What was the course about?
The course explored the “duality” between knowing the answer (for example, a counting problem) precisely versus only knowing estimates of the behavior of the answer (for example, estimating the number of solutions to an equation).
This duality appears in many ways in mathematics. For example, we can prove that there are an infinite number of primes, or an infinite number of primes in certain arithmetic progressions, but as we keep asking things of the primes, we lose the ability to prove their infinitude because the proofs we had are, up to a point, algorithmic and/or constructive. If instead, we let go of the algorithmic nature of the proofs, and we accept to know the objects exists without knowing how to produce the following ones, we can prove their infinitude.
We covered
- Prime numbers and prime numbers in arithmetic progressions.
- Patterns in large number of data (for example, Ramsey Theory).
- The probabilistic method to prove objects exist.
- Discussed examples of counting points on curves and estimating them (Gauss Examples in Cubics)
- Distributions and the Birch and Swinnerton-Dyer conjecture
What motivated me?
The duality described above has always been very impressive to me and informs much of how I approach things mathematically. In a sense, it is a validation of the importance of Statistics in pure mathematics. I am interested in this because, when I was an undergraduate, there was a lot of bias against statisticians. There was a pervasive view that statistics had no place in pure mathematics, which is simply wrong and very misguided. I wanted to discuss this point with the students and bring home that statistics is one of our most powerful tools as a civilization wherever we encounter it.
Exercise Sheets:
Invited Speakers:
Dr. Mario Diaz from Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, UNAM, México talked to us about Machine Learning and Data.
Malors 