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C ONFERENCE IN HONOUR OF D ON DAVIS
String Topological Robotics
L EHIGH U NIV., B ETHLEHEM , P ENNSYLVANIA , USA
My Ismail Mamouni
Agrégé-Doctor
CRMEF Rabat, Morocco
http ://myismail.net
[email protected]
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
1 / 49
Joint work with
Joint work with
Derfoufi Younes
Fac. Sc. Meknes, Morocco
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
2 / 49
Our Main Goal
Marrying Topological Robotics and String Topology
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
3 / 49
Hard Task
Feeling
Giant I
Giant II
Baby
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
4 / 49
Love and Peace and Mathematics
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àñ
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á ÔgQË@
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é<Ë@ úΫ ð
Translation : Let’s Go.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
5 / 49
Hard Task
Context
Closed, orientable mfld
Path-connect. space
path-conn. G-mfld
compact or not
orientable or not
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
6 / 49
Content
1
2
3
4
Brief Introduction
Topological Robotics
String Topology
Strings through Topological Robotics
Loop Motion Planning Algorithms (LMPA)
String LMPA Product
Acknowledgements
Summer School Announcement
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
7 / 49
Brief Introduction
Topological Robotics
Robotics
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
8 / 49
Brief Introduction
Topological Robotics
Robotics
Motion Planning Algorithm (MPA)
Given a robot, a motion planning algorithm is a set of
rules determining to the robot how to move from one
state to another.
X , The configuration space of a robot is just the set
of all robot states.
A motion is just a path from a point to another in this
configuration space.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
9 / 49
Brief Introduction
Topological Robotics
Topological Robotics
Once X and PX topologized
Farber’s Topological Approach (2003)
An MPA is any continuous section s :
X 2 −→ PX of the double evaluation
π : PX −→ X 2
.
γ 7−→ (γ(0), γ(1))
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
10 / 49
Brief Introduction
Topological Robotics
Topological Robotics
Farber’s Key Queries on MPA
1
Existence
2
Continuity
MPA exist whenever X path-connected
MPA are continuous iff X is contractible
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
11 / 49
Brief Introduction
Topological Robotics
Topological Robotics
Farber’s Key Remark on MPA
Continuity of motion planning means that close
initial-final pairs (A, B) and (A0 , B 0 ) produce close
movements s(A, B) and s(A0 , B 0 ).
discontinuity will result in the instability of behaviour
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
12 / 49
Brief Introduction
Topological Robotics
Topological Robotics
Farber’s Key Ingredient : Topological Complexity
TC(X) := minimal necessary number k (or infinity) to cover X × X by k + 1 open sets X × X = U0 ∪ · · · ∪ Uk on
each for each there exists a local MPA si : Ui −→ PX (i.e.,
π ◦ si = idUi ).
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
13 / 49
Brief Introduction
Topological Robotics
Topological Robotics
Interpretation
TC measures the discontinuity of any motion planner and
the algorithmic complexity of finding a motion planning algorithm.
Low TC
Farber (2003) : TC(X) = 0 iff X is contractible ;
Grant-Lupton-Oprea (2013) : TC(X) = 1 iff X is
homotopy equivalent to some odd-dimensional
sphere ;
Farber (2003) : TC(Sn ) = 2, whenever n is even.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
14 / 49
Brief Introduction
Topological Robotics
Topological Robotics
Interpretation
TC measures the discontinuity of any motion planner and
the algorithmic complexity of finding a motion planning algorithm.
Low TC
Farber (2003) : TC(X) = 0 iff X is contractible ;
Grant-Lupton-Oprea (2013) : TC(X) = 1 iff X is
homotopy equivalent to some odd-dimensional
sphere ;
Farber (2003) : TC(Sn ) = 2, whenever n is even.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
14 / 49
Brief Introduction
String Topology
String Topology
Chas’s and Sullivan’s Main Motivation (1999)
Study algebraic structures on the homology
of the loop space of a
manifold.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
15 / 49
Brief Introduction
String Topology
String Topology
Chas’s and Sullivan’s Main Goal
Define transversely an intersection product at level of
chains of loop.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
16 / 49
Brief Introduction
String Topology
Intersection Product at level of manifolds
Recalling
Y , Z some submanifolds of respective co-dimension
i and j, are said to intersect transversely in X if for all
x ∈ Y ∩ Z , one have Tx Y ⊕ Tx Z = Tx X .
