Talks

December 8, Tuesday

Andrea Bertozzi  (University of California, Los Angeles)

Total variation minimization on graphs for semisupervised and unsupervised machine learning

Abstract: Total variation (TV) minimization has made a big impact in image processing for applications such as denoising, deblurring, and segmentation. The TV functional has a geometric meaning in Euclidean space related to the constraints on perimeter of regions.  In a graphical setting we can define the graph TV functional and connect it to the graph min cut problem.  This allows us to develop methods for machine learning involving similarity graphs for high dimensional data.  I will talk about semi-supervised learning, unsupervised learning, and a the connection to modularity optimization for community detection on networks. I will intruce a graph version of the Ginzburg-Landau energy and discuss its gamma convergence to graph TV.  This will motivate the development of fast methods with dynamic thresholding for solving penalized graph cut problems.

 

Atsushi Nakayasu (Kyoto University)

Convexity preserving properties for Hamilton–Jacobi equations in geodesic spaces

Abstract: We study convexity preserving properties for a class of time-dependent Hamilton–Jacobi equations in complete geodesic spaces. Convexity preserving properties for nonlinear evolution equations are well known in the Euclidean space. We extend the classical results for first order equations to the Busemann spaces such as a junction by using a recently developed theory of viscosity solutions on geodesic spaces. This talk is based on a joint work with Qing Liu (Fukuoka University).

 

Mi-Ho Giga (The University of Tokyo)

Motion of graph-like curves by crystalline surface diffusion

Abstract: We consider a kind of crystalline surface diffusion equations for a graph-like curve y=u(x,t). We are particularly interested in a gradient flow in H^{-1} metric of a crystalline energy with density W(u_x). Here, the energy density W assumed to be convex, piecewise linear and coercive. This kind of flows is often used to model relaxation of a crystal surface by surface diffusion. 

Although, the problem is a 4-th order singular diffusion equation, the initial value problem is uniquely solvable for any periodic H^{-1} initial data by the theory of maximal monotone operators initiated by Y. Komura (1967) and developed to H. Brezis and others.

We introduce a special class of piecewise linear functions. Introducing notion of a "firm crystal", we prove that the solution stays firm locally-in-time, if initially it is firm. In this case we show that the original equation is reduced a system of ordinary differential equations like a second-order crystalline flow. This is a joint work with Y. Giga.

 

Claudio Marchi (University of Padova)

First order Mean Field Games on networks

Abstract: The theory of Mean Field Games studies the asymptotic behaviour of differential games (mainly in terms of their Nash equilibria) as  the number of players tends to infinity. In these games, the players are rational and indistinguishable: each player aims at choosing its trajectory so to minimize a cost which depends on the trajectory itself and on the distribution of the whole population of agents. We focus our attention on deterministic Mean Field Games with finite horizon in which the states of the players are constrained in a network (in our setting, a network is given by a finite collection of vertices connected by continuous edges  which cannot self-intersect). As in the Lagrangian approach, we introduce a relaxed notion of Mean Field Games equilibria and we shall deal with probability measures on trajectories on the network instead of probability measures on the network. This is a joint work with: Y. Achdou (Univ. of Paris), P. Mannucci (Univ. of Padova) and N. Tchou (Univ. of Rennes).

 

December 9, Wednesday

Yoshikazu Giga (University of Tokyo) 

A finer singular limit of a single-well Modica-Mortola functional and its applications 

Abstract : We characterize the Gamma limit of a single-well Modica-Mortola functional under graph convergence in one-dimensional space which is finer than L¹ convergence.  As an application, we give an explicit representation of a singular limit of the Kobayashi-Waren-Carter energy, which is popular in materials science. We also establish compactness under the graph convergence. For these purposes we change  an independent variable of a function by  introducing arc-length parameter of its graph, which we call an unfolding of a function.  This is a joint work of  Jun Okamoto (University of Tokyo) and Masaaki Uesaka (University of Tokyo, Arithmer Inc. ). 

 

Ayato Mitsuishi  (Fukuoka University)

Certain min-max values related to the p-energy and packing radii

Abstract: Juutinen, Lindqvist and Manfredi proved the first eigenvalue of the Dirchlet p-Laplacian raised to power of (1/p), on a domain of Euclidean space, converges to the inscribed radius of the domain, when p goes to infinity. Grosjean also proved that the first positive eigenvalue of p-Laplacian raised to the power of (1/p), on a closed Riemannian manifold, tends to the inverse of the half of the diameter of the space. We generalize these results to the case of ``a kind of k-th eigenvalue'' defined by a min-max procedure, for general metric measure spaces.

