# Geneseo Mathematics Colloquium Schedule

## Spring 2009

Thursday, February 5, 4:00 - 4:50 pm
Newton 203
Mark Steinberger, University at Albany

### Topological Equivalence of Matrices

Two n x n real matrices A and B are said to be topologically similar if there is a continuous, 1-1, onto function f: Rn → Rn whose inverse is continuous, such that  f(Ax)=Bf(x) for all x in Rn. If f were the function induced by a matrix P, i.e., if f(x)=Px for all x in Rn, then P must  be invertible and PAP-1=B, so A and B are similar in the usual sense. Similarity of matrices can be thought of as saying that the transformations induced by A and B differ by a linear change of variables. So topological similarity means that A and B differ by a topological change of variables.

Thursday. February 26, 4:00 - 4:50 pm
Newton 204
Patrick Rault, SUNY Geneseo

### Arithmetic Geometry

How many rational points are on the unit circle? That is, how many points on the circle x2+y2=1 have coordinates which are rational numbers? In answering this question, we will find an algorithm that gives all Pythagorean Triples!

More generally, arithmetic geometry is the study of integral and rational points on curves. In this talk, we will generalize each of the following concepts: fractions, discriminants of polynomials, and the aforementioned Pythagorean Triple method.

We will end with several related unsolved problems, and an upcoming research and travel funding opportunity to participate in a 2009-2010 research course. This small 3-credit course would be ideal for those who have taken Math 319 or 330 and are planning to attend graduate school.

Corequisites: Any students who have taken or are taking Elementary Linear Algebra (Math 233) should enjoy this talk.

Wednesday, March 4, 3:00 - 3:50 pm
Newton 203
Bronlyn Wassink, Utica College (SUNY Geneseo Alumna)

### Functions, Rubber Bands, and Trees

There is a group of special functions, Thompson's Group F, that can be represented in many different ways. After showing exactly which functions are in Thompson's Group, we will explore both the rubber band model and the tree pair model for this group. This exploration will involve defining these models, showing how to quickly and easily change from one model to the next, and how to compose two functions in this group using only trees. No background knowledge is required to understand this talk!!

Thursday, March 26, 4:00 - 4:50 pm
Newton 204
Michael Starbird, University of Texas

### To Infinity and Beyond

Infinity is big. For thousands of years, people also thought it was incomprehensible--an idea so vast that understanding it was beyond the scope of people's finite minds. But a child's method of sharing "one for me, one for you", an Infinite Inn, a barrel containing infinitely many Ping-Pong balls, and a game called Dodge Ball combine to take us to infinity. And beyond.

Wednesday, April 1, 3:00 - 3:50
Newton 203
Vijay Sookdeo, University of Rochester

### The Exotic World of p-Adic Numbers

Under the usual notion of an absolute value, we may "complete" the set of rational numbers (fractions) to obtain all the real numbers. In 1902, Kurt Hensel found different ways of taking absolute values on the rational numbers, and thereby discovered the "p-adic numbers". This talk will introduce some of the bizarre and wonderful properties in this strange new world. Among them are: all triangles are isosceles, every point in a disk is the center, and two disks can only (non-trivially) intersect in one way. We will also sketch the proof of an important theorem concerning recurrence relations (the Skolem-Mahler-Lech Theorem) which exploits the exotic features of p-adic numbers.

Thursday, April 23, 4:00 - 4:50 pm
Newton 204
Keary Howard and Friends, SUNY Fredonia

### Teachers' Masters Capstone Projects in Secondary and College Mathematics

Does cash money motivate in a college math classroom? What is the 'right' amount of high school geometry homework? Can your iPod improve math performance? Are ninth graders really faster than a calculator? Join us as we attempt to explore these questions in mathematics education from a quantitative perspective.

Thursday, April 30, 4:00 - 4:50 pm
Newton 204
Scott Russell, SUNY Geneseo Computer Science

### Beyond Confidentiality: Cryptographic Applications of Homomorphic Encryption Schemes

Most people are aware that encryption schemes such as RSA and DES are useful tools for protecting the privacy of confidential information, e.g. your credit card or bank account number. Homomorphic encryption schemes provide additional algebraic properties beyond those necessary for confidentiality. Consequently, these schemes are oft-used building blocks in solutions to a variety of other cryptographic problems. We will briefly explain the exclusive-or-homomorphic Golwasser-Micali encryption scheme whose security is derived from the Quadratic Residuosity Assumption. Then we'll demonstrate how to use Golwasser-Micali encryption to construct solutions to a couple of cryptographic problems.