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

Updated slides from a 2019 talk on

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

Slides for a talk given on Oct. 2, 2018, in Notre Dame's logic seminar, entitled

Slides for a talk in the logic seminar at Notre Dame on April 18, 2017 are linked here. The title is

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

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

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.

*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*:(born November 8, 1997);

Emma Jae Wilson(born March 25, 2000);

Grace Demena Wilson(born February 10, 2001);

Noah Riley Alonzo

Sophia deMena Alonzo (born September 28, 2003).

This page was last modified on April 23, 2020.