Mathematics

Clay Lecture at the INI, June 2022. The lecture is based on joint work with Mark Green and Colleen Robles. Theme of the lecture is global properties of period mappings with applications to the geometry of completions of moduli spaces.

Algebraic geometry is the study of the geometry of algebraic varieties, defined as the solutions of a system of polynomial equations over a field k. When k = C the earliest deep results in the subject were discovered using analysis, and analytic methods (complex function theory, PDEs and differential geometry) continue to play a central and pioneering role in algebraic geometry. The objective of these talks is to present an informal and illustrative account of some answers to the question in the title.

Almost all of the deep results in Hodge theory and its applications to algebraic geometry require understanding the limits in a family of Hodge structures. In the literature the proofs of these results frequently use the consequences of the analysis of the singularities acquired in a degenerating family of Hodge structures; that analysis itself is treated as a "black box." In these lectures an attempt will be made to give an informal introduction to the subject of limits of Hodge structures and to explain some of the essential ideas of the proofs. One additional topic not yet in the literature that we will discuss is the geometric interpretation of the extension data in limiting mixed Hodge structures and its use in moduli questions.

Existence theorems are a central part of algebraic geometry. These results frequently involve linear problems where positivity assumptions are used to prove existence of solutions by establishing the vanishing of obstructions to that existence. Beginning with Riemann (algebraic curves), Picard (algebraic surfaces) and continuing into more recent times (Lefschetz, Hodge, Kodaira-Spencer and many others since this work) it has come to be understood that the vanishing theorems are intimatedly related to the topology of algebraic varieties. What remains is the case that although the results are about algebraic varieties, analytic tools are needed to establish them. Moreover the property of positivity also appears in other aspects where analytic methods are needed, an example being the proof of the Iitaka conjecture which is central in the classification of algebraic varieties. The purpose of these lectures is to present, sometimes from an historical perspective, some of the principal aspects of the theory.

This is the story of some of the mathematical work of two mathematicians, Jean Victor Poncelet and Niels Henrik Abel. They were contemporaries in the early 19th century who never met and who were not even aware of each other's work. However, between them Poncelet and Abel laid the cornerstones of the modern field of algebraic geometry, a field that is central to current work in geometry, arithmetic and theoretical physics. In this talk I will try to explain what each of them did, Poncelet in geometry and Abel in analysis, and how the fusion of their work revealed one of the deepest aspects of mathematics. This fusion is captured by an amazing property of playing billiards on a table formed by two ellipses.