Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Optimal Transport: Cyclically Monotone Sets

In course

MATH-476: Optimal transport

The first part is devoted to Monge and Kantorovitch problems, discussing the existence and the properties of the optimal plan. The second part introduces the Wasserstein distance on measures and develops applications of optimal transport to PDEs, functiona ...

Description

This lecture discusses the concept of cyclically monotone sets in the context of optimal transport. It covers the definition of cyclically monotone sets, their properties, and their relevance in optimal transport theory. The lecture also explores the theorem related to optimal transport with cyclically monotone sets and provides examples to illustrate the concept.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructors (2)

Maria Colombo

Xavier Fernandez-Real Girona

Official source

Related concepts (32)

Definition

A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitions (which try to list the objects that a term describes). Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions.

Lexical definition

The lexical definition of a term, also known as the dictionary definition, is the definition closely matching the meaning of the term in common usage. As its other name implies, this is the sort of definition one is likely to find in the dictionary. A lexical definition is usually the type expected from a request for definition, and it is generally expected that such a definition will be stated as simply as possible in order to convey information to the widest audience.

Circular definition

A circular definition is a type of definition that uses the term(s) being defined as part of the description or assumes that the term(s) being described are already known. There are several kinds of circular definition, and several ways of characterising the term: pragmatic, lexicographic and linguistic. Circular definitions are related to Circular reasoning in that they both involve a self-referential approach. Circular definitions may be unhelpful if the audience must either already know the meaning of the key term, or if the term to be defined is used in the definition itself.

Theoretical definition

A theoretical definition defines a term in an academic discipline, functioning as a proposal to see a phenomenon in a certain way. A theoretical definition is a proposed way of thinking about potentially related events. Theoretical definitions contain built-in theories; they cannot be simply reduced to describing a set of observations. The definition may contain implicit inductions and deductive consequences that are part of the theory. A theoretical definition of a term can change, over time, based on the methods in the field that created it.

Proof by contradiction

In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally valid. More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved.

Related lectures (47)

Functions: Definitions and Notations

Covers the generalities of functions, including the definition of an application between sets and the uniqueness of elements in the image set.

Approximation Algorithms

Covers approximation algorithms for optimization problems, LP relaxation, and randomized rounding techniques.

Introduction to Convexity

Introduces the key concepts of convexity and its applications in different fields.

Proofs: Logic, Mathematics & Algorithms

Explores proof concepts, techniques, and applications in logic, mathematics, and algorithms.

Unconstrained Optimization Theory