When investigating geometric spaces, symmetric spaces form an interesting yet manageable class. One step more complex than spaces of constant curvature, these spaces are of considerable interest in physics. We shall present a couple of examples and a nice way of thinking about more general symmetric spaces.
A Lie group or algebraic group in positive characteristic has a naturally associated structure on its tangent space and the identity. Namely that of a lie algebra, g. From this lie algebra one can produce what is called a universal enveloping algebra, Ug. If the group is commutative this algebra is the same as the symmetric algebra, Sg, on the tangent space. Locally, there is only one commutative group of a fixed dimension, namely vector addition. If we call this the "trivial local group" we will see how an arbitrary universal enveloping algebra, Ug, occurs as a deformation of, Sg. Moreover, in studying such deformations we will see how one is naturally lead to studying "multiplicative Poisson structures" on Sg.
I'll present a relatively friendly introduction to the wonders and mysteries of ideal classes and the ideal class group, especially those of cyclotomic number fields. I'll discuss why people care about the properties of these groups and a process for finding ``annihilators'' for them using Gauss sums. Specifically, we'll see the relationship between ideal class groups, Fermat's last theorem and the recently-proven Catalan conjecture. Finally, time-permitting, I'll describe the theorems of the three annihilator-busters in the talk's title.
We will define an elliptic curve E, defined over the complex field, and its endomorphism ring Hom(E,E). We will discuss those elliptic curves with "special" endomorphisms. Time permitting, I'll share my own interest in studying these curves. This introductory talk is accessible to all graduate students.
For eons, graph theorists have studied the spectra of graphs (i.e., the eigenvalues of their adjacency matrices); more recently, number theorists have done the same with graph zeta functions. As so often happens in mathematics, these two constructs seem to be related. How much so? Using charm, proof, and free bagels, I will attempt to convey that in the instance of a regular graph, knowing one is precisely knowing the other.
I will discuss the existence of the thermodynamic limit and the correlation limits for the models in question.
We shall discuss a question fundamental to the very foundations of our subject, yet a question which so many mathematicians are more afraid of than things that go bump in the night. We shall present a history of some of the most commonly accepted schools of mathematics and how they originated (or at least my own opinion of how they came about). To finish, I will explain the reason why this question no longer gives me nighmares.
Jill Newby, subject specialist at the Science-Engineering library on campus, will speak on the kinds of services and resources that are available from the library. Find out how to most effectively navigate MathSciNet and other databases, keep up-to-date with the latest publications, navigate the Library website to locate books and journal articles, and request materials not available in the Library.
The motivation for our problem is to construct a model for an efficient communication network. This model could be the telephone network of a country, the wiring of big parallel computers, or the neuronal system of a human body. Starting from the design of communication networks, one passes to a problem about graphs which is solved by some fundamental results of number theory and graph theory. In particular, we will discuss some special types of graphs that are good models for an efficient communication network. Of primary concern are explicit constructions of expanders. In particular, we introduce the zig-zag product, a recently-developed method for efficiently constructing expanders.
Gauge field theory is a branch of mathematical physics that uses differential geometry and topology to describe the evolution of physical fields (like the electric and magnetic fields) which have internal degrees of freedom or symmetries. In this talk we will tour the framework of the theory in the simple case of electromagnetism. As an example we will compare and contrast the Coulomb electric field of a point charge to the Dirac magnetic monopole. In doing this we will encounter something called the Hopf fibration which gives a very interesting way to think about the 3-sphere. This talk should be accesible to anyone who is taking, or has taken the geometry/topology core course and wants to see some really neat pictures.
Elliptic curves have become increasingly important over the past two decades in cryptography. When implementing certain elliptic curve cryptosystems, it is important to be able to choose an elliptic curve with group order divisible by a large prime factor. In 1985, Schoof presented an algorithm for computing the number of points on an elliptic curve over finite fields. His algorithm ran much faster than existing point-counting algorithms and has become the basis of most current efficient schemes for point-counting.
Given an elliptic curve defined over the rationals, we try to find rational points on the curve. More generally we could look for points in E(K), the group of points on E with coordinates in some number field K. After reviewing the group structure of E/K, I'll discuss some known results and interesting conjectures. This material is fundamental for the non-specialist.
We will provide a brief introduction to the theory of formal languages, including a partial account of their classification. We will see how presentations of groups give rise to languages, and then give some results relating the characterstics of a group to the language it determines.
Have you been dying to put all that Algebraic Geometry knowledge to good use? To bad use? Continuing on this semester's Elliptic Curve theme, I'll describe how elliptic curve cryptography works. Which curves are secure? What are some of the known attacks? We'll see why elliptic curve point counting is so important to cryptographers. Time permitting, I'll discuss hyperelliptic cryptosystems and their security.
The goal of the talk will be to present a collection of fun results taken from ``elementary'' number theory using techniques from what some might call combinatorial number theory. In particular, we will see how to make crazy dice and perfect rulers using techniques available to college algebra students. This should be a fun and relaxing end-of-the-semester talk to take your mind away from elliptic curves and connections on Riemann surfaces and focus it squarely on important subjects like bagel-eating and coffee-drinking.