Math Club

(Lecture) The World of p-adic Numbers

• Axel Turnquist
• May 11, 2012 at 4:30 PM
• Location: Savery 158

Abstract:

The world of p-adic numbers is considerably different to that upon which our intuition is built. In our everyday intuition, we have notions of metrics and norms by which we can acquire some meaning out of the "closeness" of two entities on a space S, by defining a function S->R. Our geometric intuitions do not work in the p-adic metric and as a result some strange and fascinating results emerge. Can an analogue to the historical construction of the complex numbers be made for p-adic numbers? Can we define analysis? The answer turns out to be yes.

Topics: Metrics and metric spaces, Norms, P-adic metric, Non-Archimedean norms, Ostrowski's Theorem, Algebraic closure, Constructing C, P-adic construction, P-adic numbers, integers, units, P-adic series, P-adic arithmetic, Hensel's Lemma, Building up Omega

There will be pizza and drinks provided during the talk. We look forward to seeing you here!

What are NURBS?

• Richard Fuhr
• February 10, 2012 at 4:30 PM
• Location: Savery 157

Abstract: What are NURBS? NURBS curves and surfaces are widely used in computer-aided design, in technical illustration software, and in other computer graphics applications. In the upcoming presentation, we will take a look at what factors motivated the use of NURBS what NURBS entities are why they are well-suited for computer applications what some of their mathematical properties are what practical challenges we have encountered As usual, there will be pizza and drinks provided. We look forward to seeing you here!

DeBruijn Sequences and Card Tricks - Rescheduled

• Monty McGovern
• January 27, 2012 at 4:30 PM
• Location: Smith 105

Abstract: Update: Monty's lecture has been postponed to Friday Jan. 27th because of the snow. This Friday the Math Club is pleased to host a lecture by Monty McGovern on DeBruijn Sequences and Card Tricks. We will be meeting in Savery 156 at 4:30. Here is Monty's abstract: This talk is mostly shamelessly lifted from my colleague Sara Billey. I will start with a cool card trick and gradually explain it how it works mathematically. The tools used along the way will range from graph theory to abstract algebra. As always, there will be free pizza. We hope you'll join us.

Lecture

• Prof. McGovern
• August 05, 2011 at 4:30 PM
• Location: Savery 155

Abstract: Once again I will give a talk on matching people up (this time not necessarily with other people). Instead of taking preferences into account, I will merely try to make as many acceptable matches as possible, subject to the usual rule that no two people get the same match. Along the way I will prove one of many max-min principles that pervade much of combinatorics and game theory.

The Birch and Swinnerton-Dyer Conjecture

• William Stein
• April 22, 2011 at 4:30 PM
• Location: Padelford C 401

Abstract: I will give an introduction to the Birch and Swinnerton-Dyer Conjecture, which is considered by many to be one of the most central open problems in number theory. The conjecture, which was discovered based on numerical computations in the 1960s, asserts that the rank of a certain finitely generated abelian group equals the order of vanishing of a certain complex analytic function at the point 1.

How to Have a Stable Marriage

• Prof. Monty McGovern
• February 25, 2011 at 4:30 PM
• Location: SMI 105

Abstract: This seemingly thorny social problem holds no terrors for a mathematician. I will show in fact how to pair up ALL the men and women in a society so as to avoid any elopements. I will then give a real-life application, to medical school graduates and the internship programs which are their next step.

A Gentle Introduction to Category Theory: Architecture of the Universe or Abstract Nonsense?

• Luke Wolcott
• November 19, 2010 at 4:30 PM
• Location: Smith 105

Abstract: Some people call mathematics the language of nature. Some mathematicians call category theory the language of mathematics. This broad theory attempts to capture the shape of all of mathematics, and is notoriously abstract. (Search "general abstract nonsense" on Wikipedia and category theory comes up!) But it is also remarkably simple. I'll give you a gentle introduction to this very important and powerful theory, and will show with lots of examples how simple and natural it is. At the same time, you'll get a taste of what higher mathematics is about. I'll also mention some of the recent applications of category theory to theoretical computer science.

Higher-dimensional Geometry

• Prof. Sándor Kovács
• October 29, 2010 at 4:30 PM
• Location: SMI 105

Abstract: We will explore how using higher dimensional spaces can help us understand the geometry of plane curves.

On Numbers and Games

• Prof. Monty McGovern
• October 22, 2010 at 4:30 PM
• Location: 105

Abstract: This talk is taken from J.H. Conway's wonderful book with the same title. I will introduce a large class of two-player games without chance, many of which unexpectedly turning out to be numbers (!) in disguise. Moreover, the numbers can be used to work out the best strategies for the games and determine who wins them with best play. Along the way we will meet not only all the real numbers we know and love, but rather exotic numbers like the square root of infinity cubed plus one.

One million monkeys and typewriters versus one random number

• Jim Gill
• October 08, 2010 at 4:30 PM
• Location: Smith 115

Abstract: It is often said that if you give a million monkeys a million typewriters and enough time, they will produce the complete works of Shakespeare. We will examine this claim. Actual data with actual monkeys will be presented, however only higher order primates will be present at the talk.

Convexity in the plane

• Prof. Scott Osborne
• May 14, 2010 at 4:30 PM
• Location: Smith 107

Abstract: A set in the plane (or 3-space, or...) is "convex" if, when it contains two points, it contains the entire line segment between them. The subject here will be convex subsets of the plane, with a few mild restrictions (e.g. "bounded"). Such sets have areas and perimeters, and a surprising result says that the perimeter is the integral of the width. Try this on a rectangle: The perimeter does not grow in coordination with the width; it jumps most when the rectangle is narrowist. I'll also talk about the "edge" characterizations of convexity, such as [Pun Alert!] "If you drive your car on a plain always turning left, and wind up where you started (including the direction you point), then the region in the plain enclosed by your path is convex."

The Story of Pseudodiagrams and Knot Games

• Prof. Allison Henrich
• April 30, 2010 at 4:30 PM
• Location: Smith 107

Abstract: In the mathematical study of knots, we consider a knot to be essentially a knotted piece of rope with its ends glued together. One knot is the same as another if you can pull, bend, tighten or stretch one knot to get the other. Rather than playing with rope, we usually draw pictures of knots to analyze their properties. The main question in knot theory, then, is the following: Given two diagrams of knots, how can we tell if they represent the same knot or two different knots? In our SMALL REU at Williams College last summer, my students and I studied objects called pseudodiagrams that are related to knot diagrams. A pseudodiagram is a diagram of a knot that may be missing some information about which strand is over and which strand is under at certain crossings. We spent the summer analyzing properties of pseudodiagrams, which gave us an idea for several types of games you might play with knots. Not only are these games fun to play, but they are related to the fundamental question of knot theory. In this talk, we will learn all about pseudodiagrams and play some games!

Fibonacci Numbers and Chinese Nim

• Prof. William Monty McGovern
• April 02, 2010 at 4:30 PM
• Location: Smith 107

Abstract: Almost nine centuries ago Leonardo of Pisa, better known as Fibonacci (meaning son of a dunce), wrote a book with an enormous number of problems, of which by far the most famous one involved the breeding of rabbits. It gave birth to the famous sequence 1,1,2,3,5,8,..., of which each term is the sum of the preceding two. I will briefly derive a formula for the n-th term of this sequence together with an unexpected application to a variation of Nim which will show that this sequence is but one piece of a huge jigsaw puzzle: the entire set of positive integers can be written as a disjoint union of Fibonacci-type sequences.