Grothendieck inequalities and semidefined programming
Alexander Grothendieck is one of the most influential mathematicians of the 20th century, his work in geometry, topology, and arithmetic revolutionized those areas with such intensity that today it is difficult to follow a logical thread between the immediate results before him and those that preceded him. However, his professional career as a mathematician did not begin in the areas mentioned (and perhaps for which he is best known today). He began his career studying Banach spaces and made fundamental contributions, one of which is the socalled Grothendieck inequality. The applications of these inequalities are very powerful and have permeated various areas of knowledge due to their depth and their applications. This course aims to study the implications of these algorithms in three fundamental areas: Machine Learning, Convex Optimization and Graph Theory.
Course syllabus
Course one
1. Linear programming

Basic tools of linear programming

Convex sets and general results

Algorithms

Applications in Machine Learning

Grothendieck inequalities: statements and applications
2. Applications to Graph Theory (Szemeredi)
3. Semidefined programming
4. Kernel methods
5. Demonstration of Grothendieck inequalities using Kernel
Course two
1. An introduction to the probabilistic method

Probability Elements

Guassian vectors in higher dimensions

Probabilistic proof of Grothendieck's inequalities
2. Applications of semidefined programming

Combinatorial optimization

Relationship to Shannon's ability

MAX CUT or similar problems

Applications to Machine Learning