If you pour water on a statue, the water will descend in predictable ways. Amazingly, if you know exactly how streams of water on a statue can break, you can reconstruct the statue's shape. This fact is a toy example of Morse theory, a field of topology that tries to understand shapes of things in terms of streams of least resistance (these streams are sometimes called gradient flows). I want to introduce this theory in some fun, concrete examples, and as time allows, talk about a mysterious, recent discovery that connects two disparate branches of math: The geometry of Morse theory encodes exactly the algebra of associativity (yes, like associativity of multiplication).
Topology Day: Streams of water, the shapes of things, and the geometry of associativity Sponsored by the Meyer Fund
Generalized indiscernibles can be built in first-order theories by generalizing the combinatorial Ramsey’s Theorem to classes with more structure, which is an active area of study. Trying to do the same for infinitely theories (in the guise of Abstract Elementary Classes) requires generalizing the Erdos-Rado Theorem instead. We discuss various results about generalizations of the Erdos-Rado Theorem and techniques (including large cardinals and forcing) to build generalized indiscernible.
Building generalized indiscernibles in AECs with set theory
The field of algebraic topology has exposed deep connections between topology and algebra. One example of such a connection comes from algebraic K-theory. Algebraic K-theory is an invariant of rings, defined using tools from topology, that has important applications to algebraic geometry, number theory, and geometric topology. Algebraic K-groups are very difficult to compute, but advances in algebraic topology have led to many recent computations which were previously intractable. Equivariant homotopy theory, a branch of algebraic topology which studies topological objects with a group action, has been particularly important in the study of algebraic K-theory. In this talk I will give a gentle introduction to algebraic K-theory and its applications, and explain the interesting role of equivariant homotopy theory in this story.
Topology Day: Algebra in topology and topology in algebra: an introduction to algebraic K-theory Sponsored by the Meyer Fund
The last couple of decades saw a considerable increase in applications of topological techniques within other disciplines, particularly in data analysis, leading to a creation of a nascent field of topological data analysis. The premise of the latter is that data have shape, and that shape matters. Mathematically, this typically translates into an assumption that the data under consideration represent a sample from some topological space whose homotopy type is of interest. For computational reasons, the focus shifts to recovering homology groups of the underlying space, and several important results have been obtained to address this important task. In this talk I will cover some of the key results which allow one to compute homology groups of a subspace of a metric space given only a finite (possibly noisy) sample from it. Not surprisingly, such results rely on certain assumptions about the global geometry/topology of the subspace. I will then show that these assumptions (and results) become more subtle if the goal is to recover local rather than global homology groups of the subspace.
Topology: Day Recovering homology from samples: from global to local Sponsored by the Meyer Fund
Topology Day: Streams of water, the shapes of things, and the geometry of associativity