Research Statement Dan Hathaway Department of Mathematics University of Michigan 1. Introduction My research is in set theory. The central theme of my work is to combine ideas from descriptive set theory and the study of cardinal characteristics. I will describe both of these areas. Then, I will describe how my research relates to both areas. ω is the set of natural numbers, ω ω is the set of functions from ω to ω, ω 1 is the first uncountable cardinal (the set of all countable ordinals), and 2 ω is the cardinality of the set R of real numbers. 2. Background 2.1. Cardinal Characteristics There are two main purposes of the theory of cardinal characteristics (of the continuum, for example). The first is to serve as an interface between mathematicians who study objects whose behavior depends on additional set theoretic assumptions beyond ZFC, and set theorists who determine the status of and relationship between axioms. A mathematician can easily master the language of cardinal characteristics without needing to learn any of the technical set theoretic constructions (such as forcing). He then can derive results from assumptions beyond ZFC that are close to (and in some cases exactly) optimal. By doing this, he maximizes the utility of his results, and he reduces his problem to the precise set theoretic issues that are relevant. Let me mention three cardinals, which appear in the theory, to refer to later. The cardinal add(l) is the smallest number of Lebesgue measure zero subsets of R whose union is not Lebesgue measure zero. The cardinal d is the smallest size of a family F of functions from ω to ω such that for each function g : ω ω there is some f F satisfying ( n ω) g(n) f(n). The cardinal b is the smallest size of a family F of functions from ω to ω such that for each function g : ω ω there is some f F satisfying ( n ω)( m n) g(m) f(m). It turns out that ω 1 add(l) b d 2 ω. The fact that add(l) b is not obvious. Let us give a specific example. One can construct, assuming CH (the Continuum Hypothesis), a simple C -algebra with finite nuclear dimension which is non Z-stable (see [5]). This, however, is not useful to another mathematician who is working under a contradictory set theoretic assumption, such as MA + CH (Martin s Axiom and the negation of CH). However, a result which follows from CH often follows from a weaker statement about cardinal characteristics. In our example, the existence of the C -algebra follows from the assumption b = 2 ω, which in turn follows from MA. Now the mathematician working under MA + CH can use the result. For one who believes in the truth of these statements, while it is not at all clear whether CH is true, b = 2 ω is more believable. At one time, Gödel believed that ω 1 < b = 2 ω. In general, the relationships between set theoretic axioms are very complicated and difficult to master. However, cardinal characteristics provide a great organization tool, and the key relationships between the cardinals can be encoded into diagrams which require almost no background knowledge to understand. This framework will help bridge the gap between foundational questions at the heart of set theory, and more mainstream mathematical questions sensitive to set theoretic assumptions. The second purpose of the theory of cardinal characteristics is to discover hidden structure (of the continuum and more complicated structures). For example, the inequality add(l) b between the cardinal numbers add(l) and b is merely the consequence of there existing a one-way Galois
Dan Hathaway page 2 of 5 connection between two binary relations. These connections are important even if one assumes CH. Finding these canonical one-way Galois connections (or proving that they do not exist) is an important step towards understanding the continuum. The deepest result I have proven shows that one of these canonical connections exists (see [7]). 2.2. Descriptive Set Theory Descriptive set theory studies well-behaved sets of real numbers. While questions about arbitrary sets of real numbers can be intractably difficult (for example, the Continuum Hypothesis), questions about easy to define sets often can be answered from ZFC alone. When they do not follow from ZFC, usually the canonical axioms beyond ZFC do settle them (for example, Projective Determinacy seems to settle all important questions about projective sets). Often, mathematicians do not need to work with arbitrary sets of real numbers, only nicely definable ones. This makes it possible to answer their questions. Also, since countably infinite structures can be encoded as real numbers, statements about the behavior of sets of real numbers can have profound consequences. A particularly important and well-behaved example is the class of all Borel sets of real numbers. The Borel sets are Lebesgue measurable and satisfy other regularity properties. Even very difficult results about them usually follow from ZFC, for example Borel Determinacy. Furthermore, there are many deep questions about their structure which remain open, which will likely by decided one way or the other by ZFC alone. This will yield a structure theory available to all