ICTS

In this series of lectures I will give an introduction to the gluing methods and their applications in geometry and nonlinear partial differential equations.

This is a series of 5 lectures to be delivered at the program "Geometric and Analysis Pde" about different types of isoperimetric-like problems.

This 5-lecture mini-course provides a rigorous introduction to the classical theory of Optimal Transport (OT), bridging the gap between discrete assignment problems and continuous mass redistribution. We begin by analyzing discrete and semi-discrete cases, establishing the foundational concepts of stability and optimality. Then we will dive into the modern Kantorovich duality framework, with a specific focus on existence and uniqueness theorems within Polish spaces and L^1 and L^2 cost functions. In the final sessions, we discuss advanced applications, including degree theory for circle maps,  branched transport models,  reversible mechanics of adhesion and separation, and introduce some new open problems.

Lecture 1: Discrete and Semi-Discrete Optimal Transport

Topics: Introduction to the Monge problem; the Kantorovich relaxation in discrete settings.
Key Concepts: Stable matchings, assignment problems, and the geometry of power diagrams in semi-discrete transport.

Recommended Prerequisites
Basic knowledge of Measure Theory and Functional Analysis.
Familiarity with Convex Analysis is helpful but not required.

This 5-lecture mini-course provides a rigorous introduction to the classical theory of Optimal Transport (OT), bridging the gap between discrete assignment problems and continuous mass redistribution. We begin by analyzing discrete and semi-discrete cases, establishing the foundational concepts of stability and optimality. Then we will dive into the modern Kantorovich duality framework, with a specific focus on existence and uniqueness theorems within Polish spaces and L^1 and L^2 cost functions. In the final sessions, we discuss advanced applications, including degree theory for circle maps,  branched transport models,  reversible mechanics of adhesion and separation, and introduce some new open problems. 

Lecture 2: The Kantorovich Duality 

Topics: The dual formulation of the transport problem.
Key Concepts: Existence of optimal plans in Polish spaces; the role of c-concavity and cyclical monotonicity.

Recommended Prerequisites:
Basic knowledge of Measure Theory and Functional Analysis. Familiarity with Convex Analysis is helpful but not required.

This 5-lecture mini-course provides a rigorous introduction to the classical theory of Optimal Transport (OT), bridging the gap between discrete assignment problems and continuous mass redistribution. We begin by analyzing discrete and semi-discrete cases, establishing the foundational concepts of stability and optimality. Then we will dive into the modern Kantorovich duality framework, with a specific focus on existence and uniqueness theorems within Polish spaces and L^1 and L^2 cost functions. In the final sessions, we discuss advanced applications, including degree theory for circle maps,  branched transport models,  reversible mechanics of adhesion and separation, and introduce some new open problems.

Lecture 3: Optimal Transport with L^1 and L^2 Costs

Topics: Analysis of the transport problem under specific cost functions.
Key Concepts: Brenier’s Theorem for quadratic costs; geometric properties of the L^1 (Monge-type) cost and the ray structure. Beckmann’s problem and application to Congested transport

Recommended Prerequisites:
Basic knowledge of Measure Theory and Functional Analysis.
Familiarity with Convex Analysis is helpful but not required.

This 5-lecture mini-course provides a rigorous introduction to the classical theory of Optimal Transport (OT), bridging the gap between discrete assignment problems and continuous mass redistribution. We begin by analyzing discrete and semi-discrete cases, establishing the foundational concepts of stability and optimality. Then we will dive into the modern Kantorovich duality framework, with a specific focus on existence and uniqueness theorems within Polish spaces and L^1 and L^2 cost functions. In the final sessions, we discuss advanced applications, including degree theory for circle maps,  branched transport models,  reversible mechanics of adhesion and separation, and introduce some new open problems. 

Lecture 4: Flows in measure spaces, circle maps

Topics: Transport in 1-D and periodic domains, 
Key Concepts: Circle maps, rotation numbers and metric derivatives

Recommended Prerequisites
Basic knowledge of Measure Theory and Functional Analysis.
Familiarity with Convex Analysis is helpful but not required.

This 5-lecture mini-course provides a rigorous introduction to the classical theory of Optimal Transport (OT), bridging the gap between discrete assignment problems and continuous mass redistribution. We begin by analyzing discrete and semi-discrete cases, establishing the foundational concepts of stability and optimality. Then we will dive into the modern Kantorovich duality framework, with a specific focus on existence and uniqueness theorems within Polish spaces and L^1 and L^2 cost functions. In the final sessions, we discuss advanced applications, including degree theory for circle maps,  branched transport models,  reversible mechanics of adhesion and separation, and introduce some new open problems.

Lecture 5: Advanced Applications: Traffic Flow and Branched Transport

Topics: Modern extensions and physical models.
Key Concepts: Application to branched transport (Steiner tree-like structures), the reversible model of adhesion and separation, circle maps, ect... 

Recommended Prerequisites:
Basic knowledge of Measure Theory and Functional Analysis.
Familiarity with Convex Analysis is helpful but not required.