Overview
Professor Hodel is currently working on a number of projects.
LOGIC, TOPOLOGY, MATROID THEORY: WHAT DO THEY HAVE IN COMMON? A quick answer: fundamental concepts in each of these branches of mathematics can be described in terms of a closure operator that satisfies certain axioms. These ideas are due to Tarski, Kuratowski, and Whitney respectively. We take a somewhat different approach and use the closely related notion of a nearness relation. This gives a somewhat more geometric approach to the ideas under discussion; numerous applications of this approach are obtained.
A THEORY OF CONVERGENCE BASED ON KAPPA-NETS. This theory is very similar to convergence based on nets, but the directed set in question is always the collection of all finite subsets of some infinite cardinal kappa. This restriction allows one to take advantage of the eligence of the filter approach while retaining the intuitive advantage of nets. An immdediate application: a simplified cryptomorphic description of a topology in terms of convergence. This theory also yields characterizations of various covering properties (for example, Lindelof spaces) in terms of "generalized" kappa nets that have cluster points.
INTRODUCTION TO MATHEMATRICAL LOGIC I am currently working on a second edition of my logic book.
Current Appointments & Affiliations
Associate Professor Emeritus of Mathematics
·
2008 - Present
Mathematics,
Trinity College of Arts & Sciences
Recent Publications
Book review: G. Tourlakis, Lectures in Logic and Set Theory (Volume 2), 592 pages.
Journal Article Review of Modern Logic · 2004 CiteReflection theorems for cardinal functions
Journal Article Topology and Its Applications · January 1, 2000 This paper is a systematic study of reflection theorems for cardinal functions. There are four sections: theory of reflection; reflection theorems for c, e, s, hd, hL, L, d, and nw; reflection theorems for χ, t, ψ, and psw (assuming compactness); reflectio ... Full text CiteBook review: The Mathematical Legacy of Edward Cech, edited by Katetov and Simon, 441 pages.
Journal Article Modern Logic · 1996 CiteRecent Grants
Duke University Noyce Scholarship - Phase II
Inst. Training Prgm or CMECo Investigator · Awarded by National Science Foundation · 2013 - 2020The Robert Noyce Fellows at Duke University
Inst. Training Prgm or CMECo-Principal Investigator · Awarded by National Science Foundation · 2005 - 2009View All Grants
Education, Training & Certifications
Duke University ·
1962
Ph.D.
Davidson College ·
1959
B.S.