mathematicians free from the complication of independence results. Much can be modeled using only Borel sets. For example, by encoding each finitely generated group as a real number, the binary relation A R R, defined by r 1 Ar 2 iff the groups coded by r 1 and r 2 are isomorphic, is Borel. The structure theory for Borel sets can then be used to show that A is maximally difficult among all similar Borel isomorphism problems. Various hierarchies can be put on the Borel functions. It is useful to consider the Baire hierarchy. This is formed by starting with the continuous functions and repeatedly taking the pointwise limits of functions previously constructed. The functions which are the pointwise limits of continuous functions are called Baire class one. My work shows that when passing from continuous functions to Baire class one functions, there is a dramatic change in how much information can be encoded. Beyond the Borel sets, but still not considering arbitrary sets of reals, there are the projective sets. These are the sets that can be obtained from the open sets by taking complements, countable intersections and unions, and continuous images. While a working mathematician may naturally encounter sets of reals which are low level in the projective hierarchy but not Borel, he is unlikely to encounter a set of reals that is not projective unless he tries. The structure theory for these sets cannot be settled by ZFC alone. However, in contrast to the situation with other independent statements, we seem to be able to answer all important questions about these sets by using the axiom of Projective Determinacy. This axiom is canonical (and arguably correct). Thus, we expect to be able to find a compelling answer to any natural question about the projective sets. Developing the structure theory from this axiom, however, takes work. 3. My Research So far, I have investigated combinatorial questions, which one might ask in the theory of cardinal characteristics, but about the objects studied in descriptive set theory. An example is the following theorem. Recall that d is the smallest size of a family of functions from ω to ω such that every function is everywhere dominated by one in the family. One can easily see that ω 1 d 2 ω. More surprisingly, I have shown the following: Theorem 3.1. Let W be the set of all well-founded trees on ω. Then the cofinality of this set ordered by inclusion is d.
Dan Hathaway page 3 of 5 The more technical version of this theorem is stating that there is a one-way Galois connection between two relations. Namely, the relation associated to d is above the inclusion ordering on W. We may represent this using a diagram as follows: ω ω ω ω W W, which is shorthand for saying there exist φ : W ω ω and φ + : ω ω W satisfying ( T W)( g ω ω) φ (T ) f T φ + (f). The partially ordered set described in the theorem is closely connected to the set of continuous functions from Baire space ω ω to N ordered by everywhere domination. When we then look instead at the set B 1 of Baire class one functions (which are not necessarily continuous but which are the pointwise limits of the sequences of continuous functions), we get a completely different situation. Namely, if we let A c B denote that the set A ω is constructible from B ω (a notion that generalizes computable in a natural way), then we have the following: B 1 P(ω) c B 1 P(ω). Hence, the transition from continuous to Baire class one causes a phase transition of the associated relations from relatively simple to extremely complicated. Here is a specific version of the result (see [7]): Theorem 3.2. For each A ω, there is a Baire class one function f : R ω such that if g : R ω is any function satisfying ( x R) f(x) g(x), then A is 1 1 definable using g as a predicate. The method of proof for this theorem is quite powerful, and I have used it to prove seemingly unrelated results. For example, I have discovered a new proof that if λ and κ are cardinals satisfying λ κ = λ, then the cofinality of the set of functions from λ to κ ordered by everywhere domination is 2 λ (see [6]). The old proof used independent families of functions. The following is a direct consequence of one of my underlying lemmas. We use the notion of weak distributivity as defined in [9]. Theorem 3.3. Let λ be an infinite cardinal. distributive, then it is (λ, 2)-distributive. If a complete Boolean algebra is weakly (λ ω, ω)- There is a priori no reason to expect the above theorem to be true. It came as a complete surprise and was discovered by accident. It uses the fact that the well-foundedness of trees is absolute. The theorem does not hold when we replace ω with ω 1 even if we take λ = ω 1 and require the Boolean algebra to be (ω, 2)-distributive. This is because of the (consistent) existence of a Suslin algebra. Pushing the technique further, we get a family of results which sound interesting in their own right, such as the following which uses a (small) large cardinal: Theorem 3.4. Let κ be a weakly compact cardinal. If a complete Boolean algebra is weakly (2 κ, κ)- distributive and is (α, 2)-distributive for all α < κ, then it is (κ, 2)-distributive.
