# PMASS Colloquia 2012

 Amie Wilkinson, University of Chicago Thursday, March 22 2:30pm The first digit ABSTRACT Write down a list of numbers and look at their first digits. Is there a pattern? I will discuss methods from dynamical systems that can be used to understand the first digits of numbers encountered in everyday life.
 Vladimir Retakh, Rutgers University Thursday, April 19 2:30pm Commutative and Noncommutative Symmetric Functions ABSTRACT The classical theory of commutative symmetric functions started a long  time ago when people realized the connection between coefficients of  polynomial equations and their roots. Nowadays, symmetric functions  appear in almost every area of mathematics and the theory continues to  grow. The theory of noncommutative symmetric functions is much younger but but  we can already count a number of interesting results and applications in  this direction. The talk is dedicated to basic notions and connections of the both  theories of symmetric functions.
 John Roe, The Pennsylvania State University Thursday, April 5 2:30pm Being right more often can make you more wrong ABSTRACT Suppose that we know the values of a function f(x) at (n+1) different x-values, say in the interval [0,1]. A polynomial function p(x), of degree n, can be put through these (n+1) points and it is natural to use p as an approximation to f for other x-values also. This idea, called "polynomial collocation", is one of the oldest in numerical analysis.  I'll explain some of the techniques that were used in the B.C. (=before computer) era to carry out these calculations using pencil and paper. Then, I'll show you a surprising fact: as n increases (so that the number of points at which the approximation is "right" gets larger), the difference between p(x) and f(x) for other x-values can increase without bound. In brief, being right more often can make you more wrong.
 Mark Levi, The Pennsylvania State University Thursday, January 19 2:30pm Physical discoveries and proofs of mathematical theorems ABSTRACT Physics often provides mathematics not only with a problem, but also with the idea of a solution.  Some calculus problems can be solved more quickly without calculus, by using physics instead.  Quite a few theorems which may seem somewhat mysterious become completely obvious when given a proper physical incarnation.  This is the case for some “elementary” theorems (the Pythagorean Theorem, Pappus' theorems, some trig identities, Euler's formula V-E+F=2, and more)  and for some less elementary ones: Green's theorem, the Riemann Mapping Theorem, Noether's theorem on conserved quantities,  and more (no familiarity with any of these is assumed). I will describe a miscellaneous sampling of problems according to the audience's preferences.
 Jana Rodriguez Hertz, IMERL, Uruguay Thursday, February 9 2:30pm Lake fish population estimates and mixing properties ABSTRACT A mark-and-recapture method of lake fish population estimate consists in collecting and tagging a random sample of, for instance, 1000 fish, and counting the tagged fish in a second random capture, after releasing and allowing a period of mixing. A number of, for instance, 10 tagged fish would give an estimate of 100,000 fish in the lake. Is this a good estimate? We will see how this depends on the way the fish get mixed.
 Jan Reimann, The Pennsylvania State University Thursday, February 23 2:30pm The Continuum Hypothesis ABSTRACT In the late 19th century the German mathematician Georg Cantor tried  to show that every uncountable subset of the real numbers can be  mapped bijectively onto the real line. He was unable to prove this,  and the question became the Continuum Hypothesis (CH). It was the  first question on Hilbert's famous problem list of 1900. Seminal works  by Gödel and Cohen showed that CH can neither be proved nor disproved  from Zermelo-Fraenkel set theory (ZF), a basic axiom system for sets  that captures most of modern mathematics. In other words, CH is  independent of ZF.  In this talk I will sketch the history of the Continuum Hypothesis,  how it influenced the development of logic and set theory in the 20th  century, and I will outline how one can show that a statement is  independent of ZF.
 Eugene Wayne, Boston University Thursday, March 15 2:30pm The Navier-Stokes equations ABSTRACT We are surrounded by fluids in motion. As a consequence, they  were among the first physical systems that physicists and applied  mathematicians tried to model and the currently accepted mathematical  model for fluid motion, the Navier-Stokes equations, are more than 150  years old. As a consequence, one might assume that all the interesting  mathematical questions about these equations have been answered. However,  when the Clay Mathematics Foundation made a list of seven "Millennium  Prize Problems" (with a one million dollar prize for their solution),  one of the seven was to show that the three-dimensional Navier-Stokes  equations has a smooth solution. I will explain the physical origin  of these equations, some of the surprising phenomena that appear  in these equations, and why they are so hard to solve.