C. Ward Henson's Home Page

2010 photo

C. Ward Henson

Professor Emeritus
Department of Mathematics
University of Illinois at Urbana-Champaign
1409 W. Green Street, Urbana, Illinois 61801-2975 USA.
email: henson(at)math(dot)uiuc(dot)edu

Slides for Henson's talk April 5, 2013, in the UC Berkeley Logic Colloquium, entitled Continuous model theory and Gurarij's universal homogeneous separable Banach space are here. This reports on recent joint work with Itai Ben Yaacov, and details are in their paper Generic Orbits and Type Isolation in the Gurarij space, a preprint of which is available in the arXiv. The main new result is this: If X is the Gurarij space and F is any finite dimensional Banach space, then there is a linear isometry J of F into X such that X has the unique Hahn Banach extension property over J(F) (introduced by Phelps, this means that every linear functional on J(F) has a unique extension to X with the same norm). Moreover, the set of all such embeddings of F into X is a dense G-delta subset of the space of all isometric linear embeddings of F into X, and it is a full orbit of the action of the automorphism group of X on this space. In W Lusky's paper proving the uniqueness of the Gurarij space, he indicated a complicated proof of the special case of this result in which F is one dimensional. Our proof of the more general result uses conceptual tools from continuous model theory together with some technical tools from convex analysis.

LOGIC AND MATHEMATICS 2011: this conference took place September 3-4, 2011, at UIUC. Invited speakers were Itai Ben Yaacov (Lyon), Gregory Cherlin (Rutgers), Julien Melleray (Lyon), Anand Pillay (Leeds), Christian Rosendal (UIC), David Sherman (Virginia), and Henry Towsner (UCLA). For more information, including the titles of talks and abstracts, look HERE.

Web pages of the Logic Program of the University of Illinois at Urbana-Champaign
Distribution list of people getting news about the logic program at UIUC. (This file gives the names of those who receive these notices by email. Anyone interested in being added to this list should send email to henson(at)math(dot)uiuc(dot)edu.)

Research Interests of CWH:

General Interests: Mathematical logic and its interactions with the rest of mathematics and computer science; nonstandard analysis and other applications of model theory in analysis and geometry; model theoretic properties of specific structures in mathematics; logical decision problems and their complexity.
Continuous First-order Logic and Model Theory of Metric Structures: Henson's main research activity at the present time is the development and application of the [0,1]-valued continuous version of first-order logic for structures from analysis, topology, geometry, etc; the properties of this logic are closely parallel to those of first-order logic applied to structures from algebra.

Articles on continuous first-order logic and the model theory of metric structures:

Model Theory for Metric Structures by Itai Ben Yaacov, Alexander Berenstein, C. Ward Henson, and Alexander Usvyatsov; in Model Theory with Applications to Algebra and Analysis, Vol. II, eds. Z. Chatzidakis, D. Macpherson, A. Pillay, and A.Wilkie, Lecture Notes series of the London Mathematical Society, No. 350, Cambridge University Press, 2008, 315--427.

Continuous first order logic and local stability by Itai Ben Yaacov and Alexander Usvyatsov, Trans. Amer. Math. Soc. 362 (2010), 5213-5259.
Model Theory of Nakano Spaces; PhD thesis of L. Pedro Poitevin. Defended in August, 2006, at the University of Illinois at Urbana-Champaign, 76 pages. This thesis treats some model-theoretic aspects of the Banach lattices known as "Nakano spaces", which are generalizations of Lp spaces, and their expansions obtained by adjoining the "convex modular" as an additional predicate. In a loose sense, these are Lp spaces in which p is allowed to vary randomly (with respect to a given measure space) over a compact subset K of real numbers >=1. When the measure space is required to be atomless and the set of values of p that occur essentially is required to be equal to K, it is shown that the corresponding theory in continuous logic is complete and stable, and it admits quantifier elimination when the convex modular is adjoined as a predicate. (Stability is explicitly proved in the thesis only when inf(K)>1; however, Poitevin's convexification technique easily yields stability even when inf(K)=1.)

See also Modular functionals and perturbations of Nakano spaces by Itai Ben Yaacov (Journal of Logic and Analysis 1 (2009), 1--42) in which some questions are answered that were left open in Poitevin's thesis. In particular, it is shown in Ben Yaacov's paper that in any Nakano Banach lattice, the modular is a definable predicate in the sense of continuous first order logic. Furthermore, any Nakano space is stable, even when inf(K) = 1.
Fraisse Theory for Metric Structures; PhD thesis of Konstantinos Schoretsanitis. Defended in November, 2007, at the University of Illinois at Urbana-Champaign, 81 pages. This thesis takes some first steps toward developing analogues of results of Fraisse in the setting of continuous first-order logic. Let L be a continuous signature for bounded metric structures that has a finite number of predicate symbols and constants, but no function symbols. Results are proved in this thesis that characterize separably categorical L-structures whose theories admits quantifier elimination using properties of the category of their finite substructures (with embeddings as the morphisms). In the metric setting these characterizations have a fundamental metric character that has no counterpart in the results of Fraisse.
Model Theory of R-trees; PhD thesis of Sylvia Carlisle. Defended in May, 2009, at the University of Illinois at Urbana-Champaign, 93 pages. This thesis treats the theory of R-trees as metric structures in the setting of continuous first-order logic. This theory has a model companion, the theory of "richly branching" R-trees; the model companion theory has quantifier elimination, is complete, and is stable (but not superstable and not categorical in any cardinality). Most of the results in this thesis concern the model theory of isometries of R-trees. This divides between hyperbolic isometries and elliptic ones, with each basic theory being easily axiomatizable in continuous logic. It turns out that each of these theories has a model companion and the complete theories extending the model companions are also very well behaved from the model-theoretic point of view; in particular, they are stable and the independence relation for each such complete theory is characterized in the thesis in a natural way using familiar properties of R-trees and their isometries.