Distances are changed during the transformation.
) The torus of revolution is not an isometric embedding of the
square flat torus
science received a particular attention in the last decades (for
instance when it comes to shapematching). Many practical questions
have been addressed related to the comprehension of thegeometry of
an object and its spectral data. Whereas progress has been made
regarding for in-stance the parametrization of meshed surface,
manifold with a large genus (that is with manyholes) are still very
challenging to manipulate or cluster. We expect that progresses in
the math-ematical understanding of the link between the geometry
and analytic or dynamical propertiesmay also lead to significant
progresses in these fields.
2 ObjectivesThe general goal of this project is to combine the
knowledge of theoretical and applied mathe-maticians in order to
study optimal manifolds with respect to complex criteria, under
complexgeometrical constraints. The main difficulty to achieve this
purpose lies in that the comprehen-sion of extremal manifolds
requires fine mathematical parametrization. Related tools have
beendeveloped in the areas of topology and dynamical systems but
are still beyond the reach of ap-plied mathematicians or computer
scientists. This project plans to benefit from the expertise oftwo
teams in Grenoble University in the related fields of theoretical
and applied mathematics toproduce original and promising studies in
this direction: we indeed believe that the researchersof the
Fourier Institute, of the LAMA and of the Jean Kuntzmann Laboratory
involved in thisproject have the exact complementary skills which
will make it possible to develop new ap-proaches in the numerical
approximation of optimal or critical manifolds and the analysis
ofshape optimization problems under complex geometrical
constraints.
We focus our project on three problematics in which both
theoretical and applied mathemati-cians are involved. By this
proposal, our team targets progress on understanding deeper
thefollowing three different fields of metric structures, dynamical
systems and spectral theory.
3 Metric complexity [D.Bucur - G. Besson - E. Oudet - B.
Thibert]Metrics are the relevant mathematical tools to generalize
the notion of length, shortest path andcurvatures for general
surfaces. In the last decades, the theoretical point of view on
these classicalgeometrical notions generated a wide range of
applications in many fields like data analysis inhigh dimension,
mesh matching, optimal transportation, etc.
We develop in the following paragraphs the part of our project
related to the understandingof abstract metrics. More precisely, we
focus here on the identification of specific geometricalobjects
which help to describe and quantify metric structures.
Figure 1: Two approximate isometric embeddings of a flat torus
(the left illustration comes from[29] and the central picture has
been obtained by from [4]).
3
-
Une approche spectrale
-
Une approche spectrale
Thorme doptimalit de Nadirashvili
max
|S|=, genre(S)=1(S) = (Tequi)
-
Problme aux valeurs propres pour des surfaces singulires.
Approche paramtrique spectrale. Approche par boug de maillage
dangereuse.
Une approche spectrale
Thorme doptimalit de Nadirashvili
max
|S|=, genre(S)=1(S) = (Tequi)
-
Problme aux valeurs propres pour des surfaces singulires.
Approche paramtrique spectrale. Approche par boug de maillage
dangereuse.
Une approche spectrale
Thorme doptimalit de Nadirashvili
max
|S|=, genre(S)=1(S) = (Tequi)
Comment paramtrer des surfaces de genres suprieurs de manire
compatible avec le calcul scientifique. Classes conformes et
reprsentants.
-
Problme aux valeurs propres pour des surfaces singulires.
Approche paramtrique spectrale. Approche par boug de maillage
dangereuse.
Une approche spectrale
Distance godsique, valuation quantitative de la rgularit du
plongement.
Thorme doptimalit de Nadirashvili
max
|S|=, genre(S)=1(S) = (Tequi)
Comment paramtrer des surfaces de genres suprieurs de manire
compatible avec le calcul scientifique. Classes conformes et
reprsentants.
-
science received a particular attention in the last decades (for
instance when it comes to shapematching). Many practical questions
have been addressed related to the comprehension of thegeometry of
an object and its spectral data. Whereas progress has been made
regarding for in-stance the parametrization of meshed surface,
manifold with a large genus (that is with manyholes) are still very
challenging to manipulate or cluster. We expect that progresses in
the math-ematical understanding of the link between the geometry
and analytic or dynamical propertiesmay also lead to significant
progresses in these fields.
