In this talk, I will give some classical examples of Von Neumann Regular Rings (VNRR) and discuss some interesting properties that such rings have. In particular, I will give a sufficient condition for a VNRR to be a division ring. Undergraduates with some background in Algebra are encouraged to attend.
Outreach activities are a fun, rewarding way to add some spark to your resume during graduate school. We will present a variety of ways that you can get involved in outreach activities with local high school students through a developing program coordinated by graduate students in the department. Two types of outreach activities, the development and presentation of weekday mathematics workshops for high school students and "special guest" visits to local calculus classes, will be discussed. Students with VIGRE support are especially encouraged to get involved this year. Come and learn how you can get involved!
This is partly designed as an introduction to some upcoming talks I will give in the geometry seminar, but it should also be entertaining all by itself. I will discuss "Kirby calculus", a technique for drawing pictures of 4-manifolds using knots and links; this is old-fashioned topology (no PDE's, no moduli spaces, no connections on weird bundles) but it continues to be a useful tool in modern work on 4-manifolds. I will also talk about how to add a little extra structure and use Kirby calculus to think about symplectic 4-manifolds.
One aspect of modern number theory is interactions among various areas of mathematics: algebra, analysis, geometry, etc. In this talk, I will illustrate this aspect by discussing a relation between the Birch and Swinnerton-Dyer conjecture (one of the most fascinating conjectures in modern number theory, with an award of a million dollars attached to it) and the congruent number problem (an old problem systematically studied by Arab Scholars of the 10th century). No prior knowledge of number theory is assumed.
Attached to each genus 1 curve C, defined over the rationals Q, is its Jacobian, J(C), which is easily described abstractly as equivalence classes of certain divisors on C, but turns out to admit the structure of an elliptic curve over Q, and hence *must* be representable as an explicit Weierstrass equation y^2 = f(x). Given equations for C, how does one find a Weierstrass equation for J(C)? Surprisingly, results from classical invariant theory of the late 1800s answer this very question in select cases.
Rather than presenting a single subject, this talk is intended to be a spicy gumbo of set and category theories. I wish to give a brief intuitive introduction to category theory, or as some might say (but probably wouldn't), the theory of arrows. Then, by way of a few examples, I will show how we can quickly run into some problems with a naive set theory. After discussing a few remedies, I will give a few more examples of the categorical perspective -- such as how to make sense of/generalize a topology, a set, the natural numbers, and propositional calculus. Hopefully in the end we will have a sense of how we can take category theory instead of set theory as a fundamental notion in mathematics.
In this talk I will discuss a character theoretic method for attempting to prove non-generation of a group G from a finite number of group elements g1,...,gr, called Scott's Criterion. After giving a brief proof of the result, I will discuss some examples and how this may be helpful in certain branches of mathematics.
Consider the following system of polynomial equations:
If I gave you a manifold
with bumps and wrimples all about
Would you ever claim so bold
the eigenfunctions figured out?
"Symmetry, you silly Jeff"
is what a few would plainly state.
But that is like a seafood chef
with only beef upon the plate.
So in this talk we'll sit and chew
and I'll define the basica.
In fifty minutes, plus a few,
the frumious generica.
Topoi are categories which possess certain characteristic properties of the category of sets.
"The general mathematician, who regards category
theory as `generalized abstract nonsense,' tends
to regard topos theory as generalized abstract
category theory."
-- P. Johnstone (Topos Theory, 1977)
However, Johnstone goes on to say that the true value of topos theory is
as a tool for understanding concepts in several areas of mathematics, and
providing such concepts with concrete structure. In this talk, we present
an introduction to topoi, including definitions and examples, in an
attempt to defend Johnstone's point of view.
Cable theory has been very useful in computational neuro-physiology. Equivalent circuits have been used to model and study conduction of signals and elicit physiologically relevant properties of nerve cells. I will describe some anatomical and physiological features of neurons and go through the derivation of the cable equation in this context.
This past summer I was calculating Brauer character values to identify the character in GAP which corresponds to a particular irreducible matrix representation. During this process I caused an error in GAP which I eventually worked around by using Conway polynomials. In this talk I will define all the big words above and discuss where this error occurred and how Conway polynomials fixed the problem.