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Graduate Seminar
Jason Rute Carnegie Melllon University Title: Randomness and Computable Analysis Abstract: Probabilists often work with various infinite processes. One of the easiest processes to understand is a fair coin that is flipped infinitely often. While one would expect such a sequence of flips to have certain propertiessuch as an (asymptotically) equal number of 0's (tails) and 1's (heads)there are obvious counterexamples, for example the string of all tails (000...). Such counterexamples are somehow not "random" enough.One could naively define a "random" string as one which avoids all probability zero events, but such a definition is quickly seen to be vacuous. Instead, if we limit ourselves to "computable" probability zero events, such a definition is not only consistent, but also interesting and robust.In this talk I will talk about the basics of computable analysis, and at the end use it to define a type of algorithmic randomness called MartinLof randomness. Computable analysis is not just a means to an end; it is itself an interesting field that makes precise the computational concerns of much of modern and classical mathematics. As an example, I will give a computation interpretation of the notions of point, open set, continuous function, and null set in a variety of well known spaces.If time allows, I will then talk about some of the properties of MartinLof random sequences. Date: Tuesday, January 18, 2011 Time: 6:00 pm Location: Wean Hall 8220 Submitted by: Chris Almost 