Speaker: Dr. David S. Gunderson. Professor of Mathematics, Department of Mathematics, University of Manitoba
This talk is an invitation to infinite Ramsey theory,
accessible to most mathematicians. Most combinatorists are familiar with Ramsey theory regarding finite structures, and many are aware of some infinitary techniques often used to solve Ramsey questions in the finite, for example, ultrafilters, harmonic analysis, and ergodic
techniques. However, it seems that few combinatorists are familiar with much infinite Ramsey theory. On the other hand, topologists, analysts, and set theorists seem to regularly use Ramsey theory in the infinite, but it seems that only a few basic theorems find application.
I survey some infinite Ramsey-type theorems (with few or no proofs) and hope to reveal some surprises. One surprise to me is that, often,
topology is required to refine an infinite Ramsey-type statement before finding a proof for that statement. My expertise is not infinite Ramsey
theory, and I claim no expertise in topology, but I hope to bring the audience to the point where it is clear that topology might help, or even be required, to further advance the field of Ramsey theory.
For those not familiar with Ramsey theory, a typical theorem has the form: for any r (number of colours), H (small structure or set) and G
(medium), there exists a (large) F so that for any r-colouring of the H-substructures of F, there exists a G-substructure in F all of whose H-substructures are monochromatic. For example, the pigeonhole principle is such a theorem (where H is a single vertex). In Ramsey’s original theorem, r is finite, H, G, and F are simply sets, where H is finite, and G is finite or countably infinite.
About the speaker:
David Gunderson received his BSc and MSc at U of Calgary, and received his PhD from Emory (Atlanta), supervised by Vojtech Rodl. He has had post doctoral positions in Bielefeld (under Walter Deuber), Howard University (under Neil Hindman), and McMaster (under Alex Rosa). Presently, he is an assistant professor at U of Manitoba. He has given
numerous invited lectures across North America and Europe.
Dr. Gundarson’s interests are primarily in combinatorics, particularly Ramsey theory, as well as extremal combinatorics and finite geometry. As
a hobby, he makes mathematical models from wood, with a major display at UofM. (see
http://home.cc.umanitoba.ca/%7Egunderso/pages/other_pages/mathematical_models.htm).
If you are planning to attend this seminar, please contact a SHARCNET staff member at your local institution as AG rooms will only be opened for these seminars where someone has indicated they plan to attend.
