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Hommage Daniel Bennequin

Analyse Complexe Discrète
by

Christian Mercat

on 16 July 2013

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Transcript of Hommage Daniel Bennequin

coboundary

wedge product

Hodge star

Laplacian

singularities Quasi-conformality
Discrete
Conformality

Morera Theorem
Harmonic functions minimise the Dirichlet energy

Hodge decomposition
Quad-integrability 3D
Discrete
Geometry

Conformality
Finite
Conformal Field

Integrability
Darboux
Transform

Discrete
Riemann Surfaces

Image
Analysis

Discrete
Geometry

Polynome de Taylor (ordre 8)
Holomorphicity conformality
Diagonal ratio preserving maps
Linear Equation
on each face
Local Similitude
Discrete Conformal Blocks
Discrete Fusion Algebra
Discrete Holomorphicity
Paul Pearce, University of Melbourne, Australia
2000-2001
1994-1998
Daniel Bennequin, Univ. Louis Pasteur, Strasboug
and Ising model
Cardy & Rajabpour generalized to Parafermions (arXiv:0708.3772)
Fusion table of the Ising model, observed in the discrete case
is a Dirac spinor at criticality
Depends
on a path
Fermion
Discrete
spin
Structure
Z-invariant Ising model
Schroeder, Desbrun et al. (CalTech)
Bobenko, Pinkall, Polthier (Berlin), Gotsman, Ben-Chen (Technion), Gu, Yau (Harvard), Alliez, Cohen-Steiner (INRIA Sophia)
Morera Theorem
Lelong-Ferrand, Duffin, Novikov, Kiselman
Linear
Quadratic
Alexander Bobenko, TU-Berlin
2002-2003
A. Bobenko, Yu. Suris, TU-Berlin/TU-Munich
Local Similarity
Local Homography
Diagonal ratio preserving maps
Crossratio preserving maps
Linear Equation
on each face
Quadratic Equation
on each face
Morera theorem
Morera theorem
Lelong-Ferrand, Duffin, Novikov
Surface de Lawson
Tore de Wente
Residue Theorem
Green function
Cauchy kernel
Lozenge Case
Kenyon 2002
parallelogram & lozenge
Thurston
Circle patterns
Polynomials
Exponential
Frank Nijhoff
Krichever construction
Doliwa, Grinevich,
Nieszporski, Santini
N-solitons discrete function
Isomonodromic Solutions
Zero curvature Representation
Mobile frame
isomonodromic
if fixed
poles & orders
The Green function is an isomonodromic solution
Configuration: spins
Interaction Constants
Ferromagnetic Boltzmann Weights
Observable
Correlation
Statistic Criticality =
Integrability of Discrete Complex Analysis
for the discrete conformal structure
Simply laced Lie Algebras
q-deformed integers
Perron-Frobénius eigenvector
Verlinde Formula
Ocnéanu graphs
Fusion Algebra
Pasquier (ADE) model
Fused Boltzmann weights
Baxter, Perk, McCoy
Mass = criticality defect
elliptic deformation of circular functions
Smirnov & Chelkak, Bazhanov, Cimanosi & Reshetikhin, Bohle & Pedit & Pinkall, de Tilière & Boutillier, Kenyon
criticality is visible at the finite level: it is the compatibility with holomorphy
http://images.math.cnrs.fr/Applications-conformes.html
Comm. Math. Phys. 218(1):177-216 (2001)
Integrability : parallelogram
Criticality => Derivation & integration of holomorphic functions
Dilation Coefficient
Discrete Complex Structures on Surfel Surfaces
DGCI2008 LNCS 4992
Discrete Riemann Surfaces, in Handbook of Teichmueller theory, IRMA 2007
A. Bobenko, Ch.M., M. Schmies: Period Matrices of Polyhedral Surfaces,
in Riemann Surfaces a computational approach
A.B., Ch.M., Yu. S. Crelle 583:117-161 (2005)
BSMF 132(2):305-326 (2004)
Diffusion & convolution
F. Rieux, Ch. Fiorio, Ch.M. CompImage 2010
cotan formula Laplacian
Pinkall & Polthier 1993
vertices equiprobability
adjacent faces equiprobability
on Z, recover binomials
équipe ARITH LIRMM
McCoy & Costa Santos
Itzykson, Zuber, Petkova, Coquereaux, Jones, Ocnéanu
Belavin, Polyakov, A&Al Zamolodchikov, A&V Fateev, Vl&Vi Dotsenko, Neveu, Itzykson, Zuber, di Francesco, E&H Verlinde, Roche, Buffenoir
Criticality at the thermodynamic limit
Slopes
Standard deviation
Standard deviation
Slopes
Fuzzy discrete line
Dispersion of standard deviations with respect to fuzzy parameters
Anniversaire de Daniel Bennequin 16/3/2012
What is geometry?
Rumold (son of Gerardus) Mercator 16th century
Without geometry, life would be pointless
points cloud
polygons soup
Registration
Mesh vs Voxel
Georg Friedrich Bernhard Riemann 1826-1866
Riemann uniformization
Gu & Yau
Monica Hurdal
walking, singing the world into being
exp:
Kanisza
Petitot, Sarti, Morel, Masnou
An object as stochastic inputs to reconcile
Colin Cartade (Clermont-Ferrand)
DGCI 2011
fiber bundle on V1
Edelsbrunner Persistent Homology
Full transcript