10 August 2017:
This week's progress:
1) Jason Erbele wrote an entertaining blog article about the conference he attended at the Perimeter Institute:
* [Hopf Algebras in Kitaev’s Quantum Double Models: Mathematical Connections from Gauge Theory to Topological Quantum Computing and Categorical Quantum Mechanics](https://jasonmaths.wordpress.com/2017/08/09/haikqdmmcfgtttqcacqm-postmortem/).
2) Nina Otter gave a talk at a week-long conference in Sapporo, Japan, called Applied Algebraic Topology 2017:
* New invariants for multi-parameter persistent homology.
> **Abstract.** Topological data analysis (TDA) is a field that lies at the intersection of data analysis, algebraic topology, computational geometry, computer science, and statistics. The main goal of TDA is to use ideas and results from geometry and topology to develop tools for studying qualitative features of data. One of the most successful methods in TDA is persistent homology (PH), a method that stems from algebraic topology, and has been used in a variety of applications from different fields, including robotics, material science, biology, and finance.
> PH allows to study qualitative features of data across different values of a parameter, which one can think of as scales of resolution, and provides a summary of how long individual features persist across the different scales of resolution. In many applications, data depend not only on one, but several parameters, and to apply PH to such data one therefore needs to study the evolution of qualitative features across several parameters. While the theory of 1-parameter persistent homology is well understood, the theory of multi-parameter PH is hard, and it presents one of the biggest challenges of TDA.
> In this talk I will briefly introduce persistent homology, give an overview of the complexity of the theory in the multi-parameter case, and then discuss how tools from commutative algebra give invariants able to capture homology classes with large persistence.
> No prior knowledge on the subject is assumed. This talk is based on joint work with Heather Harrington, Henry Schenck, and Ulrike Tillmann.
3) I also gave a talk at this conference:
* [The rise and spread of algebraic topology](http://math.ucr.edu/home/baez/alg_top/)
> **Abstract.** As algebraic topology becomes more important in applied mathematics, it is worth looking back to see how this subject has changed our outlook on mathematics in general. When Noether moved from working with Betti numbers to homology groups, she forced a new outlook on topological invariants: namely, they are often functors, with two invariants counting as "the same" if they are naturally isomorphic. To formalize this it was necessary to invent categories, and to formalize the analogy between natural isomorphisms between functors and homotopies between maps it was necessary to invent 2-categories. These are just the first steps in the "homotopification" of mathematics, a trend in which algebra more and more comes to resemble topology, and ultimately abstract "spaces" (for example, homotopy types) are considered as fundamental as sets. It is natural to wonder whether topological data analysis is a step in the spread of these ideas into applied mathematics, and how the importance of "robustness" in applications will influence algebraic topology.
Quite a few other talks used category theory! But as this slide from Tom Leinster's talk, they did so with a bit of hesitancy: