[CMC T2-3]

CMC conference:

Optimal transport and related topics

Date: July 31 - August 4, 2017     Place: Rm. 1503, KIAS

Title/Abstract Home > Title/Abstract
Myoungjean Bae
- TITLE: Prandtl-Meyer reflection configurations, Transonic shocks, and Free boundary problems
- ABSTRACT: Prandtl (1936) first employed the shock polar analysis to show that, when a steady supersonic flow impinges onto a solid wedge whose angle is less than a critical angle (i.e., the detachment angle), there are two possible steady configurations: the steady weak shock solution and strong shock solution, and then conjectured that the steady weak shock solution is physically admissible since it is the one observed experimentally. The fundamental issue of whether one or both of the steady weak and strong shocks are physically admissible has been vigorously debated over several decades and has not yet been settled in a definite manner. In this talk, I will address this longstanding open issue and present the recent analysis to establish the stability theorem for the steady weak shock solutions as the long-time asymptotics of the Prandtl-Meyer reflection configurations for unsteady potential flow for all the physical parameters up to the detachment angle. This talk is based on a joint work with Gui-Qiang G. Chen and Mikhail Feldman.

Jaeyoung Byeon
- TITLE: Variational construction of single peak solutions for a singularly perturbed Neumann problem.
- ABSTRACT: In this talk  I would like to introduce a variational construction of solutions with a peak on the boundary for a singularly perturbed Neumann problem using a transplantation argument.

Sun-Sig Byun

- TITLE: Calderon-Zygmund theory for non-uniformly parabolic systems
- ABSTRACT: We discuss the natural Calderon-Zygmund theory for solutions of

non-uniformly parabolic systems in divergence, and proving that the (spatial) gradient of solutions is as integrable as that of the assigned nonhomogeneous term in the divergence.

Hi Jun Choe
- TITLE: Malliavin calculus of L'evy process by subordination
- ABSTRACT: A chaotic expansion for subordinate of L'evy process is developed. The chaotic expansion is expressed in term of It^o multiple integral exapansions like Sole et al. Considering jumps due to original process and subordinator, we find a mixed chaotic expansion whcih provides defintion of Malliavin derivatives. Also, Malliavin derivatives are characterized by the quotient of increment like Sole et al. and Clark-Ocone formula is derived.

Marco Cuturi
- TITLE: Generative Modeling with Optimal Transport
- ABSTRACT: We present in this talk recent advances on the topic of parameter estimation using optimal transport, and discuss possible implementations for "Minimum Kantorovich Estimators". We show why these estimators are currently of great interest in the deep learning community, in which researchers have tried to formulate generative models for images. We will present a few algorithmic solutions to this problem.

Mikhail Feldman
- TITLE: Short-time existence of smooth solutions for semigeostrophic system with variable Coriolis parameter.
- ABSTRACT: The semigeostrophic (SG) system is a model of large scale atmosphere/ocean flows. Previous results were obtained for the SG system with constant Coriolis parameter, by rewriting the problem in the "dual variables" and using Monge-Kantorovich mass transport techniques. A more physically realistic SG model has variable Coriolis parameter. Dual space is not available in this case. We work directly in the original "physical" coordinates, and show existence of smooth solutions for short time. This is a joint work with M. Cullen and J. Cheng.

Wilfrid Gangbo
- TITLE: On intrinsic differentiability in the Wasserstein space P2(Rd).
- ABSTRACT: We elucidate the connection between different notions of differentiability in P2(Rd): some have been introduced intrinsically by Ambrosio–Gigli–Savar ́e, the other no- tion due to Lions, is extrinsic and arises from the identification of P2(Rd) with the Hilbert space of square-integrable random variables. We mention potential applications such as uniqueness of viscosity solutions for Hamilton-Jacobi equations in P2(Rd), the latter not known to satisfy the Radon–Nikodym property. (This talk is based on a work in progress with A Tudorascu).

Yuxin Ge
- TITLE: Regularity of optimal transport on compact nearly spherical manifolds
- ABSTRACT: In this talk, I describe some stability results on the positivity of MTW tensor on the manifolds close to sphere in the high dimensions. As application, we will prove the regularity issue of optimal transport maps on such manifolds.

Nassif Ghoussoub
: Ballistic cost and the iteration of mass transports
- ABSTRACT: I investigate deterministic and stochastic dynamic optimal mass transport problems associated to ballistic-related cost functionals on phase space. This leads to Hopf-Lax formulae on Wasserstein Space, and to links with Mean Field Games. It also suggests a more general and useful theory for iterating mass transports. 

Hyung Ju Hwang
- TITLE: A generalized Ginzburg-Landau model for nonlinear relaxation oscillation of magnetized plasma boundary with shear flow

- ABSTRACT: The boundary of high-temperature plasma confined by a toroidal magnetic field structure often undergoes quasi-periodic relaxation oscillations between high and low energy states . On the tokamak, the oscillation cycle consists of a long quasi-steady state characterized by field-aligned filamentary eigenmodes, an abrupt transition into non-modal filamentary structure, and its rapid burst (via magnetic reconnection) leading to the boundary collapse. A reduced MHD model including the effects of time-varying perpendicular flow shear, turbulent transport, and external heating, which has the form of a generalized complex Ginzburg-Landau equation, has been developed to understand the nonlinear oscillation. The model shows that the amplitude and shear rate of the flow across the plasma boundary are the key parameters determining the nonlinear oscillation. Numerical solutions for a range of the flow amplitude and shear revealed that there exists a critical flow level below which steady-state solutions can exist.

