Program

• First Summer School (June 27-July 1, 2016)
  

  Start: Monday, June 27th at 12.45
  End: Friday, July 1st at 2.30	→ Program: click here

  
  Three 7-hours lectures by
  • Dorin Bucur (U. Savoie)

    Shape optimization of spectral functionals ↓Abstract

    In these lectures, isoperimetric type inequalities involving the spectrum of the Laplace operator (with some boundary conditions) will be seen from a shape optimisation point of view. Depending on the boundary conditions, the analysis of those problems (existence of solution, regularity, qualitative properties) is either related to a free boundary problem of Alt-Caffarelli type, or to a free discontinuity problem. I will make an introduction to this topic and present recent results, with a focus on Robin boundary conditions. In particular I will detail a monotonicity formula which is the key point for the (Ahlfors) regularity of the optimal sets.

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  • Guido De Philippis (CNRS & ENS Lyon)

    The selection principle: the use of regularity theory in proving quantitative inequalities ↓Abstract

    I will first introduce the topic of quantitative inequalities and give some examples. Then I will present a general technique to derive them based on the regularity theory for solutions of variational problems. The course will be mainly focused on the (quantitative) isoperimetric inequality and on the (quantitative) Faber-Krahn inequality

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  • Filippo Santambrogio (U. Paris-Sud)

    Optimal transport, optimal curves, optimal flows ↓Abstract

    The course will consist in an introduction to optimal transport theory with a special attention to the comparison between Eulerian and Lagrangian point of views, and between statical and dynamical approaches. Optimal flow versions of some issues of the problem will also be presented, and this will lead at the end of the course to the study of some traffic equilibrium problems.

    The course will consist of four lectures, roughly divided as follows:

    • Basic theory of Optimal Transport

      The problems by Monge and Kantorovich.

      Convex duality and Kantorovich potentials.

      Existence of optimal maps (Brenier Theorem) for strictly convex costs.

    • Wasserstein distances

      Definitions of the distances W_p induced by optimal transport costs.

      The duality between W_1 and Lipschitz functions and the topology induced by the distances W_p.

      The continuity equation and the curves in the space W_p.

    • Curves of measures and geodesics in the Wasserstein space

      From measures on curves to curves of measures and back.

      Constant-speed geodesics in the Wasserstein space.

      The Benamou-Brenier dynamical formulation of optimal transport.

    • Minimal flows

      An Eulerian formulation of the Monge problem with cost |x-y| (p=1): the Beckmann problem.

      From measures on curves to vector flows and back.

      Extensions to traffic congestion models.

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  Second Summer School (July 11-15, 2016)

  Start: Monday, July 11th at 8.30
  End: Friday, July 15th at 6	→ Program: click here

  
  Three 6-hours lectures by
  • Daniel Cremers (Munich)

    Variational Methods for Computer Vision ↓Abstract

    Variational methods are among the most classical and established methods to solve a multitude of problems arising in computer vision and image processing. Over the last years, they have evolved substantially, giving rise to some of the most powerful methods for optic flow estimation, image segmentation and 3D reconstruction, both in terms of accuracy and in terms of computational speed. In this tutorial, I will introduce the basic concepts of variational methods. I will then focus on problems of geometric optimization including image segmentation and 3D reconstruction. I will show how the regularization terms can be adapted to incorporate statistically learned knowledge about our world. Subsequently, I will discuss techniques of convex relaxation and functional lifting which allow to computing globally optimal or near-optimal solutions to respective energy minimization problems. Experimental results demonstrate that these spatially continuous approaches provide numerous advantages over spatially discrete (graph cut) formulations, in particular they are easily parallelized (lower runtime) and they do not suffer from metrication errors (better accuracy).

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  • Jérôme Darbon (CNRS & ENS Cachan)

    On Optimization Algorithms in Imaging Sciences and Hamilton-Jacobi equations ↓Abstract

    The course will consist of two parts

    • Total variation minimization and maximal flows in graphs

      Applications to image processing

      Anisotropic mean curvature flow

    • Optimization in image processing, Hamilton-Jacobi equations, and optimal control

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  • Quentin Mérigot (CNRS & U. Paris-Dauphine)

    Computational optimal transport ↓Abstract

    Optimal transport has been used as a powerful theoretical tool to study partial differential equations, differential geometry and probability for a few decades. In comparison, its use in numerical applications is much more recent, not because of lack of interest but rather because of computational difficulties. The simplest discretization of the optimal transport problem lead to combinatorial optimization problems for which can only be solved with superquadratic cost. On the other hand, the partial differential equations arising from optimal transport are fully non-linear Monge-Ampère equations, for which there did not exist robust and efficient numerical solvers until recently. This course will present a variety of approaches to solve optimal transport and related problems, with applications in mind, such as:

    • Entropic penalization and Wasserstein barycenters
    • Benamou-Brenier algorithm, gradient flows and simulation of non-linear diffusion equations
    • Monge-Ampère equation, computational geometry and convexity constraints
    • Semi-discrete optimal transport and inverse problems in geometric optics
    • Measure-preserving maps, optimal quantization and Euler's equation for incompressible fluids