Hence, Y ∩ Z is a submanifold of X of co-dimension
i + j.
Intersection Product : [Y ∩ Z ] = [Y ] · [Z ].
Here [−], denotes the homological fundamental class that represents
the named submanifold.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
17 / 49
Brief Introduction
String Topology
Intersection Product at level of loop spaces
1
Transeversality in LX := X S is a little specific
1
A path in LX is a path
of loops ;
2
A velocity vector in this
case is a vector field.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
18 / 49
Brief Introduction
String Topology
String Topology
Chas’s and Sullivan’s Key ideas
1
Consider two families of
closed oriented curves
2
At each point of
intersection of two
curves, form a new
closed curve by going
around the first curve
and then going around
the second.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
19 / 49
Brief Introduction
String Topology
String Topology
Chas’s and Sullivan’s Mathematical Approach
1
1
consider γ : ∆i −→ X S and σ : ∆j −→ X S some
1
simplices of X S such that γ(1) : ∆i −→ X and
1
σ(1) : ∆j −→ X S intersect transversally in X ;
compute the intersection product γ(1).σ(1) at each
point (s, t) ∈ ∆i × ∆j such that γ(1)(s) = σ(1)(t)
perform the composition of the loops γ(s) and σ(t) to
1
obtain, the (i + j − n)-simplex γ • σ : ∆i+j−n −→ X S
(loop product) ;
1
1
set H∗ (X S ) := H∗+n (X S ; Z) (loop homology ) ;
extend the loop product to loop homology product ;
define a structure of associative graded commutative
algebra : (H∗ (X ), •).
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
20 / 49
Strings through Topological Robotics
Loop Motion Planning Algorithms (LMPA)
Loop Motion Planning Algorithms (LMPA)
Derfoufi, M. (2015)
LMPA on a manifold X is any continuous section s : X ×
1
X −→ X S of the loop evaluation
π LP : X S
γ
1
.
−→ X × X
7−→ (γ(0), γ( 21 ))
Interpretation
Input : Pair (A=departure, B=target) ;
Output : A goings and comings motion by requiring a
come-back to the departure point.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
21 / 49
Strings through Topological Robotics
Loop Motion Planning Algorithms (LMPA)
Loop Motion Planning Algorithms (LMPA)
Derfoufi, M. (2015)
LMPA on a manifold X is any continuous section s : X ×
1
X −→ X S of the loop evaluation
π LP : X S
γ
1
.
−→ X × X
7−→ (γ(0), γ( 21 ))
Interpretation
Input : Pair (A=departure, B=target) ;
Output : A goings and comings motion by requiring a
come-back to the departure point.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
21 / 49
Strings through Topological Robotics
Loop Motion Planning Algorithms (LMPA)
Loop Motion Planning Algorithms (LMPA)
Applications Areas
The motion of a drone like an unmanned warplane
or a guided TV camera ;
The famous NP-complete problem of vehicle routing
with pick-up and delivery
Derfoufi, M. (2015)
LMPAs exist iff X is contractible.
Notation
MLP (X ) denotes the set of all LMPAs on X
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
22 / 49
Strings through Topological Robotics
Loop Motion Planning Algorithms (LMPA)
Loop Motion Planning Algorithms (LMPA)
Applications Areas
The motion of a drone like an unmanned warplane
or a guided TV camera ;
The famous NP-complete problem of vehicle routing
with pick-up and delivery
Derfoufi, M. (2015)
LMPAs exist iff X is contractible.
Notation
MLP (X ) denotes the set of all LMPAs on X
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
22 / 49
Strings through Topological Robotics
Loop Motion Planning Algorithms (LMPA)
Loop Motion Planning Algorithms (LMPA)
Applications Areas
The motion of a drone like an unmanned warplane
or a guided TV camera ;
The famous NP-complete problem of vehicle routing
with pick-up and delivery
Derfoufi, M. (2015)
LMPAs exist iff X is contractible.
