Index | This Week | Current Schedule & Abstracts | Past Talks | Graduate Page
We will characterize the automorphism groups of elliptic curves defined over the field of complex numbers. The automorphism groups turn out to be a semidirect product similar to the holomorph of the underlying group. We will conclude with a brief discussion of the moduli space of all complex elliptic curves. The bagels provided at the colluquium will serve as a visual aid
This talk will be a "gentle" introduction to ultrafilters, and ultrapowers. Many fields of mathematics make use of these abstract and un-tangible objects, making them an essential tool for many disciplins. However ultrafilters and ultrapowers are eternally obscure because their existence is coupled to the Axiom of Choice, and the equivalence of ultrapowers is related to the Continuum Hypothesis. After an introduction, we will survey various properties and examples of each, concluding with the ultrapower construction of the Hyperreal Numbers.
Given the fanfare surrounding Perelman's recent proof of the Poincare conjecture, it may come as a surprise that the analogous result for number fields has been known for half a century (and was no less celebrated at the time). This talk will focus primarily on setting up the impressively deep analogy between 3-manifolds and number fields, and will conclude with a proof of the number-theoretic version of the Poincare conjecture.
String theory predicts that the universe is ten dimensional. The hidden 6 dimensions are wrapped up in Calabi-Yau manifolds. Calabi-Yau manifolds are defined in terms of Chern classes. In this talk, we will define Chern classes and give some constructions. We will then define Calabi-Yau manifolds and explain how Calabi-Yau three-folds (which have six real dimensions) find their way into string theory. We will conclude with the results of some computations which show that a quintic three-fold in 4 dimensional complex projective space is Calabi-Yau.
We will address the mathematics of closet doors. Specifically we answer the questions, "How much clear floor space is required to open and close a bifold closet door, and what is the boundary curve?" We then generalize the problem to address n-fold doors and show a connection to Archimedes' construction of an ellipse using the Trammel of Archimedes.
When we learn about quantum mechanics at the graduate level, it's easy to become lost in all the technicalities which arise (necessarily!) in the infinite-dimensional case: unbounded operators, domain restrictions, singular measures, abstract spectral theory, etc. Fortunately, at the AMS/MAA joint meetings in San Diego this winter I saw a splendid little talk about a two-state quantum mechanical system (involving oscillation of solar neutrinos, as it happens) which is completely manageable at the elementary level. As well, I suddenly realized a few weeks ago that the Schroedinger equation can be solved numerically on a discrete mesh using the discrete Laplacian and a simple Runge-Kutta method, using just a couple dozen lines of Matlab code. I'll show you the results of both experiments. This means actual numbers -- decimal places and all -- and some nice Matlab plots. As you will see, things become much clear in finite-dimensional cases -- the math reduces to straightforward notions in probability, finite-dimensional linear algebra, and ODEs. We will see, tangibly, demonstrations of the fundamental notions of quantum mechanics: state spaces, time evolution, and Hermitian and unitary operators.
Tensegrities, named for their tensional integrity, are structures which stay rigid through a combination of tension and compression. We will discuss some of the concepts involved in the rigidity, energy, and stresses associated with these structures, a little of the history of them, ways in which they can represent the symmetries of the group S_n, and some of the places that tensegrities show up. The talk will be fairly elementary.
Modular curves are topological spaces produced by the action of a group on the upper half plane (this is not uncommon, think about the torus being given by an action of Z^2 on the complex plane). Modular curves (just like tori) are Riemann surfaces (i.e. smooth manifolds of real dimension two with the extra condition that the change of coordinate maps are holomorphic). The plan is to show that any modular curve is a Hausdorff, connected, Riemann surface that can be compactified. I will try to skip some of the gory details, but I will give you plenty of motivation. More importantly, the audience will see some of the theorems they have learned in their graduate topology and graduate algebra classes in action.
This will be a continuation of the talk from last week. Again I will try to give motivation for the concepts that will arise and not get bogged down in calculations. The talk should be short and filled with pictures. Tom has also asked me to talk about some applications, so the talk will end with a "fairy tale" description of a very sophisticated application of modular forms.
First, I will give a relatively gentle introduction to cluster expansions, which refers to a technique for controlling measures on spaces of large dimension that are defined by a density with respect to a reference measure. After that, I will present a connected graph identity that may lead to an alternative proof of the Fernandez-Procacci result.
For almost a century mathematicians and scientists from many different fields of research have used continuum limits of random processes to study phenomena described by simple (random) rules that dictate the behavior of a giant ensemble of locations on an infinitesimal grid. For example, Brownian motion (a fractal) can be described as a certain limit of random walks on a discrete rectangular mesh. The existence of these fractal patterns as limits of discrete processes sensitively depends on the values of the probabilities that define these processes; in technical jargon, this is called critical phenomena. Other examples include self-avoiding random walk, random spanning trees, percolation exploration processes and Ising model interfaces. We will survey several of these discrete processes with interesting random fractal limits and identify what they have in common.
There is a simple, natural characteristic which distinguishes the sphere, the Euclidean plane, and the hyperbolic plane as geometric spaces: curvature. This simple number (and its more complicated generalizations, the curvature tensors) is what drives much of Riemannian geometry. For example, theorems such as Bonnet-Myers or Cartan-Hadamard require no more than that the curvature is uniformly bounded away from zero. Yet these theorems are inadequate to describe spaces of mixed positive and negative curvature, many of which are no more pathological than the torus. In this talk, I will give a basic introduction to curvature, give the intuition behind these powerful local-to-global theorems, and give some interesting examples of mixed curvature spaces. This talk will be accessible to those without a background in Riemannian geometry, and should still be worthwhile to those who do.
The natural numbers* arise frequently in mathematics, having applications in fields ranging from non-linear ordinary differential equations to non-linear partial differential equations**. Many mathematicians are foolishly content with only a mediocre understanding of these rather intricate objects, leaving the foundational underpinnings of the theory subject to such whimsical farces as intuition, emotion, and set theory***. The more enlightened among us recognize the importance of establishing a solid base to the theory. In this talk, we will cover topics from arithmetic such as ``decimal notation'' and ``carrying the one'' from a true mathematician's point of view. This talk should be accessible to everyone****, and will illuminate some insightful new ways of teaching arithmetic to elementary school children*****. (*)-by which I mean the natural numbers object in a suitably chosen topos category (**)-and possibly beyond (***)-``naive'' indeed... (****)-with some background in group cohomology (*****)-who you do not like
This page is maintained by John Kerl