Research - Algebra and Number Theory
Number theory researchers at UVM enjoy participation in the stimulating Québec-Vermont Number Theory Seminar, an extremely active and high-quality gathering of mathematicians from several institutions in Quebec and Vermont that has run since 1984. At least twice a month, a seminar day will present two or three speakers discussing recent work. Often these are top researchers describing cutting-edge results of major significance. Outside of the lectures seminar participants talk informally and develop collaborations. In this arena, Professors Dummit, Foote and Sands of UVM have collaborated with R. Murty and J. Labute of McGill, H. Kisilevsky, K. Murty, and X. Roblot of Concordia, and L. Simons of St. Michael's College. For more details, see Québec-Vermont Number Theory Seminar.
David Dummit, Professor, works in algebraic number theory and arithmetic algebraic geometry; his principal area of research currently is in the area of Stark's Conjectures in algebraic number theory. Recently supervised graduate student theses have dealt with problems in modular forms and in computing units in totally real quintic fields. Dave is currently UVM's Associate Vice-President for Research and Graduate Education, and he remains active as a teacher and researcher.
Richard Foote, Professor, works in the area of group theory and its applications. His early research and continuing touchstone is the Classification of Finite Simple Groups. He has explored applications of group theory to problems in algebraic number theory, topology, and digital signal processing. He is currently working with Gagan Mirchandani in UVM's School of Engineering on discrete and algebraic notions of multiresolution analysis and wavelets, and the applications of group theory and algebra to image processing and edge detection. His interests also include Complex Systems and their relations to group theory and theoretical physics.
David Dummit and Richard Foote are coauthors of the internationally used graduate text Abstract Algebra, Third Edition, published by John Wiley & Sons, Inc.
Jonathan Sands, Professor, works in the area of algebraic number theory. His current research focuses on Stark's conjectures for abelian and non-abelian Galois extensions of number fields, and generalizations of these conjectures involving towers of number fields or algebraic K-theory. These conjectures link certain algebraic invariants to L-functions which are defined analytically. Together with a number of co-authors, he has also published theoretical and computational results on the subjects of Leopoldt's conjecture and Iwasawa theory.
John Voight, Assistant Professor, works in the area of number theory and arithmetic geometry, especially their algorithmic aspects. His current research is following three threads. First, he has investigated methods for computing zeta functions of varieties over finite fields using Dwork cohomology, with applications in coding theory and cryptography. Secondly, he has been working in the areas of quaternion algebras and Shimura curves, and he is currently developing computational tools in analogy with algebraic number fields and modular curves, respectively. Finally, John is interested in quadratic forms, and has published work answering the question of when two binary positive definite quadratic forms represent almost the same primes.