Notation
MLP (X ) denotes the set of all LMPAs on X
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
22 / 49
Strings through Topological Robotics
String LMPA Product
First Obstacle
In Farber’s context
LMPAs has sense iff X is contractible
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
23 / 49
Strings through Topological Robotics
String LMPA Product
First Obstacle
In Farber’s context
LMPAs has sense iff X is contractible
Solution : Lubawski’s & Marzantowicz’s definition of MPA
PX is replaced by PX ×X /G PX
:=
{(γ, τ )
=
∈
PX × PX ,
G.γ(0)
G.τ (1)}.
Paths can have a discontinuity point, but
this discontinuity is controlled by a group
action.
"motion planner" in the sense of Lubawski-Marzantowicz is stable in the Farber’s sense : small
changes of either the initial or terminal point result in small changes of the path between them).
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
23 / 49
Strings through Topological Robotics
String LMPA Product
LMPA product
Derfoufi, M. (2015)
For any LMPA’s s1 and s2 , we set :
µ(s1 , s2 )(A, B)(t) = s1 (A, B)(t)
if 0 ≤ t ≤ 21
= s1 (A, B)(3t − 1) if 12 ≤ t ≤ 23
= s2 (A, B)(3t − 2) if 32 ≤ t ≤ 1
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
24 / 49
Strings through Topological Robotics
String LMPA Product
LMPA product
Key Inspiration Remark
two free loops are composable iff they have a
common base point.
two LMPAs are composable iff they have two
common base points.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
25 / 49
Strings through Topological Robotics
String LMPA Product
Next Aim
Inspire from Laudenbach’s work
He extends the theory
of string topology to all
manifolds, compact or
not, orientable or not.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
26 / 49
Strings through Topological Robotics
String LMPA Product
Laudenbach context
Small simplices
Equip X with a chart A.
A p-simplex σ : ∆i −→ X is said to be small iff there
exists U(σ) ∈ A such that σ(∆i ) ⊂ U(σ) ;
U(σ) is chosen once for all ;
P
Small p-chain : any linear combination nk σk of
many finitely p-small simplices σk , with integer
coefficients nk ∈ Z ;
Small bi-simplex : σ × γ : ∆i × ∆j −→ X × X with σ
and γ are both small.
Small
bi-chains : any linear combination
P
nk (σk × γk ) of many finitely small bi-simplices
σk × γ, with integer coefficients nk ∈ Z.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
27 / 49
Strings through Topological Robotics
String LMPA Product
Laudenbach context
Small Transverse bi-Simplices
A small bi-simplex σ × γ : ∆i × ∆j −→ X × X is said to be
transverse whenever the map σ × γ and all its faces are
transverse to the diagonal map ∆X
Recall
Two maps are said to be transverse, when their image are,
as submaniflods.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
28 / 49
Strings through Topological Robotics
String LMPA Product
Laudenbach context
Small Transverse bi-Simplices
A small bi-simplex σ × γ : ∆i × ∆j −→ X × X is said to be
transverse whenever the map σ × γ and all its faces are
transverse to the diagonal map ∆X
Recall
Two maps are said to be transverse, when their image are,
as submaniflods.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
28 / 49
Strings through Topological Robotics
String LMPA Product
Our Key Idea
Bridge from simplex of MLP (X ) to that of X 2
the bi-evaluation
ev : MLP (X ) −→ X 2
,
s
7−→ (s(−, −)(0), s(−, −)(1/2))
relates any i-simplex Σ : ∆i −→ MLP (X ) of MLP (X ) to
the i-simplex σ := ev (Σ) : ∆i −→ X 2 of X 2 .
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
29 / 49
Strings through Topological Robotics
String LMPA Product
Transeversality at level of LMPA chains
A (i, j)-bi-simplex
Σ × Γ : ∆i × ∆j −→ MLP (X ) × MLP (X )
is said to be small (resp. transverse) in MLP (X ), when
the associated (i, j)-bi-simplex
ev (Σ × Γ) := ev (Σ) × ev (Γ) : ∆i × ∆j −→ X 2 × X 2
is also in X 2 .
