Bellman Functions: The dynamic programming point of view

DATES: 07 - 11 November 2022 (5 sessions)
TIME: 9:30-11:30 (a total of 10 hours)
LOCATION: Maryam Mirzakhani Seminar Room at BCAM


ABSTRACT:
We will give an introduction to the technique of using Bellman functions in Harmonic Analysis. The approach will be elementary and will take a practical point of view with many examples. A general numerical scheme will be presented to approximate these Bellman functions, which will be used to find or guess properties of the functions with the help of a computer. The course will be divided into the following chapters


CONTENTS:
1. Dynamic programming: from computer science to mathematics
We will describe some famous examples of problems that can be solved with dynamic programming (Fibonacci numbers, sorting, etc.). These problems exemplify very well the overlapping subproblem structure typical of the dynamic programming approach. The discussion will be blended together with a historical perspective of the use of the method in mathematics.
2. A basic example: the height of sparse collections
Here we will describe the concept of sparse collection, and the more flexible cousin: Carleson sequences. These sequences have an associated height function which will serve as one of the main characters in the following chapters. We will start looking at the problem of finding quadratic bounds for the integral of such height functions, and give a definition of the Bellman function associated with the problem.
3. Universality of the Bellman function
We will continue with the problem from chapter two, and show a certain universality of Bellman functions. This will give us the biggest hint for how to find it. After some properties, we will start to use the computer to approximate the Bellman function and then explicitly find it.
4. Exponential decay of level sets
Here we will introduce a more general problem of the height function: find estimates for its level sets. The Bellman function approach can also be used for this problem. We will set up the optimization problem and find an explicit expression.
5. Level sets for sparse operators
Continuing with the example from the previous chapter, we introduce the harder problem of studying the level sets of sparse operators, and how Bellman functions can help us here.
6. Muckenhoupt weights
We will introduce the Ap class of weights, and talk about various elementary properties. Then we will talk about how to use the Bellman function approach to give quantitative bounds for embeddings between these classes.

PREREQUISITES:
We will assume familiarity with a standard measure theory graduate course. Some familiarity with elementary algorithms can be useful, but is not necessary.

REFERENCES:
V. Vasyunin, & A. Volberg (2020). The Bellman Function Technique in Harmonic Analysis (Cambridge Studies in Advanced Mathematics)
F. Nazarov, S. Treil, A.Volberg. The Bellman functions and two-weight inequalities for Haar multipliers. Journal of the American Mathematical Society, vol. 12, no. 4. 1999
S. Dasgupta, C. Papadimitriou, U. Vaziarni (2008). Algorithms (McGraw-Hill)


*Registration is free, but mandatory before 2 November 2022. To sign-up go to https://forms.gle/1qPf2SJmBk2Hmve39 and fill the registration form.

• BCAM_Course_2022_11_07