JMM 2017 Day 2

I didn't go to quite as many talks this day, as I had to attend a reception for graduate students who had received money to attend, and I gave my talk on this afternoon as well.

Talk 1: "Descriptive Set Theory and Borel Probabilities" by Chris Caruvana

In Chris's research, he is studying the space of Borel probability measures. Since this is a Polish space, one can talk about comeager collections of measures in a coherent way.  In particular, the sets of reals which are measure zero for a comeager collection of measures is an object of interest. This forms a countably closed ideal, and it at least contains the meager sets. Thus there is an associated notion of "smallness" that comes from these sets. With this notion of smallness comes an analogue of being Baire Measurable, that is your difference with a Borel set is small. Using this analogue, Chris was able to prove an automatic continuity result. There are interesting open questions floating around about exactly what relationship between these sets and universally null sets is consistent.

Talk 2: "Rationales for Mathematics and its Significance in Recently-Excavated Bamboo Texts from Ancient China" by Joseph W. Dauben

A whole slew of ancient Chinese mathematical texts have been recovered recently. They were written on flattened bamboo strips, and some are really well preserved. They tend to follow a student-master dialogue style, and covered a pretty wide range of arithmetic and word problems. It's interesting that they seemed to have a very Pythagorean view of things, where everything is number (and in particular expressible with fractions? That point wasn't totally clear). In one of the dialogues, the student is upset, because he can't seem to master literature and math at the same time. The teacher tells him to learn math if he has to choose, because "math can help you understand literature, but literature cannot help you understand math." This was a fun talk, it was definitely cool to see pictures of the bamboo strips. I don't know much about ancient Chinese math, so I didn't understand a lot of the speaker's references, but I'm kind of curious now to look at some of the important texts that have been uncovered (translated into English of course).

Talk 3: "Modal logic axioms valid in quotient spaces of finite CW-complexes and other families of topological sets" by Maria Nogin

This talk mostly dealt with realizing the semantics of modal logic in certain topologies. In particular point topologies (where open means containing a particular point), excluded point topologies (open means missing a particular point) and quotients of finite CW-complexes. Different extensions of S4 can be realized across these spaces. I've never looked into this kind of thing too much, beyond understanding the relationship between compactness in topological products and compactness in classical logic. I remember a colleague once looking into this sort of representation for intuitionistic logic. I think this level of understanding of the semantics can be helpful for these more complicated logics.

Talk 4: "On a variant of continuous logic" by Simon Cho

Another pure logic talk. This one had more of a non-standard analysis bent. Simon was looking at something called geodesic logic, which expands on continuous logic by loosening the requirement for the semantics to consist of continuous functions, but adds linear and geodesic structure into the semantics. Using this, uniform convergence results can be obtained for functions which are not continuous. In some sense, this is an extension of ideas presented in a really good sequence of blog posts by Terry Tao on non-standard analysis, convergence and ergodic theory. I really enjoy seeing where this line of thought is progressing. In some sense it may be easiest to approach intractable analysis problems by first figuring out what the appropriate logic is to study the problems with.