2 ObjectivesThe general goal of this project is to combine the
knowledge of theoretical and applied mathe-maticians in order to
study optimal manifolds with respect to complex criteria, under
complexgeometrical constraints. The main difficulty to achieve this
purpose lies in that the comprehen-sion of extremal manifolds
requires fine mathematical parametrization. Related tools have
beendeveloped in the areas of topology and dynamical systems but
are still beyond the reach of ap-plied mathematicians or computer
scientists. This project plans to benefit from the expertise oftwo
teams in Grenoble University in the related fields of theoretical
and applied mathematics toproduce original and promising studies in
this direction: we indeed believe that the researchersof the
Fourier Institute, of the LAMA and of the Jean Kuntzmann Laboratory
involved in thisproject have the exact complementary skills which
will make it possible to develop new ap-proaches in the numerical
approximation of optimal or critical manifolds and the analysis
ofshape optimization problems under complex geometrical
constraints.
We focus our project on three problematics in which both
theoretical and applied mathemati-cians are involved. By this
proposal, our team targets progress on understanding deeper
thefollowing three different fields of metric structures, dynamical
systems and spectral theory.
3 Metric complexity [D.Bucur - G. Besson - E. Oudet - B.
Thibert]Metrics are the relevant mathematical tools to generalize
the notion of length, shortest path andcurvatures for general
surfaces. In the last decades, the theoretical point of view on
these classicalgeometrical notions generated a wide range of
applications in many fields like data analysis inhigh dimension,
mesh matching, optimal transportation, etc.
We develop in the following paragraphs the part of our project
related to the understandingof abstract metrics. More precisely, we
focus here on the identification of specific geometricalobjects
which help to describe and quantify metric structures.
Figure 1: Two approximate isometric embeddings of a flat torus
(the left illustration comes from[29] and the central picture has
been obtained by from [4]).
3
Problme aux valeurs propres pour des surfaces singulires.
Approche paramtrique spectrale. Approche par boug de maillage
dangereuse.
Une approche spectrale
Distance godsique, valuation quantitative de la rgularit du
plongement.
Thorme doptimalit de Nadirashvili
max
|S|=, genre(S)=1(S) = (Tequi)
Comment paramtrer des surfaces de genres suprieurs de manire
compatible avec le calcul scientifique. Classes conformes et
reprsentants.
-
science received a particular attention in the last decades (for
instance when it comes to shapematching). Many practical questions
have been addressed related to the comprehension of thegeometry of
an object and its spectral data. Whereas progress has been made
regarding for in-stance the parametrization of meshed surface,
manifold with a large genus (that is with manyholes) are still very
challenging to manipulate or cluster. We expect that progresses in
the math-ematical understanding of the link between the geometry
and analytic or dynamical propertiesmay also lead to significant
progresses in these fields.
2 ObjectivesThe general goal of this project is to combine the
knowledge of theoretical and applied mathe-maticians in order to
study optimal manifolds with respect to complex criteria, under
complexgeometrical constraints. The main difficulty to achieve this
purpose lies in that the comprehen-sion of extremal manifolds
requires fine mathematical parametrization. Related tools have
beendeveloped in the areas of topology and dynamical systems but
are still beyond the reach of ap-plied mathematicians or computer
scientists. This project plans to benefit from the expertise oftwo
teams in Grenoble University in the related fields of theoretical
and applied mathematics toproduce original and promising studies in
this direction: we indeed believe that the researchersof the
Fourier Institute, of the LAMA and of the Jean Kuntzmann Laboratory
involved in thisproject have the exact complementary skills which
will make it possible to develop new ap-proaches in the numerical
approximation of optimal or critical manifolds and the analysis
ofshape optimization problems under complex geometrical
constraints.
We focus our project on three problematics in which both
theoretical and applied mathemati-cians are involved. By this
proposal, our team targets progress on understanding deeper
thefollowing three different fields of metric structures, dynamical
systems and spectral theory.
