Looking Back at the 2010s

Dec 27 2019, 18:53

The decade is drawing to a close, and while it is entirely arbitrary, it's a great excuse to look back at this decade.

At the end of 2009 I started my senior year as an undergraduate. I both read the first part of "Introduction ot Cardinal Arithmetic" to get a hold on the basics of set theory, and also took my first course on set theory (I'm omitting the introductory course from my freshman year since that one covered very very basic set theory). I've studied with the wonderful Matti Rubin, and it was a fantastic course. Too bad that it focused almost solely on the axioms (i.e. how the axioms are not provable from others, etc.) and that we only spent a short time dealing with actual set theoretic topics (e.g. Solovay's theorem on partitions of stationary sets, etc.)

Methods in Higher Forcing Axioms: The inevitable conclusion

Sep 17 2019, 17:42

The meeting in Norwich is over. Here are my thoughts.

It felt haphazard, without a concrete plan. And that was great. In the first day, Tadatoshi Miyamoto and David (Asperó) presented two problems in the morning and in the early afternoon. We then had a discussion about them, and it was just a general discussion, that went very well.

Methods in Higher Forcing Axioms

Sep 09 2019, 16:12

Methods in Higher Forcing Axioms (or MEHIFOX, for short) is an experimental workshop hosted in Norwich by David Asperó and myself. You can find the website, right here. This workshop is sponsored by the London Mathematical Society, and the School of Mathematics in UEA.

The idea is to have a workshop, where we actually work. This is contrary to the normal use of "workshop" (in set theory, at least, but I believe in most mathematical areas) which means a very small conference where almost all the participants are also speaking.

7 Easy Hacks to Improve Your Math Skills

Aug 22 2019, 19:55

Everybody wants to improve their mathematical skills! And quickly, too! Since it's so hard to do just that, I've written down some quick and dirty hacks for quickly improving your mathematical skills!

1. Get a graduate-level degree in mathematics!

Getting a PhD in mathematics is not really about getting the PhD itself. It's more about getting much better at learning mathematics. So if you get a PhD in mathematics it will help you better your ability to study more mathematics and improve your skills.

New notes online!

Jul 22 2019, 17:40

I have posted two new lecture notes online. The one is from a course in functional analysis I took in the autumn of 2015/16 with Prof. Matania Ben-Artzi, and the second is from the course I taught in axiomatic set theory in the autumn of 2016/17.

Just as a general caveat for the set theory notes, since all the students in the course were also my students in the basic set theory course that I taught with Azriel Levy (yes, that Azriel Levy, and yes it was quite an awesome experience) and there I managed to cover some fairly nontrivial things in that course, these notes might feel as if there are some gaps there, or that I skip here and there over some information.

In praise of Replacement

Mar 06 2019, 22:20

I have often seen people complain about Replacement axioms. For example, this MathOverflow question, or this one, or that one, and also this one. This technical-looking schema of axioms state that if \(\varphi\) defines a function on a set \(x\), then the image of \(x\) under that function is a set. And this axiom schema is a powerhouse! It is one of the three component that give \(\ZF\) its power (the others being power set and infinity, of course).

You'd think that people in category theory would like it, from a foundational point of view, it literally tells you that functions exists if you can define them! And category theory is all about the functions (yes, I know it's not, but I'm trying to make a point).