## Lectures on Optimal Transportation (2019-2020)

In the academic years of 2019 and 2020, I gave lectures on optimal transportation in seminar and online courses in multiple universities.

The courses focused on the fundamental theories and computational algorithms for optimal transportation, Monge-Ampere equation and convex geometry. The topics include Monge-Kantorovich prime, dual problem, Brenier theory; Monge-Ampere equation theory, including existence, uniqueness, regularity and stability; Alelxandrov convex geometry, Brunn-Minkowski inequality, Alexandrov mapping lemma, Minkowski problem; computational algorithms, including direct numerical methods for solving Monge-Ampere equation, Oliker-Prussner algorithm, variational method; applications, including parameterization in graphics, registration in vision, shape comparision in medical imaging and generative models in deep learning.

The lecture notes are available upon request. The textbook based on the lectures will be published in 2021. Please forward your comments and criticisms to gu@cs.stonybrook.edu.

### Lecture Notes

#### Optimal Transportation Theory

1. Monge problem, Kantorovich problem

This lecture introduces the classical Monge problem and its relaxation, Kantorovich problem, proves the existence of the solution to Kantorovich problem.

2. Kantorovich Dual problem

This lecture introduces the dual formulation of the optimal transport problem, c-transform.

3. Brenier Theory

This lecture proves Brenier theory, Legendre-Fenchel transformation.

4. Cyclic monotone condition

This lecture introduces the cyclic monotone condition, and proves the equivalence between the Kantorovich problem and the dual problem formulation.

5. Polar decomposition

This lecture introduces Brenier's polar decomposition theorem.

#### Alexandrov Convex Geometry

6. Alexandrov Mapping Lemma

This lecture introduce Spenere lemma, Brouwer fixed point theorem, domain invariance and Alexandrov mapping lemma.

7. Brunn-Minkowski inequality

This lecture explains Brunn-Minkowski inequality, Isoperimetric inequality.

8. Minkowski problem

This lecture focuses on the Minkowski problems with prescribed curvatures.

#### Monge-Ampere Equation

9. Convex function and Alexandrov Theorem

This lecture introduce Alexandrov regularity theorem for convex functions.

10. Existence of Alexandrov solution to Monge-Ampere equation

This lecture proves the existence of Alexandrov solution to Monge-Ampere equation.

11. Uniqueness of Alexandrov solution to Monge-Ampere equation

This lecture proves the uniqueness and stability of Alexandrov solution to Monge-Ampere equation.

12. Regularity of Alexandrov solution to Monge-Ampere equation

This lecture introduces the regularity of the solution to Monge-Ampere equation.

#### Computational Algorithms

13. Computational geometric algorithms

Convex hull, upper envelope, Delaunay Refinement algorithms.

14. Numerical algorithms to solve Monge-Ampere equation

Explicit, semi-implicit solution methods, linearization of Monge-Ampere operator.

15.Oliker-Prussner method

Oliker-Prussner method for solving Monge-Ampere equation with Dirichlet condition.

16.Semi-discrete optimal transportation

Semi-discrete optimal transportation theorem and algorithm.

17.Variational Principles

Variational princple for semi-discrete optimal transportation.

18.Spherical Optimal transportation

Spherical optimal transportation algorithm.

19.Optimal transportation in Optics

Optimal transportation theory in optics.

20.Optimal transportation in Optics Algorithms

Optimal transportation algorithms in optics.

#### Optimal Transportation in Deep Learning

Optimal transportation for generative models in deep learning

This lecture focuses on the fundamental concepts and algorithms generative models in deep learning and the applications of optimal transport in generative model, including manifold distribution principle, manifold structure, autoencoder, Wasserstein distance, mode collapse and regularity of solutions to Monge-Ampere equation.

#### Direct Applications

a. Area-preserving surface parameterization

This lecture introduces area-preserving surface parameterization algorithms.

b. Volume-preserving solid parameterization

This lecture introduces volume-preserving solid parameterization algorithms.

c. Registration

This lecture introduces surface registration based on optimal transportation.

d. Medical imaging

The lecture notes are updated constantly, more lecture notes will be added to cover broader scope.