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
30 / 49
Strings through Topological Robotics
String LMPA Product
Key Remark I
Hence W = ev (Σ × Γ)−1 (∆X 2 ) is (i + j − 2n)-dimensional
orientable sub-manifold of ∆i ×∆j , with corner that can be
triangulated.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
31 / 49
Strings through Topological Robotics
String LMPA Product
Key Remark I
Hence W = ev (Σ × Γ)−1 (∆X 2 ) is (i + j − 2n)-dimensional
orientable sub-manifold of ∆i ×∆j , with corner that can be
triangulated.
Illustrative Diagram
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
31 / 49
Strings through Topological Robotics
String LMPA Product
For x ∈ W , put (Σ × Γ)(x) = (s1 , s2 ) ∈ MLP (X ) × MLP (X ).
Illustrative Diagram
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
32 / 49
Strings through Topological Robotics
String LMPA Product
For x ∈ W , put (Σ × Γ)(x) = (s1 , s2 ) ∈ MLP (X ) × MLP (X ).
Key Remark II
Note that
ev (Σ × Γ)
ev (Σ × Γ)
=
∈
(s1 (−, −)(0), s1 (−, −)(1/2), s2 (−, −)(0), s2 (−, −)(1/2))
∆X 2
s1 and s2 are composable in MLP (X ) since they have two common base points,
namely A = s1 (−, −)(0) = s2 (−, −)(0) and
B = s1 (−, −)(1/2) = s2 (−, −)(1/2).
Illustrative Diagram
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
32 / 49
Strings through Topological Robotics
String LMPA Product
Intersection LMPA product
Derfoufi, M. (2015)
Σ.Γ := µ((Σ × Γ)|W )
is well defined at level of transverse bi-simplices of
MLP (X ) and can be extended
P naturaly and linearly at level of transverse bi-chain Σk × Γk , where each simplex
Σk × Γk is transverse.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
33 / 49
Strings through Topological Robotics
String LMPA Product
Loop Motion Homology
Twisted boundary operator
at level of p-simplices of MLP (X ) : ∂Σ
:=
P
p
i
i=0 εi (−1) Fi Σ, where εi is the sign of the Jacobian of
the coordinates change U(ev (Fi Σ)) −→ U(ev (Σ)). Here
Fi denotes the i-th face.
Loop Motion Homology
H∗ (MLP (X ), ∂)
coefficients are taken in Z.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
34 / 49
Strings through Topological Robotics
String LMPA Product
Loop Motion Homology
Twisted boundary operator
at level of p-simplices of MLP (X ) : ∂Σ
:=
P
p
i
i=0 εi (−1) Fi Σ, where εi is the sign of the Jacobian of
the coordinates change U(ev (Fi Σ)) −→ U(ev (Σ)). Here
Fi denotes the i-th face.
Loop Motion Homology
H∗ (MLP (X ), ∂)
coefficients are taken in Z.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
34 / 49
Strings through Topological Robotics
String LMPA Product
String LMPA Homology Product
Derfoufi, M. (2015)
[Σ].[Γ] = [Σ.Γ].
Well defined : Sketch of Proof
1
Claim 1 : Any product bi-cycle can be represented, up to
a small boundary preserving homotopy, by a small ad
transverse product bi-cycle, and that it does not depend
on the choice of the homological representants.
2
Claim 2 : The homological class [Σ.Γ] does not depend on
the choice of the homological representants.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
35 / 49
Strings through Topological Robotics
String LMPA Product
String LMPA Homology Product
Derfoufi, M. (2015)
[Σ].[Γ] = [Σ.Γ].
Well defined : Sketch of Proof
1
Claim 1 : Any product bi-cycle can be represented, up to
a small boundary preserving homotopy, by a small ad
transverse product bi-cycle, and that it does not depend
on the choice of the homological representants.
2
Claim 2 : The homological class [Σ.Γ] does not depend on
the choice of the homological representants.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
35 / 49
Strings through Topological Robotics
String LMPA Product
String LMPA Homology Product
Derfoufi, M. (2015)
[Σ].[Γ] = [Σ.Γ].
Well defined : Outlines of Proof
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
36 / 49
Strings through Topological Robotics
String LMPA Product
String LMPA Homology Product
Derfoufi, M. (2015)
[Σ].[Γ] = [Σ.Γ].