 

Renjin Jiang (Tianjin University)

Gradient estimates for heat kernel and harmonic functions and applications

Abstract:  In this talk, I shall report some recent results regarding regularity of harmonic functions and heat kernels on manifolds as well as metric measure spaces. The progress started from the regularity problem of harmonic functions in metric geometry, and it turns out some new phenomenon about heat kernel and harmonic functions appears in the study. We shall also discuss its applications to the Riesz transform. The talk is based on  some recent result joined with T. Coulhon, P. Koskela, F. Lin and A. Sikora. 

 

Fabio Camilli (University of Rome)

A regularity theory for viscous Hamilton–Jacobi equations with Caputo time-fractional derivative

Abstract: We present  a regularity theory for classical  and weak solutions to viscous Hamilton–Jacobi equations with Caputo time-fractional derivative. The approach relies on a combination of gradient bounds for the time-fractional Hamilton–Jacobi equation obtained via Evans's nonlinear adjoint method and sharp estimates in Sobolev and Holder spaces for the corresponding linear problem.

 

December 10, Thursday

Andrzej Swiech (Georgia Institute of Technology) 

Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures

Abstract: We will discuss convergence of viscosity solutions of Hamilton-Jacobi-Bellman (HJB) equations corresponding either to deterministic optimal control problems for systems of n particles or to stochastic optimal control problems for systems of n particles with a common noise, to the viscosity solution of a limiting HJB equation in the space of probability measures. The limiting HJB equation is interpreted in its ``lifted" form in a Hilbert space, which has a unique viscosity solution. The key step is in obtaining uniform continuity estimates for viscosity solutions of the approximating problems. When the Hamiltonian is convex in the gradient variable and equations are of first order, it can be proved that the viscosity solutions of the finite dimensional problems converge to the value function of a variational problem in $\mathcal{P}_2(\R^d)$ thus providing a representation formula for the solution of the limiting first order HJB equation. We will also briefly discuss an intrinsic definition of viscosity solution on the Wasserstein space for our class of second order equations. This is a joint work with W. Gangbo and S. Mayorga.

 

Asuka Takatsu (Tokyo Metropolitan University)

Elliptic and parabolic boundary value problems on rotationally symmetric domains 

Abstract： 
We study power concavity of solutions to elliptic and parabolic boundary value problems on rotationally symmetric, strongly convex open metric balls in a Riemannian manifold. We show that if the radius of a ball is small enough, then the first Dirichlet eigenfunction on the ball is positive power concave. As an application, we prove that  the heat kernel on the hyperbolic space is strictly log-concave.This is joint work with Kazuhiro Ishige (The University of Tokyo) and Paolo Salani (University of Florence). 

December 11, Friday

Wilfrid Gangbo (UCLA)

Well-posedness and regularity for an H1-projection problem.

Abstract: We prove the existence, uniqueness, and regularity of minimizers of the H1-projection of measure-preserving maps. Our result introduces a new criteria on the uniqueness of the minimizer, based on the smallness of the Lagrange multiplier. As an application, we introduce a minimizing movement scheme to construct L^r-solutions of the Navier-Stokes equation (NSE) for a short time interval. Our scheme is an improved version of the split scheme introduced in Ebin-Marsden (EM), and allows us to solve the equation with L^r initial data (r > 3) as opposed to H^{13/2} initial data requirement in the (EM) work. (Talk based on a joint work with M. Jacobs and I. Kim).

 

Xiaodan Zhou (OIST)

Eikonal Equations on Metric Spaces

Abstract: In this talk, we focus on the eikonal equation, a special case of the Hamilton-Jacobi equation. We compare several known notions of metric viscosity solutions and show the equivalence of these notions in a complete length space. By using the induced intrinsic (path) metric, we reduce the metric space to a length space and show the equivalence of these solutions to the associated Dirichlet boundary problem. Without utilizing the boundary data, we also localize our argument and directly prove the equivalence for the definitions of solutions. Regularity of solutions related to the Euclidean semi-concavity is discussed as well. This talk is based on joint work with Qing Liu (Fukuoka University) and Nageswari Shanmugalingam (University of Cincinnati).