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.
professional email: cwhenson(at)illinois(dot)edu -- new in early 2018;
("old" addresses w-henson(at)illinois(dot)edu and henson(at)math(dot)uiuc(dot)edu will also
work; they are treated as aliases, and mail to either of them will be appropriately forwarded.)
personal email: cwhenson(at)gmail(dot)com
home address and phone: 2925 Lincoln Way, San Francisco, California 94122; +1-415-661-4559

Selected Slides from Recent Talks by Henson:

Slides for a talk given in November 2019 at Notre Dame and the University of Illinois (Urbana) on Model theory of R-trees and of ultrametric spaces. (Slides last revised on 11-14-2019.)

Updated slides from a 2019 talk on Ultraproducts as a tool in the model theory of metric structures, given in several places. (Slides last revised on 4-23-2020.) A subset of these slides were used for a 4-22-2020 zoom seminar "at" Cornell.)

This mostly expository talk lays out how understanding ultraproducts of members of a class C of metric L-structures can be an important practical tool for understanding the full class of models of the theory of C -- including not only the models themselves, but also the definable predicates and (especially) definable sets in those models, and often also toward getting an explicit set of axioms for that theory. Ultraproducts have proved to be much more important in the model theory of structures from functional analysis and geometry than in classical model theory of discrete, algebraic structures. There are several examples at the end, including an aspect of the speaker's program with Yves Raynaud to verify many new examples of uncountably categorical Banach spaces. The basic tools discussed here were developed in conversations with Bradd Hart and Isaac Goldbring. For basic background and some proofs of definability results, see:
             BenYaacov, Berenstein, Henson, Usvyatsov; Model theory for metric structures, Section 9, published in 2008.
             Goldbring, Spectral gap and definability, section 2; preprint 2018.
             Hart, slides for talks at ASL annual meeting and BIRS workshop in 2018.

Slides for a 2019 talk on Uncountably Categorical Banach Spaces, with some new examples (joint work with Yves Raynaud).

Slides for a talk given on Oct. 2, 2018, in Notre Dame's logic seminar, entitled "On the model theory of group actions on probability measure spaces". This is based on a collaboration with Alex Berenstein. The talk was also given in Urbana's model theory seminar on Sept. 28, 2018.

Slides for a talk in the logic seminar at Notre Dame on April 18, 2017 are linked here. The title is Uncountably Categoricity for Structures Based on Banach Spaces and the main new results have to do with examples of uncountably categorical Banach spaces that have been constructed/verified in joint work with Yves Raynaud, Univ. of Paris 6. A paper by Henson and Raynaud has been published in Commentationes Mathematicae and is available as arxiv:1606.03122. Talks were given at earlier stages of the project in Lyon, France, at UCLA, and at a Midwest Model Theory Day at UIC. In the background is a weak form of a theorem of Baldwin-Lachlan, for classical structures, that has been proved for Banach structures by Shelah and Usvyatsov, in which Hilbert space plays a role analogous to that of a strongly minimal set.

Slides for a talk given in a Special Session on Continuous Model Theory at the ASL Annual Meeting, March 20-23, 2017, at Boise State University, are linked here. The title is Continuous model theory, and the purpose was to provide some background for the other talks in that special session. The slides contain about twice the material that could be presented in the talk, and also give a short list of references at the end.

Slides for talks given in Paris (Analysis Seminar, Nov. 12, 2015) and in Berkeley (Model Theory Seminar, Nov. 18, 2015) are linked here. The title is Banach lattice methods for proving axiomatizability of Banach spaces. The talk discussed new methods based on a study of disjointness preserving linear isometries between Banach lattices, introduced by Raynaud, and a main theorem proved first by Raynaud and then strengthened by Henson. This is part of an ongoing collaboration.

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
The distribution list for seminar announcements and other information from the logic group in Urbana is now handled by an automated system. Interested people should contact Philipp Hieronymi (phierony(at)illinois(dot)edu)for information about how to sign up for these messages.

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 C*-algebras by Farah, Hart, Lupini, Robert, Tikuisis, Vignati, Winter, preprint (2016), 141 pages. arXiv:1602.08072v3.
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.
Grandchildren:
Emma Jae Wilson (born November 8, 1997);
Grace Demena Wilson (born March 25, 2000);
Noah Riley Alonzo (born February 10, 2001);
Sophia deMena Alonzo (born September 28, 2003).

This page was last modified on April 23, 2020.