W. Hugh Woodin
Distinguished Lecture Series

A scaffold for mathematics includes both local foundations for various areas of mathematics and productive guidance in how to unify them. In a scaffold the unification does not take place by a common axiomatic basis but consists of a systematic ways of connecting results and proofs in various areas of mathematics. Two scaffolds, model theory and category theory, provide local foundations for many areas of mathematic including two flavors (material and structural) of set theory and different approaches to unification. We will discuss salient features of the two scaffolds including their contrasting but bi-interpretable set theories. We focus on the contrasting treatments of 'size' in each scaffold and the advantages/disadvantages of each for different problems.

Woodin's axiom \(AD^+\) is a strengthening of the classical Axiom of Determinacy. It is an open question whether \(AD\) implies \(AD^+\), although this implication is known to hold in the inner model \(L(\mathbb{R})\). The axioms of \(AD^+\) allow one to lift much of the known analysis of \(L(\mathbb{R})\) to larger models of determinacy. We will give a brief introduction to \(AD^+\), including a survey some of Woodin's work in the area and some interesting open questions.

We present recent work, joint with J. Wolf, in which we define a natural notion of higher-order stability and show that subsets of \(\mathbb{F}_p^n\) that are tame in this sense can be approximately described by a union of low-complexity quadratic subvarieties, up to linear error. This extends joint work with Wolf on arithmetic regularity lemmas for stable subsets of \(\mathbb{F}_p^n\) to the realm of higher order Fourier analysis.

The practice of foundations of mathematics is built around a firm distinction between syntax and semantics. But how stable is this distinction, and is it always the case that semantically presented mathematical objects in the form e.g. of a model class might give rise to a "natural logic"? In this talk I will investigate different scenarios in which an investigation of the notion of an implicit or internal logic or syntax becomes possible, from the perspective of model classes.

The determinacy axiom, AD, was introduced by Mycielski and Steinhaus over 60 years ago as an alternative to the Axiom of Choice for the study of arbitrary sets of real numbers. The modern view is that determinacy axioms concern generalizations of the borel sets, and deep connections with large cardinal axioms have emerged.

Further a specific technical refinement of AD, this is the axiom AD\(^+\), has also been isolated. The further connectionswith large axioms have implicitly led to a duality program, which is the AD\(^+\) Duality Program.

The central open problems here are intertwined with those of the Inner Model Program, and this is distilled into a specific conjecture, the Ultimate-\(L\) Conjecture. This conjecture, and its related problems, are now arguably the key problems in both AD\(^+\)-theory and the Inner Model Program.

There are many open problems in the model theory of wellfounded models of \(ZFC\). Cohen's method of forcing, including the generalization to class forcing, is the main technique for building models of \(ZFC\).

But consider the following problem. Suppose \(ZFC + \phi\) has a unique wellfounded model. Must that model satisfy \(V = L\)? No counterexample can be a nontrivial class forcing extension of any other model, and moreover that model of \(ZFC + \phi\) must be countable and belong to \(L\) (by Shoenfield absoluteness). This suggests that any such model must satisfy \(V = L\). Rephrased, if \(\text{ZFC} + \phi\) is both \(\beta\)-consistent and \(\beta\)-categorical then the natural conjecture is that \[ ZFC + \phi \vdash_{\beta} V = L. \] However relativising to a real and assuming large cardinals, there are counterexamples, and in the strongest possible sense. This is joint work with Peter Koellner.