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Worldwide Lecture Seminar Series: Pablo Soberón REGISTER Description The Worldwide Lecture Seminar Series Presents Pablo Soberón - Northeastern University Friday, March 16, 2017 Coffee, tea, cookies: 3:30pm Talk: 4-5pm Worldwide Center of Mathematics -- Cambridge, MA, USA Abstract: During this talk we will discuss some robust variations of Tverberg’s theorem. The aim is to seek partitions of a finite set of points in R^d such that the convex hulls of the parts intersect, even if our set of points is going to be modified later on. Surprisingly, random partitions give sharp results. These are variations of Tverberg’s theorem which behave like weak epsilon-nets for convex sets.  

The first conception of this episode was to prove that the rationals are a dense within the reals, which is an algebraic proof showing that between any two real numbers, there is a rational number. This proof does not define the real numbers, and treats them as some empirical fact that you know; yet, once the real numbers are constructed, the proof is really trivial. The proof used in this episode utilizes an analytic definition of dense sets: if a set `A’ along with its limit points equals the `B’, then `A’ is a dense set within `B’. You will see that we construct the reals in such a way that the rationals are dense within the reals. But first, a little background. First, we construct the natural numbers using Peano’s Axioms, and the integers can be constructed many different ways from the natural numbers (think including additive inverses). From the integers, the rational numbers are all ratios of two integers. These ratios can be thought of as finite decimal expansions, and we will ...