The EPFL Mathematical Institute Invites you to the

Journée Georges de Rham 2017

Wednesday March 8, 2017 - EPFL Room CO-2

                Program :

15:20 - 15:30
Opening and welcome

15:30 - 16:30
Stéphane Mallat  (École Normale Supérieure, Paris)

Mathematical Mysteries of Deep Networks

16:30 - 17:15
Coffee Break

17:15 - 18:15
Sergei Tabachnikov (PennState University) 

Flavors of bicycle mathematics

Dinner at Gina Restaurant (if interested in coming please contact )

Stéphane Mallat is one of the world experts in wavelet theory, and more generally in the fields of signal processing and applied harmonic analysis. The recent years have seen major advances in the field of deep machine learning, in particular related to the development of so-called deep convolutional networks. H​e will introduce mathematical tools for the analysis of the properties of the convolutional networks, which scatter data with a cascade of linear filters and of non-linearities, and which have yielded spectacular results, in particular for image and sound recognition.​

He will speak about :  Mathematical Mysteries of Deep Networks

Abstract : Deep convolutional networks provide state of the art classification and regression results over many high-dimensional problems including image, audio bio-medical signal recognition, with spectacular results. They are able to approximate high-dimensional functionals without suffering from the curse of dimensionality. We shall review their architecture and concentrate on the analysis of their mathematical properties, which are mostly not understood. Applications will be shown for image and audio processing, but also to approximate complex stochastic processes for statistical physics, and quantum chemistry.

Sergei Tabachnikov received his Ph.D. from the Moscow State University in 1987. He is particularly famous for his work on the theory of billiards, but his research interests include differential topology, global analysis and differential geometry, dynamical systems and ergodic theory, etc. Tabachnikov is famous for his excellent expository style exemplified in his books "Geometry and billiards" and "Mathematical Omnibus" (with D.B.Fuks). He is a fellow of the American Mathematical Society.

He will speak about :  Flavors of bicycle mathematics

Abstract: This talk concerns a naive model of bicycle motion: a bicycle is a segment of fixed length that can move so that the velocity of the rear end is always aligned with the segment. Surprisingly, this simple model is quite rich. Here is a sampler of the problems that I plan to discuss.

1) The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and its terminal position; the monodromy map sending the initial position to the terminal one arises. This circle mapping is a Moebius transformation, a remarkable fact that has various geometrical and dynamical consequences. Moebius transformations belong to one of the three types: elliptic, parabolic and hyperbolic. I shall outline a proof of a 100 years old conjecture: if the front wheel track of a unit length bicycle is an oval with area at least π then the respective monodromy is hyperbolic.

2) The rear wheel track and a choice of the direction of motion uniquely determine the front wheel track; changing the direction to the opposite, yields another front track. These two front tracks are related by the bicycle (Backlund, Darboux) correspondence which defines a discrete time dynamical system on the space of curves. What do pairs of curves in the bicycle correspondence have in common? It turns out, infinitely many quantities (the perimeter length, the total curvature squared,…) I shall explain that the bicycle correspondence
is closely related with another, well studied, completely integrable dynamical system, the filament (a.k.a binormal, smoke ring, local induction) equation.

3) Given the rear and front tracks of a bicycle, can one tell which way the bicycle went? Usually, one can, but sometimes one cannot. The description of these ambiguous tire tracks is an open problem, intimately related with Ulam's problem in flotation theory (in dimension two): is the round ball the only body that floats in equilibrium in all positions?

Organisation:  Clément Hongler et Marc Troyanov @epfl