Department
Home Page
About Us:
• Contact Info.
• Faculty
• Staff
• Teaching Assistants
Academic Program:
• Admission
• Undergraduate
• Graduate
• Certificates
Courses:
• Course Description
• Class Web Pages
• Class Syllabi
Research:
• Colloquium
• Interdisciplinary Seminar
• Seminars
• Steve Goldman Lectures in Mathematical Physics
Links:
| • Completing Online GEP Courses |
| • Best Jobs |
| • Newsletter |
| • Math Lab |
| •Math Placement Test |
| • Careers |
| • American Mathematical Society |
| • Mathematical Association of America |
10/26/04 Colloquium
DR. YIANNIS VOURTSANIS
University of Central Florida
Department of Mathematics
University of Central Florida
Department of Mathematics
The product operation on structures and its consequences in developments in Mathematics
Abstract: The work represented in this lecture reflects works of Mostowski, Tarski, Los, Feferman, Vaught and myself. The product operation on arbitrary structures (semigroups, groups, rings, topological spaces, measure spaces, etc) will be considered both with respect to its general theory as well as its implications within virtually all mathematical spectrum. Applications of the product operation are well known in classical mathematics, such as, in Algebra (decomposability or representations of several classes of algebras (including groups), in Topology (the Tychonoff compactness theorem on product topologies) in Measure Theory (Fubini's product theorem) etc. In 20th century mathematical thought, the product operation, with an expanded expressive power with the use of filters and ultrafilters, has resulted in certain stunning applications such as in the proof of the powerful compactness theorem in mathematical logic. In the general theory, we give the structure of truth within a product structure as well as we present a product theoretic calculus on sentences, such as products, quotients, roots and projections of sentences, generalizing usual arithmetic. These have found a number of applications, including a new algorithm for the decidability of the theory of Boolean Algebras, first proved by Tarski. In addition, the above provide methods for a better understanding and perhaps eventually settling important currently open hypotheses in mathematics, such as the celebrated Continuum Hypothesis which, furthermore, has been proved to be independent of the axioms of Set Theory ZFC. Some of these methods will be also addressed in the lecture.