DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_33

Video: http://www.youtube.com/watch?v=2ebliJYcWEA

Slides: https://drive.google.com/open?id=0B0QKxsVtOaTicEphTng2VE9wclU

Presentation: https://www.see.asso.fr/node/14287

Creative Commons Attribution-ShareAlike 4.0 International

Abstract:

In this paper we propose a method to characterize and estimate the variations of

a random convex set Ξ0 in terms of shape, size and direction. The mean n-variog

ram γ(n)Ξ0:(u1⋯un)↦E[νd(Ξ0∩(Ξ0−u1)⋯∩(Ξ0−un))] of a random convex set Ξ0 on ℝ d r

eveals information on the n th order structure of Ξ0. Especially we will show that considering the mean n-variograms of the dilated random sets Ξ0 ⊕ rK by an homothetic convex family rKr > 0, it’s possible to estimate some characteristic of the n th order structure of Ξ0. If we make a judicious choice of K, it provides relevant measures of Ξ0. Fortunately the germ-grain model is stable by convex dilatations, furthermore the mean n-variogram of the primary grain is estimable in several type of stationary germ-grain models by the so called n-points probability function. Here we will only focus on the Boolean model, in the planar case we will show how to estimate the n th order structure of the random vector composed by the mixed volumes t (A(Ξ0),W(Ξ0,K)) of the primary grain, and we will describe a procedure to do it from a realization of the Boolean model in a bounded window. We will prove that this knowledge for all convex body K is sufficient to fully characterize the so called difference body of the grain Ξ0⊕˘Ξ0. we will be discussing the choice of the element K, by choosing a ball, the mixed volumes coincide with the Minkowski’s functional of Ξ0 therefore we obtain the moments of the random vector composed of the area and perimeter t (A(Ξ0),U(Ξ)). By choosing a segment oriented by θ we obtain estimates for the moments of the random vector composed by the area and the Ferret’s diameter in the direction θ, t((A(Ξ0),HΞ0(θ)). Finally, we will evaluate the performance of the method on a Boolean model with rectangular grain for the estimation of the second order moments of the random vectors t (A(Ξ0),U(Ξ0)) and t((A(Ξ0),HΞ0(θ)).

DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_32

Video: http://www.youtube.com/watch?v=7FfYJ94VhAk

Slides: Philippon_Heights of toric.pdf

Presentation: https://www.see.asso.fr/node/14286

Creative Commons Attribution-ShareAlike 4.0 International

Abstract:

We present a dictionary between arithmetic geometry of toric varieties and convex analysis. This correspondence allows for effective computations of arithmetic invariants of these varieties. In particular, combined with a closed formula for the integration of a class of functions over polytopes, it gives a number of new values for the height (arithmetic analog of the degree) of toric varieties, with respect to interesting metrics arising from polytopes. In some cases these heights are interpreted as the average entropy of a family of random processes.

DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_31

Video: http://www.youtube.com/watch?v=Pcu3oKN0Xnk

Slides: Elbaz-Vincent_finite polylogarithm.pdf

Presentation: https://www.see.asso.fr/node/14285

Creative Commons Attribution-ShareAlike 4.0 International

Abstract:

We show that the entropy function–and hence the finite 1-logarithm–behaves a lot like certain derivations. We recall its cohomological interpretation as a 2-cocycle and also deduce 2n-cocycles for any n. Finally, we give some identities for finite multiple polylogarithms together with number theoretic applications.

DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_30

Video: http://www.youtube.com/watch?v=iUPjhUcHvxM

Slides: Marcolli_information algebra.pdf

Presentation: https://www.see.asso.fr/node/14284

Creative Commons Attribution-ShareAlike 4.0 International

Abstract:

In this lecture we will present joint work with Ryan Thorngren on thermodynamic semirings and entropy operads, with Nicolas Tedeschi on Birkhoff factorization in thermodynamic semirings, ongoing work with Marcus Bintz on tropicalization of Feynman graph hypersurfaces and Potts model hypersurfaces, and their thermodynamic deformations, and ongoing work by the author on applications of thermodynamic semirings to models of morphology and syntax in Computational Linguistics.