Joint Mathematics Colloquium ICMAT-UAM-UC3M-UCM

We learn in school many interesting integer sequences and their significance. In this Colloquium, I will survey two such sequences, and explain their unexpected miracles.

The first one is the Catalan numbers. I will explain how this integer sequence leads us to two differential equations, their mirror-symmetric relations, and the quantization behind the scenes. We present Laurent polynomials that give solutions to these equations.The miracle here is that these Laurent polynomials know the topologocal invariants of the moduli space of point Riemann surfaces, calculated by Harer-Zagier, Witten, and Kontsevich.

The second example is analogous to the first one in the sense that it is an integer sequence of genus 0 Gromov-Witten invariants of a particular algebraic 3-fold. We have again two differential equations. The contrast is that although we know mirror symmetric and quantization relations between these equations, the key mechanism to calculate general GW invariants is still missing. The miracle here is that this particular integer sequence knows why Riemann zeta at 3 is irrational, the work of Apery.

Analogy is a powerful way used in philosophy, science, and mathematics to better understand one theory from similar ones. We will try to explain what we mean by analogy, we will make historical examples and we will present one of the most fruitful analogies: the analogy between number fields and function fields.

The Higher Infinite refers to the infinite cardinalities studied by set theory, as charted by large cardinal hypotheses known as large cardinal axioms. These axioms assert the existence of infinite cardinals so large that their existence cannot be proved within the standard ZFC system of set theory. Since the weakest of large cardinals, the "weakly inaccessible", were first defined and studied by Hausdorff over a century ago, a plethora of different and much stronger large cardinals have since then been identified in a great variety of contexts and taking many different forms. Indeed, after the groundbreaking results of Martin-Steel and Woodin in the 1980’s, establishing the tight connection between large cardinals and the determinacy of sets of reals, the theory of large cardinals has been expanding in multiple directions, yielding solutions to many well-known set-theoretic problems, as well as fertile applications to other areas of mathematics, from general to algebraic topology and homotopy theory, to abelian groups, etc. In this talk I will present some examples of large cardinals and will explain their role in mathematics by giving a number of examples in different areas where they have been applied to solve prominent open problems, some of them very recent.