Luis Ángel Caffarelli

Nationality: Argentine–American

Born: 1948

PhD 1972 (University of Buenos Aires)

Supervisor:

Calixto Calderón

Affiliations:

University of Minnesota 1973-1983

Courant Institute, NYU 1980-1982

University of Chicago 1983-1986

IAS Princeton 1986-1996

Courant Institute, NYU 1994-1997

University of Texas at Austin 1997-today

Research fields

Free boundary problems

Nonlinear PDE

Optimal mass transport (OMT)

Question (Monge 1781): Cheapest way to transport one mass distribution to another?

Mathematical formulation:

Example of transport map

Transport map

Polar decomposition of matrices

acts on:

via:

Bundle structure

Bundle structure

Bundle structure

"Square root" equation:

Bundle structure

Factorization theorem:

Bundle structure

Vertical gradient flow:

Bundle structure

Vertical gradient flow:

Bundle structure

Horizontal gradient flow:

Relative entropy:

Bundle structure

Horizontal gradient flow:

Lifted gradient flow on $K$ for

Bundle structure

Horizontal gradient flow:

Hessian of $F(P)$ strictly positive on $K$ $\Rightarrow$ unique limit!

Nice, but...

...what about Monge-Ampere and Caffarelli?!

?

Linear optimal mass transport

$\mathrm{P}(n) \iff$  multivariate Gaussians with zero mean

$\Rightarrow$ transport map $T:\mathbb{R}^n \to \mathbb{R}^n$ linear, $T(x) = A x$

$\Rightarrow$

Fundamental result for $L^2$ OMT

Theorem (Brenier 1987):

$\mu_0,\mu_1$ prob measures on $\Omega\subset\mathbb R^n$ (open, bounded) with $\mu_i=\rho_id \mathbf x$
Then $(*)$ has unique solution $$T = \nabla \psi$$

where $\psi:\Omega \to \mathbb R$ is convex

$\psi$ solves the weak Monge-Ampere equation $$\int_\Omega \eta \rho_1 dx = \int_\Omega (\eta\circ\nabla\psi) \rho_0 dx,\qquad \forall \eta \in C(\Omega)$$

Note: $\psi\in C^2(\Omega) \Rightarrow \rho_1\circ\nabla \psi\det(\nabla^2 \psi) = \rho_0$    (strong MA)

Fundamental result for $L^2$ OMT

Theorem (Brenier 1991):

$T \in L^2(\Omega,\mathbb{R}^n)$ such that $T_*\mu_0 = \mu_1$

Exist unique $S:\Omega\to\Omega$ with $S_*\mu_0 = \mu_0$ and convex $\psi$ such that $$T = (\nabla \psi) \circ S$$

Note: $\psi\in C^2(\Omega) \Rightarrow \rho_1\circ\nabla \psi\det(\nabla^2 \psi) = \rho_0$    (strong MA)

$T$

$\nabla\psi$

$S$

Fundamental result for $L^2$ OMT

Theorem (Brenier 1991):

$T \in L^2(\Omega,\mathbb{R}^n)$ such that $T_*\mu_0 = \mu_1$

Exist unique $S:\Omega\to\Omega$ with $S_*\mu_0 = \mu_0$ and convex $\psi$ such that $$T = (\nabla \psi) \circ S$$

Note: $\psi\in C^2(\Omega) \Rightarrow \rho_1\circ\nabla \psi\det(\nabla^2 \psi) = \rho_0$    (strong MA)

Subdifferential: $\partial \psi(x) = \{u \in\mathbb{R}^n\mid \forall y, \; \psi(y) \geq \psi(x) + u\cdot (y-x) \}$

$x$

$\psi(x)$

$\psi(y)$

$y$

Regularity obstructions

Non-convex domain

Negative curvature domain

[cf. Villani 2009]

Caffarelli's regularity theory

Theorem (Caffarelli 1992):

$\mu_0,\mu_1$ prob measures on $\Omega\subset\mathbb R^n$ (convex) with $\mu_i=\rho_id \mathbf x$

$\rho_i \in C^{k,\alpha}(\Omega), \quad 0<a\leq \rho_i(x) \leq b < \infty$
Then $(*)$ has unique solution $$T = \nabla \psi$$

where $\psi\in C^{k+2,\alpha}(\Omega)$ is convex

$\psi$ solves the strong Monge-Ampere equation $$\rho_1\circ\nabla \psi\det(\nabla^2 \psi) = \rho_0$$

Riemannian structure of OMT

[Benamou & Brenier (2000), Otto (2001)]

OMT $\leftrightarrow$ hydrodynamics

[Arnold 1966, Hamilton 1982, Caffarelli 1992, Benamou and Brenier 2000, Otto 2001]

Summary

Slides available at: slides.com/kmodin

(from an applied perspective)

Caffarelli's work enables "safe ground" for geometers to work with smooth optimal transport