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  International conference "Calculus of Variations, Geometric Measure Theory, Optimal Transportation: from Theory to Applications" (July 4-8, 2016)
  

  Start: Monday, July 4th at 12.30
  End: Friday, July 8th at 2	→ Program (updated): click here	→ Program with titles and abstracts: click here

  • Confirmed speakers
  
  
  • Giovanni Alberti (Università di Pisa, Italy)

  Structure of the boundary of integral currents and Frobenius theorem
  

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  • Jean-François Aujol (Université de Bordeaux, France)

  • Image colorization by a variational approach [Slides]
    

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  • Giovanni Bellettini (Università di Roma "Tor Vergata", Italy)

  Constrained BV functions on covering spaces and Plateau's type problems
  

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  • Virginie Bonnaillie-Noël (École Normale Supérieure de Paris, France)

  Minimal k-partition for the p-norm of the eigenvalues [Slides]
  

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  • Guy Bouchitté (Université du Sud-Toulon-Var, France)

  A duality theory for non-convex problems in the Calculus of Variations
  

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  • Blaise Bourdin (Lousiana State University, USA)

  Variational vs. phase field models of fracture
  

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  • Lia Bronsard (McMaster University, Canada)

  Minimizers of the Landau-de Gennes energy around a spherical colloid particle [Slides]
  

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  • Michael Bronstein (Università della Svizzera Italiana, Switzerland)

  Partial functional maps [Slides]
  

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  • Almut Burchard (University of Toronto, Canada)

  Extremals of the Polya-Szego inequality
  

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  • Daniel Cremers (Technische Universität München, Germany)

  Sublabel Accurate Relaxation of Nonconvex Energies
  

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  • Qiang Du (Columbia University in the City of New York, USA)

  Calculus of variations of some nonlocal problems
  

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  • Selim Esedoḡlu (University of Michigan, USA)

  Algorithms for anisotropic mean curvature flow of networks
  

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  • Ilaria Fragalà (Politecnico di Milano, Italy)

  Boundary value problems for the infinity Laplacian: regularity and geometric results [Slides]
  

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  • Adriana Garroni (Università di Roma "La Sapienza", Italy)

  Line tension for dislocations and crystal plasticity
  

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  • Young-Heon Kim (University of British Columbia, Canada)

  Optimal martingale transport in general dimensions
  

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  • Jacques-Olivier Lachaud (Université de Savoie, France)

  Convergent geometric estimators with digital volume and surface integrals
  

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  • Francesco Maggi (Abdus Salam International Center for Theoretical Physics, Trieste, Italy)

  Quantitative isoperimetric principles and applications to phase transitions [Slides]
  

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  • Maks Ovsjanikov (École Polytechnique, France)

  Functional Characterization of Shapes and their Relations
  

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  • Manuel Ritoré (Universidad de Granada, Spain)

  Isoperimetric inequalities in unbounded convex bodies
  

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  • Dejan Slepčev (Carnegie Mellon University, USA)

  Variational problems on graphs and their continuum limits [Slides]
  

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  • Jeremy Tyson (University of Illinois at Urbana-Champaign, USA)

  Densities of measures and the geometry of submanifolds in the Heisenberg group
  

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  • Bozhidar Velichkov (Université Grenoble Alpes, France)

  Lipschitz regularity for quasi-minimizers and applications to some shape optimization problems
  

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  • Max Wardetsky (Georg-August-Universität Göttingen, Germany)

  Variational Convergence of Minimal Surfaces
  

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  • Stefan Wenger (University of Fribourg, Switzerland)

  Area minimizing discs in metric spaces and applications
  

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  • Benedikt Wirth (Universität Münster, Germany)

  Optimal design of transport networks [Slides]
  

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  Accommodation for PhD students and postdoctoral young researchers
  Hotel accommodation (in shared double occupancy rooms) is offered to a maximal number of 60 PhD students or postdoctoral young researchers. Rooms are attributed in registration order. Use the pre-registration form below for application.
  

  
  

  Registration fees (Registration is closed)
  60 euros per week for permanent researchers
  30 euros per week for PhD students and postdoctoral researchers
  no registration fees for local researchers and local students
  
  Included with conference registration fees:

  • hotel accommodation for a limited number of 60 PhD students or postdoctoral young researchers (rooms are attributed in registration order)
  • daily coffee and refreshment breaks
  • social events
  • conference dinner

  
  

  Scientific Committee
  Lorenzo Brasco
  Dorin Bucur
  Antonin Chambolle
  Thierry De Pauw
  Guy David
  Vincent Feuvrier
  Antoine Lemenant
  Quentin Mérigot
  Benoit Merlet
  Vincent Millot
  Laurent Moonens
  Edouard Oudet
  Olivier Pantz
  Séverine Rigot
  Filippo Santambrogio
  
  
  Organizing Committee
  Elie Bretin
  Simon Masnou
  Hervé Pajot
  
  
  For any questions about the thematic period, please contact us at this email address.