Uniformity, Universality, and Computability Theory

We prove a number of results motivated by global questions of uniformity in computability theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups. We begin by investigating the notion of uniform universality, first proposed by Montalb\'an, Reimann and Slaman. This notion is a strengthened form of a countable Borel equivalence relation being universal, which we conjecture is equivalent to the usual notion. With this additional uniformity hypothesis, we can answer many questions concerning how countable groups, probability measures, the subset relation, and increasing unions interact with universality. For many natural classes of countable Borel equivalence relations, we can also classify exactly which are uniformly universal. We also show the existence of refinements of Martin's ultrafilter on Turing invariant Borel sets to the invariant Borel sets of equivalence relations that are much finer than Turing equivalence. For example, we construct such an ultrafilter for the orbit equivalence relation of the shift action of the free group on countably many generators. These ultrafilters imply a number of structural properties for these equivalence relations.Comment: 61 Page

Therapeutic antibodies: current state and future trends--is a paradigm change coming soon?

Antibody-based therapeutics currently enjoy unprecedented success, growth in research and revenues, and recognition of their potential. It appears that the promise of the "magic bullet" has largely been realized. There are currently 22 monoclonal antibodies (mAbs) approved by the United States Food and Drug Administration (FDA) for clinical use and hundreds are in clinical trials for treatment of various diseases including cancers, immune disorders, and infections. The revenues from the top five therapeutic antibodies (Rituxan, Remicade, Herceptin, Humira, and Avastin) nearly doubled from $6.4 billion in 2004 to$11.7 billion in 2006. During the last several years major pharmaceutical companies raced to acquire antibody companies, with a recent example of MedImmune being purchased for $15.6 billion by AstraZeneca. These therapeutic and business successes reflect the major advances in antibody engineering which have resulted in the generation of safe, specific, high-affinity, and non-immunogenic antibodies during the last three decades. Currently, second and third generations of antibodies are under development, mostly to improve already existing antibody specificities. However, although the refinement of already known methodologies is certainly of great importance for potential clinical use, there are no conceptually new developments in the last decade comparable, for example, to the development of antibody libraries, phage display, domain antibodies (dAbs), and antibody humanization to name a few. A fundamental question is then whether there will be another change in the paradigm of research as happened 1-2 decades ago or the current trend of gradual improvement of already developed methodologies and therapeutic antibodies will continue. Although any prediction could prove incorrect, it appears that conceptually new methodologies are needed to overcome the fundamental problems of drug (antibody) resistance due to genetic or/and epigenetic alterations in cancer and chronic infections, as well as problems related to access to targets and complexity of biological systems. If new methodologies are not developed, it is likely that gradual saturation will occur in the pipeline of conceptually new antibody therapeutics. In this scenario we will witness an increase in combination of targets and antibodies, and further attempts to personalize targeted treatments by using appropriate biomarkers as well as to develop novel scaffolds with properties that are superior to those of the antibodies now in clinical use

Borel circle squaring

We give a completely constructive solution to Tarski's circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If $k \geq 1$ and $A, B \subseteq \mathbb{R}^k$ are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than $k$, then $A$ and $B$ are equidecomposable by translations using Borel pieces. This answers a question of Wagon. Our proof uses ideas from the study of flows in graphs, and a recent result of Gao, Jackson, Krohne, and Seward on special types of witnesses to the hyperfiniteness of free Borel actions of $\mathbb{Z}^d$.Comment: Minor typos correcte