Research done by members of our group tends to be centered in model theory and its applications and in descriptive set theory. There are also research projects involving other members of the Mathematics Department, especially in nonstandard analysis and in aspects of group theory that have significant connections to logic.
Much of this work emphasizes the interactions between logic and other parts of mathematics. Recent successes based on these connections demonstrate that mathematical logic can provide powerful new methods for many areas of mathematics and can be the basis for breakthroughs on critical problems. Logic typically comes into the picture in two ways: (a) it provides a new set of tools; (b) it provides a new language within which to formulate results and problems.
Some Specific Projects
These brief descriptions give representative examples of the research projects that members of our group are involved in.
Van den Dries has in recent years pursued research on
(1) O-minimal structures on the real field, with connections to real analytic geometry and geometric measure theory.
(2) Asymptotic differential algebra. This is a long-term project in collaboration with M. Aschenbrenner and J. van der Hoeven.
(3) Arithmetic complexity: simple ideas from model theory and diophantine approximation yield new lower bounds on the complexity of computing various arithmetic functions.
Henson and his students developed a model theory for structures based on metric spaces, in which the key features were a restriction of language to positive bounded sentences and the introduction of an approximate concept of satisfaction. Recently a collaboration among Alex Berenstein, Itai Ben Yaacov, Henson, and Alex Usvyatsov has developed an equivalent logic for doing model theory with metric structures, a continuous version of first order logic, which is much easier to use than its precursors. Â In continuous logic, truth values are real numbers (from a bounded interval), connectives are continuous functions, and quantifiers are the operations of sup and inf. Â One focus of current work is on the application of these model theoretic ideas to specific classes of structures arising naturally in functional analysis. Various results show that this is the "correct" logic for understanding amd using ultraproducts of metric space structures (such as Banach spaces, Banach lattices, operator spaces, C*-algebras, and the like). Â The general theory has surprisingly strong parallels with ordinary first-order logic.
Jockusch (Emeritus) works in computability theory and its connections with other areas, especially combinatorics. Much of his recent work has focused on the strength and effective content of Ramsey's theorem, which is an essential tool in combinatorics, logic, and other areas. A form of this result states that for all positive integers k, n and every k-coloring of the n-element sets of natural numbers, there is an infinite set A of natural numbers which is homogeneous in the sense that all n-element subsets of A have the same color. He has investigated the extent to which there exist homogeneous sets which are "easily definable" relative to the coloring, and the strength of Ramsey's theorem as a formal statement. His other interests include degrees of unsolvability, complexity of paths through computable trees, genericity, and degrees of presentations of structures.
Solecki works in descriptive set theory and its
applications to topology and analysis. In particular, he is interested
in applications of the theory of definable equivalence relations to:
(1) the study of Polish group actions and Polishable subgroups of Polish groups;
(2) the study of indecomposable continua and the space of composants in such continua.
Topic 1 has connections, which have been to some extent explored, with the structure of ideals of subsets of $\omega$ and with maximal divisible subgroups of Polish abelian groups. Solecki has been recently also investigating Haar null subsets of Polish groups. This is a notion of smallness extending the notion of Haar null sets to not necessarily locally compact Polish groups which has applications in Banach space theory.
Research Support, Collaborations and Exchange Programs
Several NSF research grants are held by faculty in the logic group, and these provide a variety of support for the program, especially including support for graduate students (Research Assistantships, travel to meetings, etc). Faculty also regularly obtain RA support for advanced graduate students from the UIUC Campus Research Board.
Van den Dries, Pillay and Henson had an NSF grant in the Focused Research Program (2001-2005) that supported aspects of their work that are strongly interconnected with other parts of mathematics. It provided academic year support for graduate students and for two model theory postdoctoral positions, as well as for visitors and meetings (such as the Logic and Mathematics meeting held at UIUC in May, 2003).
Model theory was one of the focal points of a formal exchange
agreement between UIUC and the French research agency CNRS. This was a
vehicle for regular exchange visits between France and Urbana by people
working in model theory and closely related areas, including graduate
students. We had visits by Daniel Lascar, Zoe Chatzidakis, Jean-Marie
Lion, Elizabeth Bouscaren, Joris van der Hoeven, Michael Schmeling, Max
Dickmann, Franck Benoist, Ronald Bustamante, and Yves Raynaud; Itay
Ben-Yaacov spent the Spring semester 2002 here as a visiting student
and returned as a visiting postdoc in December 2002 and for several
months in Spring 2003. Van den Dries, Henson and Pillay have made
visits to Paris; Rahim Moosa spent the Fall semester 1999 in Paris and
this visit was important to the development of his Ph.D. thesis