Abstracts

Lectures

• Title: Impossible, Unbelievable... but True Results of Mathematics
  Speaker:Stan Wagon
  Abstract: Mathematics is full of surprising results, both concrete and abstract. Modern computing tools allow us to visualize and confirm many surprising things. In this talk I will give a sampling of several that have made a strong impression on me, such as:
  
  A drill that can be driven in a standard drill press and cut out an exact square hole. A bicycle with square wheels that one can ride perfectly smoothly. The Banach-Tarski Paradox: A ball can be decomposed into pieces and rearranged into two balls of the same size. A non-straight unicycle track that a bike can follow. How playing two losing games in a certain combination leads to a winning game. A bizarre property of Riemann's prime-counting function R(x). Surprises in Brownian motion and sparse linear algebra from the SIAM 100-digit challenge.
  
  And a cake puzzle that is easy for anyone to understand, but which causes professional mathematicians to almost always give the wrong answer with complete confidence.
  
  A round cake has icing on the top but not the bottom. Cut out a piece in the usual shape (sector of a circle with vertex at the center), pull it out, turn it upside down, and replace it in the cake to restore roundness. Do the same with the next piece; i.e., take a second piece with the same vertex angle, but rotated counterclockwise from the first one so that one boundary edge coincides with a boundary of the first piece. Remove, flip, and replace it. Keep doing this in a counterclockwise direction. The diagram shows the pieces involved in the first two moves when the central angle is 90°. For some central angles what happens is clear. For the illustrated 90° angle, eight moves brings all the icing back to the top. If the angle is 180°, it takes four moves to return to the initial state. Suppose the central angle is 181°. When does all the icing first return to the top? What if the angle is 1 radian?
  
  
• Title: Sage: Open Source Mathematical Software
  Speaker:William Stein
  Abstract: In January 2005, I started the free open source mathematical software project Sage (see http://sagemath.org/). It has since grew dramatically, with well over 100 developers and many thousands of users. In 2007, Sage was awarded first prize in the Trophees du Libre for scientific software. This talk will be explain the history and motivation behind this project, where it is headed, and demo how Sage is useful to you as a mathematician.
  
  
• Title: The Lost Notebook of Ramanujan
  Speaker:George Andrews
  Abstract:In 1976 quite by accident, I stumbled across a collection of about 100 sheets of mathematics in Ramanujan's handwriting; they were stored in a box in the Trinity College Library in Cambridge. I titled this collection "Ramanujan's Lost Notebook" to distinguish it from the famous notebooks that he had prepared earlier in his life. On and off for the past 32 years, I have studied these wild and confusing pages. Some of the weirder results have yielded entirely new lines of research. I will try to provide a gentle account of where these efforts have led. I hope to include some discussion of recent efforts to present Ramanujan's life in film, theatre, and opera.
  
  
• Title: A Computational View of the Four-Color Problem for Planar Maps
  Speaker:Stan Wagon
  Abstract: The four-color theorem is true. But what is the best way to four-color a planar graph? By implementing several algorithms one can learn good ways of doing it so that complicated planar graphs can be 4-colored. Since the problem originated as a map problem, it is natural to color maps, and that adds some complexity to the issue. The talk will give an overview of these algorithms and discuss some applications to real-world coloring problems as well as to related unsolved coloring problems.

Minicourses

• Title: Introduction to Mathematica for Mathematicians
  Instructor:Stan Wagon
  Abstract:A survey of the basic methodology of using Mathematica: plotting, 3-dimensional visualizations, numerical computing, creating dynamic manipulations, using data sets and advanced algorithms.
  
  
• Title: The Theory of Partitions
  Instructor:George Andrews
  Abstract:This will be an elementary course requiring at most familiarity with infinite series. The object will be to provide enough background so that the Polya Lecture on the following day will be placed in historical context. Partitions are representations of positive integers as unordered sums of positive integers. The minicourse will begin with the discoveries of Euler. It will mix the combinatorial aspects with the study of generating functions. We hope to include the Rogers-Ramanujan identities. These identities have a marvellous story behind them, and contain wonderful surprises.
  
  
• Title: Using Sage to Spice up your Undergraduate Courses
  Instructor:William Stein
  Abstract:Sage (http://sagemath.org) is an open source mathematical software project that I started in 2005, which has around 50 active developers and funding from NSF, Google, Microsoft, and many other organizations. I will given an overview of how easy it is to install Sage, how many goodies come with the installation, and some of the benefits of open source. I will then demo many of the interesting capabilities of Sage in the context of undergraduate mathematics education. I'll talk about how to use R with Sage for introductory statistics, how to plot 2d and 3d functions, compute limits, derivatives, and integrals, and do linear algebra, group theory, and discrete mathematics. I'll also explain how to share online Sage notebooks, create interactive web-based demos of mathematics, and setup your own sage server like the one at http://sagenb.org.