- 4 hours talk
Speaker: Young-Heon Kim (University of British Columbia & KIAS)
Title: An introduction to optimal transport
Abstract: We give a short introduction to optimal transport, which one can view as how to make an efficient interpolation between mass distributions. The theory is related to calculus of variations, partial differential equations (heat equation, Monge-Ampere equation), as well as geometry. We will discuss basic concepts and results as well as some recent progress.

Speaker: Dong Li (University of British Columbia)
Title: Norm inflation and illposedness in hydrodynamics
Abstract: The incompressible Euler equations describe the flow of an inviscid, incompressible fluid and have very rich analytic and geometric structures. The pioneering work of Lichtenstein, Gunther in 1940s and Kato-Ponce in 1980s constructed solutions to the incompressible Euler equations in function spaces above a certain regularity threshold. The intensive research in the past several decades showed that the complexity of the threshold cases is deeply connected with the inherent nonlinear and non-local structures in the Euler equations. In this minicourse I will survey some recent developments on these problems, starting from the basic wellposedness.

Speaker: Filip Rindler (University of Warwick)
Title: The positive Jacobian constraint in elasticity theory and orientation-preserving Young measures
Abstract: In elasticity theory, one naturally requires that the Jacobian determinant of the deformation is positive or even a-priori prescribed (for example in the case of incompressibility). However, such strongly non-linear and non-convex constraints are difficult to deal with in mathematical models. In this minicourse, I will present various recent results on how this constraint can be manipulated in subcritical Sobolev spaces, where the integrability exponent is less than the dimension. This setting is related to cavitation and fracture phenomena in materials. In particular, after introducing the appropriate notions, I will present a characterization of such constraint on the Jacobian determinant formulated in the language of Young measures. These objects, which I will briefly introduce, are widely used in the Calculus of Variations to model limits of nonlinear functions of weakly converging "generating" sequences. I will also discuss relations to convex integration and "geometry" in matrix space. Finally, I will show some applications to the minimization of integral functionals, the theory of semiconvex hulls and incompressible extensions.


- 2 hours talk
Speaker: Hwa Kil Kim (Seoul National University)
Title 1: Value functions in the Wasserstein spaces:finite time horizons
Abstract 1: We study analogs of value functions arising in classical mechanics in the space of probability measures endowed with the Wasserstein metric $W_p$, for $1<p<infty$. Our main result is that each of these generalized value functions is a type of viscosity solution of an appropriate Hamilton-Jacobi equation. Of particular interest is a formula we derive for a generalized value function when the associated potential energy is of the form ${cal V}(mu)=int_{R^d}V(x)dmu(x)$. This formula allows us to make rigorous a well known heuristic connection between Euler-Poisson equations and classical Hamilton-Jacobi equations. This is a joint work with Ryan Hynd.
Title 2: The Cucker-Smale-Navier-Stokes system with shear thickening
Abstract 2: We present a global existence theory for weak solutions and their asymptotic dynamics to the Cucker-Smale-Navier-Stokes system with shear thickening. Furthermore, we improve the existence result to cover more general initial data by using mass transportation technique. Part of this talk is based on a joint work with Seung-Yeal Ha, Jae-myoung Kim, and Jinyeong Park.

Speaker: Tongseok Lim (University of British Columbia)
Title: Optimal martingale transport
Abstract: Optimal martingale transport theory concerns optimal transport problems where there is martingale constraint on the transportation plans. In the first lecture, we explain the basic problem, comparing it with optimal transport without the constraint. In the second lecture, we explain some recent progress.