The EPFL Mathematical Institute Invites you to the Journée Georges de Rham 2017 Wednesday March 8, 2017  EPFL Room CO2 
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 

19:30 
Dinner at Gina Restaurant (if
interested in coming please contact jdr2017epfl@gmail.com ) 
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 socalled deep convolutional networks.
He 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 nonlinearities, 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 highdimensional problems including image, audio biomedical signal recognition, with spectacular results. They are able to approximate highdimensional 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? 