Program

More details about the program of the meeting will appear here.

Lecture Series

Julie Bergner (University of Virginia)

2-Segal spaces in algebra, homotopy theory, and algebraic K-theory

The notion of Segal space has been useful for modeling up-to-homotopy topological categories, and can be described via the so-called Segal maps. Two different approaches have led to the same generalization, known as 2-Segal spaces: while Dyckerhoff and Kapranov sought to generalize the Segal maps from a geometric point of view, Galvez-Carrillo, Kock and Tonks wanted to look at structures similar to topological categories but for which composition may be partially or multiply defined but is still associative. A key example of a 2-Segal space is the output of Waldhausen’s S-construction when applied to an exact category, and 2-Segal spaces satisfying certain finiteness assumptions provide a unifying treatment to Hall algebra constructions. In this lecture series, we will look at the definition of 2-Segal spaces from different perspectives, together with several examples. From there, we will consider Hall algebras associated to some of our examples, as well as a generalized S-construction and applications to algebraic K-theory.

Vidit Nanda (University of Oxford)

Lecture 1 (Overview)

As data and systems get more complicated, so must the tools with which we analyze them. This talk describes how algebraic topology has emerged as a powerful new technique for data analysis, and highlights a few of its recent applications.

Lecture 2 (Computation)

In theory, computing homology of finite simplicial complexes is a straightforward task: one only has to perform row and column operations to diagonalize boundary matrices. In practice, when these complexes are built around large datasets, direct matrix methods become computationally intractable. This talk focuses on a discrete version of Morse theory which allows us to drastically reduce the sizes of complexes while preserving their homology.

Lecture 3 (Persistence)

This lecture is devoted to the key ideas behind, and properties of, persistent homology. Since its inception at the turn of the millennium, persistence has fast become the premier technique in topological data analysis for two fundamental reasons. First, the algebraic objects it produces admit a complete and finite invariant, called a barcode. And second, the process of producing barcodes from data is stable under certain fluctuations (such as bounded noise) in the data. We will carefully explore the structure and stability of barcodes.

Lecture 4 (Beyond)

This lecture describes future directions of research in topological data analysis. We will discuss three areas of high activity, and describe some promising new topological tools for data analysis.