Thur, Dec 26 George Andrews (Dept. Chair & Evan Pugh Professor, Penn State) THE MAN WHO LOVED NUMBERS, Biography of Ramanujan
Thur, Oct 3 Mark Levi (Rensselaer Polytechnic Institute) STABILITY OF THE INVERTED PENDULUM AND TOPOLOGY OF THE SYMPLECTIC GROUP ABSTRACT: We give a physical demonstration, an explanation and a proof of stability of the inverted pendulum whose suspension point undergoes periodic osillations. The insight into the physical problem is gained via the topology of the group of symplectic 2×2 matrices.
Thur, Oct 31 Yu M. Suhov (Univ. of Cambridge, UK) PLAYING BILLIARD WITH A HELP OF MATHEMATICS
Thur, Nov 7 Stephen Simpson (Penn State) UNPROVABLE THEOREMS AND FAST-GROWING FUNCTIONS ABSTRACT: We comment on the role of the infinite in mathematics (Hilbert's Program). We exposit some combinatorial results whose statements are strictly finitary, but which cannot be proved without the strong use of infinite sets. The combinatorial results involve (1) coloring of finite sets, (2) embedding of finite trees, and (3) exponential notation.
Tue, Nov 12 Doug Arnold (Penn State) CONNECTING THE DOTS: THE THEORY AND PRACTICE OF INTERPOLATION ABSTRACT: If you know the value of a function at only a handful of points, what is the best way to guess to the function's value elsewhere? In other words: given a few dots on a graph, how should you connect them? This seemingly simple question inspired the rich subject known as interpolation theory. In this talk, which will be extensively illustrated with computer examples, I will survey some of the lovely, and often deep, mathematical results of this theory. We will mostly tour the classical world of polynomial interpolation, but will end with an excursion to the more modern land of piecewise polynomial interpolation and finite elements, and glimpse an application to the simulation of colliding black holes. THE COMPUTER RELATED MATERIALS for the lecture are available here
Thur, Nov 21 Gregory Swiatek (Penn State) DYNAMICS OF THE LOGISTIC FAMILY ABSTRACT: The logistic family is given by the formula f(x)=ax(1-x), 0≤x≤1, where a is a parameter in (o,4}. This is one of the most extensively researched of all examples in the dynamical systems theory. There are a few reasons for that. The family is believed to model certain phenomena in natural sciences or economics. For mathematicians, it is a very simple system that already displays a wide range of behavior, from purely deterministic to complete stochastic. For this reason the logistic family became a challenge for mathematicians: what can we rigorously prove about things that numerical experiments show, and applied scientists believe, in the logistic family? Paradoxically, the simplicity of the formula led to the use of heavy-duty tools of analysis in trying to solve these problems. In the talk, we will discuss various modes of behavior in the logistic family. This will lead us to some simply stated but very difficult to answer questions about the logistic family.