Pascal's hexagon, Desargue's triangles and the inflection points of a cubic have very pretty configurations. Come learn a little about them. We'll start with a statement of Bezout's theorem and the basic definitions.
Associated to certain nonlinear PDEs there are infinitely many surfaces in Lie algebras.
We introduce a spectral problem on real n-space which has a periodic structure. This structure has an "inside" part and an "outside" part, which interact in a certain way according to our spectral problem. The large coupling limit will correspond, in some sense, to almost completely separating the two parts. The density of states is a way of measuring the spectrum of our problem and we use it to compare the large coupling limit with what one would get if the "inside" and the "outside" were truly non-interacting. In this talk, we will describe the problem from the beginning in more detail and discuss what is known about it, including some recent results by the speaker.
I will give an overview of the process of applying for an academic job and some tips for success.
Is transcendental? Are and algebraically independent? Don't you wish you could prove big important theorems without having to do lots of work? Or knowing lots of math? A new and interesting class of numbers called periods has recently become a hot topic in mathematical circles. In an attempt to mirror the classification of algebraic numbers via their degree, we introduce a period's dimension and see where it takes us. In particular, a conjecture from Schanuel links the study of periods with that of transcendental number theory, and we come away with a new conjecture concerning the structure of logs of algebraic numbers and their powers.
In the middle of the 19th Century, William Rowan Hamilton was searching for a consistent way to multiply vectors in a 3-dimensional space. On October 16th, 1843, Hamilton realized that this task required a 4-dimensional space with some peculiar properties. Thus were the quaternions born. In this talk I'll review some of the history of the quaternions and show how they provide a natural language for rotation of vectors and how they provide a good, concrete example to reference when studying algebra and geometry. I'll also show how the quaternions are used efficiently in applied fields such as computer graphics.
The product of the eigenvalues is one characterization of the determinant of an element of End(V), where V is a finite dimensional Hilbert space. If V is infinite dimensional the spectrum of a linear operator can much more complicated, but should the spectrum consist only of eigenvalues, their infinite product may not even converge. Amazingly, one can use a modified version of Riemann's zeta-function to make sense of the determinant of a particularly important operator, the Laplacian on a compact manifold. In this talk I'll give a survey of this fascinating idea and illustrate how it can be used to give mathematical sense to the path integral in a particular class of Quantum Field Theories. I'll endeavor to introduce all the relevant details, no prior knowledge of Functional Analysis or Quantum Field Theory will be assumed.
Keywords: Quantum Field Theory, Zeta-function regularization, Heat trace, Asymptotics
The partition function of random matrices with the Gaussian distribution contains combinatoric information about specific families of labelled graphs. This information is encoded by the coefficients of the asymptotic expansion of this function for large matrices. I will present some results on the asymptotic expansion of this function. The results are proved using techniques from integrability and combinatoric arguments. The end result is a count of the labelled genus 0 graphs considered.
In 1935, just after defining the higher homotopy groups of a topological space (thus generalizing the fundamental group), Hurewicz noticed that for a certain class of topological spaces (he called them "aspherical"), the homology Hn(X) of such a space X was entirely determined by the fundamental group, G := 1(X). The same is true for the cohomology of such a space, and such a space is unique up to homotopy type for a given G. Later, Eilenberg and Mac Lane (and, independently, Eckman) gave a purely algebraic construction (using only the fundamental group G) of the cohomology of such a topological space. It turns out that this group cohomology gives us not only information about the topological space, but can be used to gain facts about the group G as well. We shall present this construction and see what information can be obtained, introducing some standard techniques of homological algebra along the way. No background apart from the core graduate Algebra and Geometry/Topology courses will be assumed.
For any (discrete) group G, suppose we have a topological space X with fundamental group G and *contractible* covering space E. Then the cohomology of the space X is determined by G, and is called the group cohomology of G. This can be computed from a cochain complex in which the n-cochains are homomorphisms from G^n (the direct product of n copies of G) to the coefficient module, often Z or R.
But if G is a topological group, we suddenly have two notions of cohomology: the cohomology of G as a topological space, and the group cohomology of G mentioned above, and there are different ways to combine the two. One could just mimic the above approach and take n-cochains to be continuous maps, but it turns out that doesn't give us the best theory in most cases. We will present the proper generalization of group cohomology to that of a topological group, and examine more closely the case where G is a Lie group.
In this talk I will outline the general Jacobi inversion problem on compact Riemann surfaces. After outlining the general method, I will demonstrate how the Jacobi inversion problem can be used to find explicit solutions to some special systems of differential equations. In particular, I will use the Jacobi inversion problem to solve the Toda lattice.
Representation theory is often used in the study of physical sciences. Such applications come about because a physical system has a symmetry group G and certain vector spaces associated with the system turn out to be RG-modules.
The vibration of a molecule is governed by various differential equations and the symmetry group of the molecule acts on the space of solutions of these equations. A few concepts from representation theory will be used to help us to better understand the molecular vibration of a molecule under the laws of classical mechanics.
Have you ever wondered about the application of elliptic curves to computer science and cryptography? Sure, they're fun and adorable, but is there more to elliptic curves than meets the eye (or the standard graduate course)? Mathematical double agents, elliptic curves posses the beauty of pure mathematics while having several marketable applications. In this talk I'll discuss the application of elliptic curve thoery to computer arithmetic and the factorization of large, LARGE, numbers. As an added bonus, I'll discuss SANDIA's intern program. Hope to see you there!
An algebra B is called a basic algebra if all simple B-modules are 1-dimensional over some division ring. These algebras are important computationally because you can study their structure on computers. Fortunately, every group algebra is categorically equivalent to a unique basic algebra. In this talk I will explain how to construct the basic algebra equivalent to a given group algebra.
A definite description is a phrase that begins with 'the' followed by some property such as 'the planet closest to the sun'. These come up frequently in mathematics. For example, 1/x is the number that when multiplied by x gives 1, and f(x) is the object assigned to x by the function f. Unfortunately, some descriptions, such as 'the tenth planet from the sun', fail to refer to anything. This means that descriptions cannot be treated like other terms and special rules are needed to deal with them. In this talk, I'll present these rules and show how some common sense restrictions make them relatively painless for everyday work in math. Applications to be considered include an analysis of mathematical definitions.