3 Metric complexity [D.Bucur - G. Besson - E. Oudet - B.
Thibert]Metrics are the relevant mathematical tools to generalize
the notion of length, shortest path andcurvatures for general
surfaces. In the last decades, the theoretical point of view on
these classicalgeometrical notions generated a wide range of
applications in many fields like data analysis inhigh dimension,
mesh matching, optimal transportation, etc.
We develop in the following paragraphs the part of our project
related to the understandingof abstract metrics. More precisely, we
focus here on the identification of specific geometricalobjects
which help to describe and quantify metric structures.
Figure 1: Two approximate isometric embeddings of a flat torus
(the left illustration comes from[29] and the central picture has
been obtained by from [4]).
3
Problme aux valeurs propres pour des surfaces singulires.
Approche paramtrique spectrale. Approche par boug de maillage
dangereuse.
Une approche spectrale
Distance godsique, valuation quantitative de la rgularit du
plongement.
Thorme doptimalit de Nadirashvili
max
|S|=, genre(S)=1(S) = (Tequi)
Comment paramtrer des surfaces de genres suprieurs de manire
compatible avec le calcul scientifique. Classes conformes et
reprsentants.
-
science received a particular attention in the last decades (for
instance when it comes to shapematching). Many practical questions
have been addressed related to the comprehension of thegeometry of
an object and its spectral data. Whereas progress has been made
regarding for in-stance the parametrization of meshed surface,
manifold with a large genus (that is with manyholes) are still very
challenging to manipulate or cluster. We expect that progresses in
the math-ematical understanding of the link between the geometry
and analytic or dynamical propertiesmay also lead to significant
progresses in these fields.
2 ObjectivesThe general goal of this project is to combine the
knowledge of theoretical and applied mathe-maticians in order to
study optimal manifolds with respect to complex criteria, under
complexgeometrical constraints. The main difficulty to achieve this
purpose lies in that the comprehen-sion of extremal manifolds
requires fine mathematical parametrization. Related tools have
beendeveloped in the areas of topology and dynamical systems but
are still beyond the reach of ap-plied mathematicians or computer
scientists. This project plans to benefit from the expertise oftwo
teams in Grenoble University in the related fields of theoretical
and applied mathematics toproduce original and promising studies in
this direction: we indeed believe that the researchersof the
Fourier Institute, of the LAMA and of the Jean Kuntzmann Laboratory
involved in thisproject have the exact complementary skills which
will make it possible to develop new ap-proaches in the numerical
approximation of optimal or critical manifolds and the analysis
ofshape optimization problems under complex geometrical
constraints.
We focus our project on three problematics in which both
theoretical and applied mathemati-cians are involved. By this
proposal, our team targets progress on understanding deeper
thefollowing three different fields of metric structures, dynamical
systems and spectral theory.
3 Metric complexity [D.Bucur - G. Besson - E. Oudet - B.
Thibert]Metrics are the relevant mathematical tools to generalize
the notion of length, shortest path andcurvatures for general
surfaces. In the last decades, the theoretical point of view on
these classicalgeometrical notions generated a wide range of
applications in many fields like data analysis inhigh dimension,
mesh matching, optimal transportation, etc.
We develop in the following paragraphs the part of our project
related to the understandingof abstract metrics. More precisely, we
focus here on the identification of specific geometricalobjects
which help to describe and quantify metric structures.
Figure 1: Two approximate isometric embeddings of a flat torus
(the left illustration comes from[29] and the central picture has
been obtained by from [4]).
3
Problme aux valeurs propres pour des surfaces singulires.
Approche paramtrique spectrale. Approche par boug de maillage
dangereuse.
Une approche spectrale
Distance godsique, valuation quantitative de la rgularit du
plongement.
Thorme doptimalit de Nadirashvili
max
|S|=, genre(S)=1(S) = (Tequi)
Comment paramtrer des surfaces de genres suprieurs de manire
compatible avec le calcul scientifique. Classes conformes et
reprsentants.
