Black holes are fundamental objects in our understanding of the universe. The mathematics behind them has surprising geometric properties, and their dynamics are governed by hyperbolic PDEs. We will see how one can answer the basic question of whether these solutions to the Einstein equation are stable under small perturbations, which is a typical requirement to be physically meaningful, and how the dispersion of gravitational waves plays a key role in the stability problem.

I will give an elementary introduction to embedding questions in symplectic geometry, mentioning some work I have been doing recently about embedding ellipsoids into balls and other shapes.

Projective algebraic varieties are special examples of manifolds. Sometimes their Betti numbers have enumerative meaning, for example they count partitions of a positive integer with some extra properties, so one wonders if theorems about the topology of algebraic varieties help in studying these counts. One remarkable theorem about the topology of algebraic varieties is the Hard Lefschetz Theorem which has numerous applications in combinatorics, for example in the work of Stanley. In this talk, I plan to explain what algebraic varieties are, what special properties their Betti numbers have, and mention some applications of these properties in combinatorics.

The Galois groups of field extensions and the fundamental groups of topological spaces are founding notions in algebra and topology respectively. In the 1960s, Grothendieck showed that these two concepts are both part of the same more general theory, which he developed through the étale fundamental group. Many years later, after Grothendieck had largely left the mathematical world, he took up this topic again in the new directions. What he developed is now known as anabelian geometry and Grothendieck-Teichmüller theory. Drawing from fields as diverse as surface topology and algebraic stacks, Grothendieck used the étale fundamental group to study the nature of algebraic varieties more closely, obtaining tantalizing links to deep Diophantine questions. In this talk I will give a gentle introduction to these ideas, complementing the mathematics with the fascinating history behind how it arose.

When you hear the word "braids", you probably think about hair. But, did you know that these types of braids (and more complicated ones, too) are an important and central object of study in mathematics? Braids are relevant to many fields, including cryptography, dynamics, and algebra. In this talk, I'll introduce braids, and tell you a bit about how they show up within low-dimensional topology, my field of research. We'll see how braids are related to knots, groups, and surfaces. There is a rich history of research in braids at Columbia and Barnard, so I'll do my best to mention other faculty members (past and present) who have studied these objects. All are welcome! In particular, I will not assume any background, just an interest in mathematics.

Hyperbolic geometry arose as a variation of the usual Euclidean geometry, by modifying the parallel axiom. This idea gave rise to an interesting geometry that is, in some sense, richer than Euclidean geometry. From a modern point of view we can use hyperbolic geometry to define interesting geometric structures on manifolds. In the case of surfaces, i.e. manifolds of dimension 2, every surface admits many hyperbolic structures, and all these structures can be put together in a space, called Teichmuller space. Teichmuller space itself carries some interesting geometric structures. I will try to give an elementary introduction to this theory, starting from some classical mathematics, to some open problems that are subject of current research.

 

I will explain how ramification phenomena, absent in characteristic zero algebraic geometry, manifest themselves in positive characteristic. I will do so through the Lefschetz fixed point formula, for which I will introduce the necessary vocabulary. From there, I may branch out to the ramification of Galois representations and, eventually, to another instance of its geometric incarnation.

Geometric flows are a class of PDEs of parabolic type. They are a fundamental tool to understand the interplay between the topology and geometry of manifolds. In this talk we will use questions connected to General Relativity to illustrate the geometric and analytic picture.

Topological data analysis applies ideas from algebraic topology to capture the shape of data. I will give a gentle introduction to this story.