Dan Hathaway page 4 of 5 The diversity of these results suggests there are further applications of the method of proof. A structure which occurs in descriptive set theory (specifically, the study of Borel equivalence relations [3]) is the set F of all Borel functions from ω ω to ω ω ordered by pointwise eventual domination. I have been studying the combinatorics of this object from the point of view of cardinal characteristics. In this direction, I have computed the cofinality of this poset by constructing a one-way Galois connection to the constructibility relation on P(ω): F F P(ω) c P(ω). It is easy to describe the Galois connection, but proving that it worked turned out to be highly non-trivial. This is atypical. The following is a version of the result, where the notion of one set of natural numbers being 1 2 in another is a strengthening of constructibility: Theorem 3.5. For each A ω, there is a Baire class one function f : R ω ω such that if g : R ω ω is a Borel function satisfying then A is 1 2 in any Borel code for g. ( r R)( c ω) f(r, c) g(r, c), Corollary 3.1. The poset F, has cofinality 2 ω and has an unbounded chain of length ω 1. I conjecture that the theorem still holds even if we do not require g to be Borel, at the expense of replacing the family of sets 1 2 in a Borel code for g with some other appropriate (countable) family of sets defined from g. The proof of this theorem works by proving a strong statement inductively on the Baire hierarchy. The theorem does not use any determinacy, which poses a challenge for extending the result beyond the Borel functions. It is possible that the conjecture will depend on axioms beyond ZFC. The proof of the theorem above actually shows a more general result. Let R ω ω ω ω be such that there is a continuous function c : ω ω ω ω such that for all y ω ω, c(y)ry. Almost all binary relations R occuring in the theory of cardinal characteristics have this property. For example, R could be a prewellordering of ω ω. Then, with respect to R, functions from ω ω to ω ω can be regarded as functions from ω ω to an ordinal. The following holds: Theorem 3.6. For each A ω, there is a Baire class one function f : ω ω ω ω such that whenever g : ω ω ω ω is a Borel function satisfying ( x ω ω) f(x)rg(x), then A is 1 2 in any code for g. Many of my results involve constructing a one-way Galois connection from a combinatorial structure to a poset which occurs in recursion theory. For example, Theorem 3.2 and Theorem 3.5 are of this form. This is reminiscent of [10] and [11] where it is shown that for each hyperarithmetical set of natural numbers A, there is a function f : ω ω such that if a function g : ω ω everywhere dominates f, then A is computable from g. This suggests a general program: find those combinatorial relations for which there exists a one-way Galois connection to a poset studied in recursion theory. I hope to develop this program, looking for combinatorial relations from descriptive set theory. 3.1. Additional Work In addition to the main work mentioned above, I am interested in a broad range of problems and have worked on various projects. In [5], my contribution was discovering an alternate characterization of the cardinal b which was suitable for the problem. In [8], Scott Schneider and I proved a combinatorial result which was an infinitary matching problem.
Dan Hathaway page 5 of 5 I am also interested in how strong axioms impact the existence or non-existence of one-way Galois connections. For example, using an immediate generalization of an argument in [1], if we assume that first-order formulas quantified over L(R) with real parameters and preserved by set sized forcing, then there cannot be unwanted one-way Galois connections in L(R). Perhaps the hypothesis can be weakened to AD L(R). Also, I am interested in the theory of Boolean ultrapowers. In the same way that there are qualitatively different ultrapowers of the standard model of arithmetic, there may be interesting differences between ill-founded Boolean ultrapowers of V arising from different forcings. References [1] Andreas Blass. Combinatorial Cardinal Characteristics of the Continuum. In M. Foreman and A. Kanamori, editors, Handbook of Set Theory Volume 1. Springer, New York, NY, 2010. [2] Andreas Blass. Needed Reals and Recursion in Generic Reals. Ann. Pure Appl. Logic 109 (2001), no. 1-2, 77-88. [3] Charles Boykin and Steve Jackson. Borel Boundedness and the Lattice Rounding Property. Advances in Logic, colume 425 of Contemp. Math., pages 113-126. Amer. Math Soc., Procidence, RI, 2007. [4] Samuel Coskey and Scott Schneider. Borel Cardinal Invariant Properties of Countable Borel Equivalence Relations. http://arxiv.org/abs/1103.2312 - in preparation. [5] Ilijas Farah, Dan Hathaway, Takeshi Katsura, and Aaron Tikuisis. A Simple C -algebra with Finite Nuclear Dimension which is Not Z-stable. http://arxiv.org/abs/1301.5030; To appear in Münster Journal of Mathematics. [6] Dan Hathaway. A Lower Bound for Generalized Dominating Numbers. http://arxiv.org/abs/1401.7948 - to be submitted for journal publication. [7] Dan Hathaway. Reducing Constructibility to Everywhere Not Domination of Borel Functions. http://arxiv.org/abs/1408.4200 - to be submitted for journal publication. [8] Dan Hathaway, Scott Schneider. Combinatorics of Reductions Between Equivalence Relations. In preparation. [9] Thomas Jech. Set Theory, The Third Millennium Edition, Revised and Expanded. Springer, New York, NY, 2002. [10] Carl Jockusch Jr. Uniformly introreducible sets. J. Symbolic Logic 33 (1968), 521 536. [11] Robert Solovay. Hyperarithmetically encodable sets. Trans. Amer. Math. Soc. 239 (1978) 99 122.