Hwa Kil Kim
- TITLE: Existence of weak solutions in Waserstein space for a chemotaxis model coupled to fluid equations
- ABSTRACT: We consider a coupled system of Keller-Segel type equations and the incompressible Navier-Stokes equations in spatial dimension two and three. We fi rst establish existence of a weak solution of a Fokker-Plank equation in the Wasserstein space under the assumption that initial mass is integrable and has fi nite entropy. As a result, we construct solutions of Keller-Segel-Navier-Stokes equations such that the density of biological organism belongs to the absolutely continuous curves in the Wasserstein space, in case that its initial mass is assumed to be bounded and integrable(joint work with Kyungkeun Kang).

Inwon Kim
- TITLE: Height constrained transport and interface motion.
- ABSTRACT: I will talk about height constrained transport for both single density and two densities. The problem can be formulated as a gradient flow in Wasserstein space, to derive a notion of discrete-time solutions. In the continuum limit the evolution of the congested zone, where the densities reach their maximum, leads to the study of the corresponding free boundary problem. One can also approximate the height constraints  by constraints on the densities $L^p$ norm for large $p$, which corresponds to a degenerate diffusion problem.  We will discuss characterization of the free boundary problem and convergence of the discrete and approximating solutions, as well as open questions.

Seick Kim
- TITLE: On C^1, C^2, and weak type-(1,1) estimates for linear elliptic operators.
- ABSTRACT: We show that any weak solution to elliptic equations in divergence form is continuously differentiable provided that the modulus of continuity of coefficients in the L1-mean sense satisfies the Dini condition.  We also prove a weak type-(1,1) estimate under a stronger assumption on the modulus of continuity. The corresponding results for non-divergence form equations are also established.

Jun Kitagawa
- TITLE: Free singularities and their stability in optimal transport
- ABSTRACT: As the regularity of optimal transport requires very rigid and strong hypotheses, one is naturally lead to the question of partial regularity, i.e. in-depth analysis of the structure of singular sets. In this talk I will discuss finer structure of the set of “free singularities” which arise in an optimal transport problem from a connected set to a disconnected set. Specifically, if the connected components of the support of the target measure are suitably separated by hyperplanes, this singular set will consist of DC (difference of convex) hypersurfaces of appropriate codimensions, and with additional constraints, pieces of the target domain can be shown to be homeomorphic to certain regions in the source domain. We prove this result via a non-smooth implicit function theorem for convex functions, which is of independent interest. I will also talk about another application of this implicit function theorem: a stability result for singular sets under appropriate perturbations of the target measure. This talk is based on joint work with Robert McCann.

Soonsik Kwon
- TITLE: A Maximizer of the kinetic energy inequality.
- ABSTRACT: We consider the kinetic energy inequality of infinitely many particle system of quantum mechanics. This is known as a dual inequality of the Lieb-Thirring inequality. We show that there exists an extremizer of the kinetic energy inequality. The argument is so called the concentration-compactness. We prove the profile decomposition in a suitable operator class and combine it with a binding inequality. Moreover, we derive the Euler-Lagrange equations of orthonormal eigenfunctions, and find certain properties of Lagrange multipliers. This result has a potential application to the Hartree-Fock equation with attractive interaction potential, from which we are motivated for this work. This is joint work with Younghun Hong and Haewon Yoon.

Robert McCann
- TITLE: On concavity of the monopolist's problems facing consumers with nonlinear price preferences
- ABSTRACT: The principal-agent problem is an important paradigm in economic theory for studying the value of private information;  the nonlinear pricing problem faced by a monopolist is a particular example.  In this lecture,  we identify structural conditions on the consumers' preferences and the monopolist's profit functions which guarantee either concavity or convexity of the monopolist's profit maximization.  Uniqueness and stability of the solution are particular consequences of this concavity.  Our conditions are closely related to criteria given by Trudinger and others for prescribed Jacobian equations to have smooth solutions,  while being simpler in many respects. By allowing for different dimensions of agents and contracts,  nonlinear dependence of agent preferences on prices, and of the monopolist's profits on agent identities,   it improves on the literature in a number of ways. The same mathematics can also be adapted to the maximization of societal welfare by a regulated monopoly, This is joint work with PhD student Shuanjian Zhang.

Shin-ichi Ohta
- TITLE: On weighted Ricci curvature of negative effective dimension"
- ABSTRACT: The weighted Ricci curvature (also called the Bakry-Emery-Ricci curvature) is a generalization of the Ricci curvature to a Riemannian (or Finsler) manifold equipped with an arbitrary measure. The weighted Ricci curvature includes a parameter N, sometimes called the effective dimension, which has been traditionally taken between the dimension of the manifold and infinity. Recently there is a growing interest in the case where N is negative. In this talk I will discuss some comparison theorems in this range of N, such as the curvature-dimension condition, spectral gap and splitting theorem (of W. Wylie).

Brendan Pass
- TITLE: A canonical barycenter via Wasserstein regularization
- ABSTRACT: I will discuss joint work with Young-Heon Kim, in which we introduce a weak notion of barycenter of a probability measure $mu$ on a metric measure space $(X, d, {bf m})$, with the metric $d$ and reference measure ${bf m}$. Under the assumption that optimal transport plans are given by mappings, we prove that our barycenter $B(mu)$ is well defined; it is a probability measure on $X$ supported on the set of the usual metric barycenter points of the given measure $mu$. The definition uses the canonical embedding of the metric space $X$ into its Wasserstein space $P(X)$, pushing a given measure $mu$ forward to a measure on $P(X)$. We then regularize the measure by the Wasserstein distance to the reference measure ${bf m}$, and obtain a uniquely defined measure on $X$ supported on the barycentric points of $mu$. We investigate various properties of $B(mu)$.