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Grothendieck inequalities and semi-defined 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 so-called 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. Semi-defined 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 semi-defined programming

  • Combinatorial optimization

  • Relationship to Shannon's ability

  • MAX CUT or similar problems

  • Applications to Machine Learning