Registration Deadline: TBA

Instructor: Professor Augusto Gerolin, University of Ottawa

Course Date: TBA
Mid-Semester Break: TBA
Lecture Time: TBA
Office Hours: TBA

Registration Fee:

• Students from our Principal Sponsoring & Affiliate Universities: Free
• Other Students: CAD$500

Capacity Limit: TBA

Format: TBA

Course Description

This graduate course is in preparations for the Thematic Program on Optimal Transport in Natural Sciences and Statistics (July-December 2026).

Optimal Transport is a theory from mathematics and economics dating back to the late 18th century that has flourished in the 90’s in pure mathematics. It has emerged as a fertile field of inquiry, and a diverse tool for exploring applications within and beyond mathematics, in such diverse fields as economics, meteorology, geometry, statistics, fluid mechanics, design problems and engineering. More recently, it has become one of the most important emerging topics in machine learning research. Several fundamental challenges in Optimal Transport and applications are still open from all analytical, computational and statistical viewpoint. Motivated by these challenges, this course aims to give a general introduction on mathematical, computational and statistical aspects of Optimal Transport Theory as well as presenting open problems. An application in Natural Science will be chosen to drive the mathematical development of the course. The options include: Density Functional Theory (Quantum Chemistry), Trajectory Inference (Biology, Omics), Quantum Information Theory.

Objectives: One of the easy course objectives is to learn ’something’ about Optimal Transport, including mathematical, computational and statistical aspects of the theory. The difficult objective is for you to learn to ask questions and then solve them. Learning Optimal Transport requires cultivating analytical and mathematical reasoning skills at a high level; by the end of this course, you will hopefully be more capable of thinking analytically and ’jump’ from theory, computations and statistics.

Make-up and Attendance policy: The primary element of the course is assignments. Depending of the size of the classroom, my intention is to assess assignments individually, so while attending lecture is not essential, to obtain marks for assignments, one must generally attend office hours or other sessions dedicated to that purpose. Marks will reflect not only the correctness of the answer but the depth of insight that led to the answer. Of course, if the number of students is very large I will assess assignments by myself and provide written feedback taking into consideration what is written.

Lecture notes: Lecture notes will be provided and expanded in highly collaborative way. They will be developed by the instructor together with the students. A advanced version of the lecture notes (in English) will be provided on overleaf every week.

Textbooks (complementary material):

1. An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows by Alessio Figalli and Federico Glaudo (2021).
2. Computational Optimal Transport by Marco Cuturi and Gabriel Peyr ́e (available online).
3. Lectures on Optimal Transport by Luigi Ambrosio, Elia Bru ́e, Daniele Semola.
4. The strong-interaction limit of density functional theory by Gero Friesecke, Augusto Gerolin and Paola Gori-Giorgi.

Catalog Description:

1. Optimal Transport (OT) Theory
1. Kantorovich formulation of Optimal Transport
2. Regularized Optimal Transport problem
3. Dual approach: existence of Entropy-Kantorovich potentials
4. Characterization of solutions of Kantorovich formulation of OT
5. Continuity and Fr ́echet differentiability of Optimal Transport functionals
2. Computational aspects
1. Linear and Semi-definite programming
2. Stochastic Gradient descent
3. Sinkhorn algorithm
4. Thereoretical guarantees of convergence and robustness
3. Applications (to be decided during the course)
1. Learning with Optimal Transport losses
2. Multi-marginal Optimal Transport and Computational Chemistry
3. Optimal Transport Barycenters
4. Optimal Transport in Gaussian spaces

Student will also recall (or learn) concepts from functional analysis, calculus of variations, convex analysis, such as lower semi-continuity, Weak*−convergence, Riesz representation theorem, Banach-Alaoglu Theorem, Convex functions, Legendre transform, the Direct Method of Calculus of Variations. The Catalog description and the mathematical tools are absurd amount of material. We will move quickly. Do not get behind. If (when) I wish to modify the schedule/content, I will notify you in class and/or online.

Prerequisites: Functional and Real Analysis courses. No programming experience is required. Functional Analysis requirement may be dropped if the student is highly motivated and master concepts of an advanced course in Analysis in n and Linear Algebra.

Homeworks: Homework assignments will be posted on the course website throughout the term. The problems designated as Hand-in are to be turned in via e-mail on the assigned due date. Additional problems, usually extracted from the textbook, will be given for extra practice but are not to be turned in and will not be graded. You are encouraged to work with others while solving homework problems, but you must write up your own solutions. Collaboration is allowed, but not plagiarism. This is the time to learn how to develop collaborative work and research. Moreover, late homework will not be accepted except in the case of an excused absence.

