C. Ward Henson
Department of Mathematics
University of Illinois at
1409 W. Green Street, Urbana, Illinois 61801-2975 USA.
professional email: cwhenson(at)illinois(dot)edu -- new in early
w-henson(at)illinois(dot)edu and henson(at)math(dot)uiuc(dot)edu
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
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:
Berenstein, Henson, Usvyatsov; Model theory for metric
structures, Section 9, published in 2008.
Spectral gap and definability, section 2; preprint
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.
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
- 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
- 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.
on continuous first-order logic and the model theory of metric
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.
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.
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.
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
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.