Well defined : Outlines of Proof
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
36 / 49
Strings through Topological Robotics
String LMPA Product
String LMPA Homology
Recall
Hi (MLP (X ), ∂)⊗Hj (MLP (X ), ∂) −→ Hi+j−2n (MLP (X ), ∂).
As Promised
H∗ (MLP (X )) := H∗+2n (MLP (X ), ∂),
endowed with the String LMPA Homology Product
Hi (MLP (X )) ⊗ Hj (MLP (X )) −→ Hi+j (MLP (X ))
is a commutative and associative graded algebra.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
37 / 49
Strings through Topological Robotics
String LMPA Product
String LMPA Homology
Recall
Hi (MLP (X ), ∂)⊗Hj (MLP (X ), ∂) −→ Hi+j−2n (MLP (X ), ∂).
As Promised
H∗ (MLP (X )) := H∗+2n (MLP (X ), ∂),
endowed with the String LMPA Homology Product
Hi (MLP (X )) ⊗ Hj (MLP (X )) −→ Hi+j (MLP (X ))
is a commutative and associative graded algebra.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
37 / 49
Strings through Topological Robotics
String LMPA Product
M. Chas, D. Sullivan, String Topology,
arXiv :math/9911159 [math.GT].
M. Farber, Topological complexity of motion
planning, Discrete Comput. Geom., vol. 29 (2003),
no. 2, 211-221.
F. Laudenbach, A note on the Chas-Sullivan product,
L’Enseignement Mathématique, Vol. 57, Issue 1-2
(2011), 3-21.
W. Lubawski, W. Marzantowicz, Invariant topological
complexity, Bull. London Math. Soc., vol. 47 (2015)
101-117.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
38 / 49
Acknowledgements
Acknowledgements
Organizers
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
39 / 49
Acknowledgements
Acknowledgements
Organizers
20 Emails with me x 100 Participants+...=....
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
39 / 49
Acknowledgements
´ Poland
Zbigniew Błaszczyk, Poznan,
For pointing out our attention to the Lubawski’s and
Marzantowicz’s work
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
40 / 49
Acknowledgements
Acknowledgements
David Chataur, Lille, France
For pointing out our attention to the Laudenbach’s work
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
41 / 49
Acknowledgements
Acknowledgements
MAAT, Moroccan Research Group
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
42 / 49
Acknowledgements
MAAT : Moroccan Area of Algebraic Topology
Born : 2012 ;
Logo :
Home Page : http ://algtop.net
Members : 3 professors, 7 PhD Students, 30 Master Students ;
Scientific production since 2012
1
Papers
Published=1
Accepted=2
Submitted=7
On Progress=3
2
3
Seminar : Monthly
Research School : Bi-Annual, Geometry, Topology in Physics and
Mathematics= GeToPhyMa.
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
43 / 49
Acknowledgements
GeToPhyMa History
2014 : on Moduli spaces, in memory of Bill Thurston
(1946-2012)
2013 : on Operads, in memory of Jean Louis Loday
(1946-2012)
2012 : General lectures
2011 : General lectures
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
44 / 49
Summer School Announcement
GeToPhyMa-2016
July 10-20, 2016 (Rabat, Morocco)
Home page : http ://algtop.net/geto16
Topic : Rational Homotopty Theory and its
Interactions
17 Speakers
Scientific Contents : 6 courses blocks, 30 hours of
courses, 14 hours of exercises or/and discussion
sessions, and 6x20mn-communications.
Excursions : One half day Casablanca visit, One half
day Rabat visit, Two days (Fes-Meknes, Atlas
Mountains) excursion
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
45 / 49
Summer School Announcement
GeToPhyMa-2016
Will Celebrate
Jim Stasheff
Dennis Sullivan
80th Anniversary
75th Anniversary
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
46 / 49
Summer School Announcement
Don Davis’s 70th birthday
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
47 / 49
Summer School Announcement
Questions or Comments are accepted in
slowly formulated
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
48 / 49
Summer School Announcement
ƒ
@Qº
My Ismail Mamouni
Lehigh Univ., Bethlehem, Pennsylvania, USA
May 22-24, 2015
49 / 49