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COLLOQUIUM Title: Reverse VC calculations: using statistics in model theory Dr. David Lippel Haverford College, Haverford, PA Abstract: Let F be a family of sets; for example, F is the family of all triangles in the plane, or the family of all spheres in space (or, more exotically, a semi-algebraic family in p-adic n-space). The Vapnik-Chervonenkis (VC) dimension of F is a measure of the combinatorial complexity of F. Once you know the VC dimension of F, theorems from computational geometry, like the Epsilon-Net Theorem, give nice geometric consequences for F. After introducing VC dimension and the Epsilon-Net Theorem, I will discuss a statistical strategy for reversing the flow of information in this theorem. Instead of starting with knowledge of the VC dimension, we merely hypothesize "dimension=d" for some value d. Then, we observe the geometric behavior of F using computer experiments and compare the observed behavior with the behavior that is predicted by the theorems (under the hypothesis "dimension=d"). If our observed results have sufficiently low probability (conditioned on "dimension=d"), then we can reject the hypothesis "dimension=d" with a high degree of confidence. Ultimately, we hope to use such methods to shed light on conjectures about VC dimension (more properly, VC density) in both the reals and the p-adics. This project is joint work with Deirdre Haskell and Nigel PynnCoates. Department of Mathematics Thursday, February 28th, 2013 3:45 p.m. 204 Morgan Hall Refreshments will be served at 3:30 p.m.

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