Exams: There will be one midterm exam and one final exam. The midterm will take place in October, during the usual class time, while the final exam will be held during the end of the term (precise date to be determined). A detailed description of the material covered by each exam will be given on the course website in due time.

Grading Policy:

The final grade will be based on the homework/quizzes, the midterm exam and the comprehensive final exam. It will be computed according to the following distribution:

• homework: 20% of your grade;
• midterm: 30% of your grade;
• final exam: 50% of your grade.

The homework and quizzes will be weighted equally. Moreover, the lowest grade will be dropped. Homework, quiz and exam scores will be posted on the course webpage so you can monitor your progress in the course. Your final letter grade will be computed according to the standard university scale: A+ (90%-100%); A (85%-89%); A- (80%-84%); B+ (75%-79%); B (70%-74%); C+ (65%-69%); C (60%-64%); D+ (55%-59%); D (50%-54%); E (40%-49%); F (0%-39%).

General Remarks/Advice: It is more important to have the “right ideas” than it is to get the “right answer”. In a course such as this one, it is more important to “think correctly” than to “work accurately”. Partial credit will often be given when you understand a problem and will always be given when you understand how to solve a problem. Omitting (or making errors in) will rarely cost you more than 25% of the total available points on a problem, but if it causes you to “misinterpret” your or get the wrong answers in subsequent problems, it could be devastating.

Don’t get behind. The volume of material in this course is staggering. It will require a daily commitment from you. Probably most of you are smart enough so that you have been able to get by studying just the night before the test. That ends here. You will need to read and take notes on the book before class. After class, you will need to go through the lecture notes, rewriting them and adding marginal comments/questions about the material. Then you will need to work through some of homework, and think critically about the material. This course will probably require a larger commitment of time and mental energy than any course you have ever taken before. Some weeks, you may find the course content easy (in which case you will probably need less time). Other weeks, you will find the course content very difficult, and you will need to budget your time accordingly.

Ask questions. Please visit me during office hours. (I get lonely). I will clarify points in lecture that were not clear (sorry!), provide guidance on homework, and more generally “shoot the breeze”, “chew the fat”, etc.. More generally, I will do everything in my power to try to help you through this course.

Don’t be afraid to criticize/comment on the course. In general, my teaching methods are flexible, and I will adapt them to your needs. What you learn is not negotiable. How you learn is entirely negotiable. I’ve designed this course based on what my opinions of what will work. I expect to change things during the term based on feedback and suggestions from you. I’m even open to totally detonating the course structure and building anew from the rubble. Thus, it is important that you make suggestions and/or tell me about portions of the course that need improvement. (Even if I do not agree with your suggestion, I will try to come up with an alternative approach that addresses your concerns.) If you feel uncomfortable giving feedback in person, slide a note under my door, get someone else to talk to me on your behalf (preserving your anonymity, if you wish), or come and talk to me yourself but raise your concerns in the third person: for example, “I don’t feel this way, but some of the people in class think your lectures are about as exciting as watching ice sublime.”

Materials Copyright: All materials generated for this class are protected by Copyright laws. Distributing copies or sale of any of these materials is strictly prohibited. The Copyright Act and copyright law protect every original literary, dramatic, musical and artistic work, including lectures by University instructors. The recording of lectures, tutorials, or other methods of instruction may occur during a course. Recording may be done by either the instructor for the purpose of authorized distribution, or by a student for the purpose of personal study. Students should be aware that their voice and/or image may be recorded by others during the class. Please speak with the instructor if this is a concern for you.

Academic Integrity: Academic fraud is an act by a student that may result in a false evaluation. Examples of academic fraud are: plagiarism, cheating of any kind or submit a work for which you are not the author, in whole or part. Any person found guilty of academic fraud will be subject to severe sanctions.

Electronic devices: Students are required to turn off all electronic devices such as cell phones, ipods, and blackberrys during class. If you need your cell phone for emergency contact (children, medical emergency) please inform the professor at the beginning of the class and use vibration mode.

Academic Accommodations Service:

Students who have a disability or functional limitation and who need adaptive measures (changes to the physical setting, arrangements for exams, learning strategies, adaptive technologies, etc.) to progress or participate fully in university life should contact me in the beggining of the course (or before the start of the course) via e-mail or in person for advice.

I will work with students with documented disabilities, and more generally with students who have temporary disabilities or other factors that inhibit their ability to perform at a high level. My goal is be maximally supportive of my students, and to treat everyone fairly. This means that while I am eager to accommodate all of you, I hold the same high expectations for all of you.

Policy – Prevention of Sexual Violence: The Fields Institute of Mathematical Sciences and the University of Ottawa will not tolerate any act of sexual violence. This includes acts such as rape and sexual harassment, as well as misconduct that take place without consent, which includes cyberbullying.