Review: Stevo Todorcevic’s CRM-Fields-PIMS Prize Lecture

After winning the 2012 CRM-Fields-PIMS Prize, Stevo Todorcevic gave a series of talks on his research: at CRM, at PIMS and at the Fields Institute.

The director of the Fields Institute asked me to write a short review on Stevo’s Fields lecture for the Winter 2013 Fields Notes, and I thought I should post my review in here, as well. Here goes.

The 2012 CRM-Fields-PIMS Prize Lecture was opened by the director of the Fields Institute Ed Bierstone who then invited Justin Tatch Moore, a former student of Todorcevic, to give a brief overview of Todorcevic’s work. Moore surveyed Todorcevic’s professional biography, and then continued with remarks about his work. He mentioned Todorcevic’s 1998 ICM lecture has served as an inspiration for some of his own work,
and invited anyone with background in analysis to read Todorcevic’s 1999 paper on compact subsets of the first Baire class, where set-theoretic forcing was applied to this field in an unconventional way (in this paper, theorems, rather than consistency results, concerning compact sets of Baire class-one functions are obtained by analyzing the corresponding objects in forcing extensions of the universe).

Todorcevic’s lecture at the Fields Institute focused on his method of minimal walks on ordinals. This method was discovered in the early 1980’s, when Todorcevic learned from Galvin about the problem of finding a Ramsey basis for structures on the first uncountable ordinal ${\omega }_1$. This is of course a part of a very general problem, but a simple form in this particular context asks whether there (consistently) exists a natural number $n=n({\omega }_1)$ such that for any symmetric function $f:[{\omega }_1{]}^2\to K$ with some finite range $K$ there exists a set $C\subseteq K$ of size at most $n({\omega }_1)$, together with an uncountable set $X$, such that $f(x,y)\in C$ for all distinct $x,y$ in $X$. Galvin and Shelah proved in 1973 that $n({\omega }_1)$ — if exists — must be bigger or equal to 4. While analyzing this problem in the early 1980’s, Todorcevic showed that several well-known basis problems in mathematics could be solved if such a number exist. For example, he showed that $n({\omega }_1)=4$ would entail that
the class of uncountable regular topological spaces has a 3-element basis, and that
the class of uncountable linear orderings has a 5-element basis. However not long afterwards, in a surprising turn of events in the year of 1984, Todorcevic  provided a negative answer to the fundamental Ramsey problem for ${\omega }_1$, exhibiting a function $c:[{\omega }_1{]}^2\to {\omega }_1$ with the remarkable property that $c$ is surjective over all uncountable squares. In the construction of such a map, he used as motivation the “walks below ${\epsilon }_0$” method which originated in proof theory, and which is one-dimensional in nature, and came up with a two-dimensional variation that goes all the way up to ${\omega }_1$ (and, in fact, could be implemented on any given ordinal). The method of “minimal walks on ordinals” was born.

After discussing the above, Todorcevic gave a technical account on the method of minimal walks, and the associated characteristic functions, highlighting two of its main features: metric triangle inequalities, and canonicity. The former turned out to be a key in many constructions (for instance, in constructions of Banach spaces with no infinite unconditional basic subsequence), whereas the latter centers around comparing derivatives of minimal walks (e.g., Lipschitz trees) with objects that were previously constructed in an ad-hoc fashion. A very interesting instance of canonicity is the fact that minimal walks may be utilized to derive (in the presence of a forcing axiom) a selective ultrafilter on the set of natural numbers which is ${\mathrm{\Sigma }}_1$-definable in the structure $(H({\omega }_2),\in )$. It follows from classical results of Nykodim, Sierpinski and Solovay that no non-principal ultrafilters on the set of natural numbers can be ${\mathrm{\Sigma }}_1$-definable in the structure $(H({\omega }_1),\in )$.

In conclusions of his lecture, Todorcevic mentioned related works of his former students. On the classification side, Moore proved that the class of uncountable regular spaces does not admit a finite basis, and that, in the presence of a standard forcing axiom, the class of uncountable linear orderings has a 5-element basis, and the class of Aronszajn orderings has a 2-element basis. On the canonicity side, Martinez-Ranero proved that the same forcing axiom implies that the class of Aronszajn orderings is well-quasi-ordered by the relation of isomorphic embeddability.