• Math 347H (D1H); MWF 11:00--11:50 in 141 Altgeld; Fundamental Mathematics.
• This is an honors section; students interested in the Honors Sequence in Mathematics should first take an honors section of Math 347.
  
• Topics include: techniques of proof, mathematical induction, binomial coefficients, rational and irrational numbers, the least upper bound axiom for real numbers, and a rigorous treatment of convergence of sequences and series.
  
• Students will regularly write proofs emphasizing precise reasoning and clear exposition.

• Math 441 (C14); MWF 10:00--10:50 in 141 Altgeld; Differential Equations.
• Basic course in ordinary differential equations; the treatment is more rigorous than that given in Math 285. 
  
• Topics include existence and uniqueness of solutions and the general theory of linear differential equations. Uniform convergence and the implicit function theorem are introduced. 
  
• All results, methods, and theorems are carefully proved in class, and students begin to learn how to approach rigorous mathematics and to write proofs.
  

• Model Theory for Metric Structures by Itai Ben Yaacov, Alexander Berenstein, C. Ward Henson, and Alexander Usvyatsov; 112 pages, to be published in a Newton Institute volume in the Lecture Notes series of the London Math. Society. This article gives an introduction to the continuous ([0,1]-valued) version of first-order logic and to model theory for metric structures, including such topics as type spaces, definability, imaginaries, quantifier elimination, separable categoricity, stability, and applications to Hilbert spaces, probability spaces, Lp spaces, and probability spaces with generic automorphisms. (Final version, in the typeseting format specified by the publisher. Last content revisions August 14, 2007; last typos fixed September 10, 2007.)
  
  
• The slides from my talk Nov. 21, 2006 at the University of Paris 6.
  
  
• The slides from my talk Oct. 6, 2007 in a special session on model theory at the AMS sectional meeting in Chicago.
  
  
• Overview of the AIM workshop on "Model Theory of Metric Structures" held September 18-22, 2006; by the organizers (Itai Ben Yaacov and C. Ward Henson).
  
  
• Some personal comments by C. Ward Henson discussing where continuous first-order logic came from and why it is the "right" version of first-order logic for application to the metric structures that arise in analysis, probability, and geometry.
  
  
• 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 proved in the thesis only when inf(K)>1.)  See also the preprint "Modular functionals and perturbations of Nakano spaces" by Itai Ben Yaacov in which some questions are answered that were left open in this thesis.  In particular, it is shown there 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.

• Spring 2008:

• Math 571 (G1); MWF 3:00--3:50 in 443 Altgeld; basic graduate course on model theory.

• Fall 2007:

• Math 424 (X1); MWF 12:00--12:50 in 345 Altgeld; Honors Real Analysis, first course in the Honors Sequence.
• Math 595 (CL); MWF 2:00--2:50 in 447 Altgeld; Continuous First-order Logic and Model Theory for Metric Structures (advanced topics graduate course).

• Spring 2007:

• Math 441; MWF 12:00--12:50 in 343 Altgeld; undergraduate course in ordinary differential equations.
• Math 571; MWF 2:00--2:50, in 443 Altgeld; basic graduate course on model theory.
  
  
• Fall 2006: sabbatical leave.
  

• Web pages of the Logic Program of the University of Illinois at Urbana-Champaign
  
• A brief statement of Henson's Research Interests.
  
• A few comments about expositions of Nonstandard Analysis.
• UIUCLogic -- email distribution list of people interested in news about activities of the logic program at the University of Illinois at Urbana-Champaign. (This file gives the names of those who receive these notices. Anyone interested in being added to this list should send email to henson(at)math.uiuc.edu.)
  
• Urbana-Champaign Logic Conference - May 21-25, 2003: Logic and Mathematics; Connections and Interactions.
  
  Photo album from the meeting. 
• Association for Symbolic Logic.

• 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).

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.

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.  (See the top of this page.)

For more information or copies of preprints and reprints, please contact Henson using the address information above.

Nonstandard Analysis:

Here is an introductory article by Henson: Foundations of Nonstandard Analysis: A Gentle Introduction to Nonstandard Extensions: this is meant to give a smooth and accessible introduction to the subject for all mathematicians, and is especially directed toward those who have no background (and perhaps no sympathy) for formal logic. The version currently posted is the final one (12/24/96). This paper can be retrieved in PDF or in PostScript format.

This article is the first chapter in a book published by Kluwer Academic Publishers: Nonstandard Analysis: Theory and Applications, edited by L. O. Arkeryd, N. J. Cutland, and C. W. Henson. This book came out of the NATO Advanced Study Institute on Nonstandard Analysis and its Applications which was organized in July, 1996, by Arkeryd, Cutland, and Henson. In its first four chapters, this book presents a careful and detailed introduction to the methodology of nonstandard analysis and the foundations of its use in analysis, topology, probability theory and stochastic analysis.The remaining eight chapters cover recent, more advanced applications in functional analysis, stochastic differential equations, mathematical physics, and mathematical finance theory. Further information about the contents of the book is given in the following documents (PostScript format): Table of Contents and Preface.

Unfortunately this book is available only at the list price (originally $183.00; now listed at the truly embarrassing price of $286.00); further information is available at the Springer web site. The Kluwer/NATO program to offer these books at a substantially reduced rate for classes or seminars does not exist any more. This editor can only offer his sincere apologies for the fact that the book is priced so far out of the reach of individuals. Certainly this limits the long-term effects of such an ASI program and largely negates the efforts of a number of instructors/authors to make their subject available to the mathematical community. Our acceptance of NATO support for this Advanced Study Institute required that we publish the book in this series with Kluwer, and we have no influence over their marketing and pricing decisions.

Association for Symbolic Logic

Henson was Publisher of the Association for Symbolic Logic until the end of 2004.  He was Secretary-Treasurer of the ASL from 1982 until the end of 2000. The ASL Secretary-Treasurer position is now held by Charles Steinhorn of Vassar College; the ASL business office can be contacted by email at asl@vassar.edu. You may also want to look at the main ASL Web page.

This page was last modified on May 23, 2008.