Applied Probability Summary
Monday - Friday, July 2 - 6, 2007
The first week of the Applied Probability group was a study in contrast. David Levin is leading the Undergraduate Faculty Program through an introduction of Mixing Times, building on his work with Markov Chains. Of course, that presupposes a knowledge of Markov Chains -- so while the first hour is interesting, the second hour, when we meet on our own, allows us to go back over the highly theoretical and abstract concepts and put them through more practical examples.
You know you're in a Markov Chain when the only thing that matters is what state you were in immediately previous and not what happened before.
Graph theory isn't just for bridges in Konigsberg - they can also be used for weather prediction (provided you have the appropriate probability distribution).
Irreducible and Aperiodic situations. The first term means you can get anywhere you want, the second term means there are no set patterns in your ways to get there.
Day 3 was our best yet. We looked at a very simple graph and determined the transition matrix and equilibrium row vector. Then we got bold and made the graph a multigraph (more then one edge between two vertices). Then, we got bolder and made the graph directional. Then we ran out of time (fortunately, because we were in over our heads!)
The week wrapped up with Day 4; we tried to make heads or tails out of the idea of couplings. We're still not sure we're comfortable with our understanding but we're going back in next week in an effort to get it solidified.
The group thus far has been challenged by the material but with Darryl's calm guidance and the group's enthusiasm and willingness to share we're making sense out of it.
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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.