Slides
for Henson's talk in the UIUC Logic Seminar on November 4, 2014,
are linked here. The
title is Generic orbits and type isolation in the Gurarij
space and the talk is a presentation of some of the content
of a preprint with the same name, written jointly with Itai Ben
Yaacov. It has been known for awhile that in
continuous model theory the Gurarij space is a Fraisse limit; it's
theory is separably categorical and has quantifier
elimination. We have written clear and simple proofs of all
those basic facts, and also have been studying the type spaces for
this theory, over an arbitrary set of parameters, which we may as
well take to be a separable Banach space E. Our main
achievement is to completely characterize isolated types in the type
spaces over E, using tools from convex analysis. This lets us
derive a lot of information about the situation in which the set of
isolated types is dense (for the logic topology) and hence there is
an embedding T of E into the Gurarij space G such that the structure
(G,T(e):e in E) is atomic; this is the same as saying there is a
generic orbit in the Polish space of all such embeddings, under the
action of Aut(G). For example this happens for any E of
dimension <= 3, for any finite dimensional E that is smooth or
polyhedral, but not for all E -- we give an E of dimension 4 such
that the isolated types over E are not dense, and we show that over
many familiar infinite dimensional spaces E, there are no isolated
types except for the obvious ones (coming from elements of E).
The tools from convex analysis that we develop should help answer
several open questions about the Gurarij space, such as: how complex
is the space of orbits of the action of Aut(G) on the unit sphere of
G? (We can derive a new result, that there are infinitely many
orbits, but this is far from giving the final story.)
Slides for Henson's talk at the Midwest Model Theory Day on
October 28, 2014, at UIC are linked here.
The title is Uncountably Categorical Banach Space Structures
and the main new results have to do with examples of uncountably
categorical Banach spaces that have been constructed/verified by
Henson in joint work with Yves Raynaud, Univ. of Paris 6.
Similar talks were given earlier this year in Lyon, France, and at
UCLA. In the background is a weak form of the
Baldwin-Lachlan theorems 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 Henson's talks in a mini course at the Hausdorff
Institute for Mathematics, part of the University of Bonn,
Germany, are linked here. The talks were given Oct. 17, 21,
and 23, 2013, for two hours each, under the general title Continuous
first-order model theory for metric structures. This
mini course was offered in the framework of a program on Universality
and Homogeneity organized at HIM during the Fall, 2013, by
Alekos Kechris, Katrin Tent, and Anatoly Vershik. (For
complete information about this program, browse the information
given here.)
The slides are here: Lecture
1 (Oct 17); Lecture
2 (Oct 21); Lecture
3 (Oct 23). (These are pdf files of beamer
slides.) Henson welcomes any comments or questions; they
should be directed to him at the email address given above.
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.
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 Analysis1
(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.
Recent teaching by CWH:
Spring 2010:
Math 414; Mathematical Logic, basic
undergraduate course in this area.
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).