## 15th SLALM

[Edit: Aug. 22, 2012]

The 15th SLALM (Latin American Symposium in Mathematical Logic) was held in Bogotá, June 4-8, 2012. There were also three tutorials preceding the main event, on May 30-June 2. I give one of the tutorials, on Determinacy and Inner model theory, 10:40-12 each day, at the Universidad Nacional. Here is the abstract:

Since the invention of forcing, we know of many statements that are independent of the usual axioms of set theory, and even more that we know are consistent with the axioms (but we do not yet know whether they are actually provable).

These proofs of consistency typically make use of large cardinal assumptions. Inner model theory is the most powerful technique we have developed to analyze the structure of large cardinals. It also allows us to show that the use of large cardinals is in many cases indispensable. For years, the main tool in the development of this area was fine structure theory.

Determinacy (in suitable inner models) is a consequence of large cardinals, and recent work has revealed deep interconnections between determinacy assumptions and the existence of inner models with large cardinals, thus showing that descriptive set theory is also a key tool.

The development of these connections started in earnest with Woodin’s core model induction technique, and has led to what we now call Descriptive inner model theory.

The goal of the mini-course is to give a rough overview of these developments.

I have written a set of notes based on these talks, and will be making it available soon.

In addition, I gave one of the invited talks during the set theory session: Forcing with ${\mathbb P}_{\rm max}$ over models of strong versions of determinacy. Here is the abstract:

Hugh Woodin introduced ${\mathbb P}_{\rm max}$, a definable poset, and showed that, when forcing with it over $L({\mathbb R})$ (in the presence of determinacy), one recovers choice, and obtains a model of many combinatorial assertions for which simultaneous consistency was not known by traditional forcing techniques. ${\mathbb P}_{\rm max}$ can be applied to larger models of determinacy. As part of joint work with Larson, Sargsyan, Schindler, Steel, and Zeman, we show how this allows us to calibrate the strength of different square principles.

This is of course related to the paper I discussed recently.