Graduate Colloquium Spring 2012

April 12, 2012

All Dot [Products] go to [Tensor] Heaven (aka I'll $\mathbb{C}$ Your Tensor and Raise You an Index)

Victor Dods 

I will talk about how linear algebra can be formulated in a very regular and flexible way using tensor products. A few motivating examples should provide enough justification for this claim, such as a clarifying approach to bilinear forms, calculating the total derivative of the matrix-inverse map, and more. I will even be drawing pictures which should make the algebraic constructions obvious. An underlying topic will be "strongly typed language", a concept borrowed from computer programming languages, in which the "type" of each object is explicitly checked in each operation. This is used to detect inherent usage errors (as a compiler would) and even to infer meaning from the type of an object. For example, matrix math is not "strongly typed" because two n-by-n matrices can be multiplied regardless of if they represent linear maps whose composition is undefined. The talk will remain relatively high level and should be accessible (and useful!) to all grad students as well as advanced undergrads who are comfortable with the dual of a vector space.

April 19, 2012

On the Euclidean Algorithm and Its History

Paul Tokorcheck 

In this talk we will give a very elementary discussion of the Euclidean Algorithm and its use in a standard Elementary Number Theory course. We will then follow with a discussion of the Algorithm's history along with that of the GCD itself, dating from Euclid up into the European Renaissance. The talk will be accessible to anyone, including small children. Keeping with tradition, Blue Ribbons are awarded to the first 11 attendees excluding small children).

April 26, 2012

Infinite volume arithmetic hyperbolic drums with low Bass notes

Michael Magee 

In this talk I will outline how one can construct infinite volume arithmetic hyperbolic manifolds with arbitrarily low first eigenvalue of the Laplacian (or Bass note). The restriction to the arithmetic setting is a main part of the problem. I will also give examples of arithmetic hyperbolic 3-manifolds arising from links in R^3. Some of the time will be dedicated to clearly introducing the concepts mentioned.

May 3, 2012

Topology of Linear Mappings, Continuously Differentiable Maps and Transversality

Yusuf Goren 

We will investigate some topological properties of linear mappings and differentiable maps over manifolds. Our aim will be to prove that transversality is a generic or at worst residual property. A basic understanding of topology, linear algebra and differentiable manifolds should be enough for the talk.

May 10, 2012

Turing Machines and Decidability

Shawn O'Hare 

Hilbert's Tenth Problem challenged mathematicians to devise devise ``a process according to which it can be determined by a finite number of operations'' whether a ``Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients" has integral roots. That such a process even exists is taken for entirely for granted. In the 1930's three equivalent models of computation were developed to answer questions like Hilbert's Tenth and hisEntscheidungsproblem of 1928. One such model are Turing machines, and in the talk I will give a formal definition of a Turing machine and demonstrate several problems which are undecidable, i.e., problems like HIlbert's Tenth that cannot be answered algorithmically.

May 17, 2012

The Kepler Problem on the Heisenberg Group

Corey Shanbrom 

The Kepler problem is among the oldest and most fundamental problems in mechanics. It has been studied in curved geometries, such as the sphere and hyperbolic plane. Here, we formulate the problem on the Heisenberg group, the simplest sub-Riemannian manifold. Key to this formulation is a 1973 result of Folland, who found the fundamental solution to the Heisenberg sub-Laplacian. We will discuss the geometry of this space and present partial results and first steps towards a solution to the Kepler-Heisenberg problem.

May 24, 2012

Curve Singularities and the Isotropy Method for the Monster Tower

Wyatt Howard 

In a recent paper titled "Points and Curves in the Monster Tower" (2010), Professor R. Montgomery and Professor M. Zhitomirskii were able to classify all of the points in a tower of manifolds, built from the plane R^{2}. They called this structure the "Monster Tower." The motivation for doing so came from Professor Montgomery's interest in trying to classify all possible Goursat distributions (meaning a subbundle of the tangent bundle). They worked primarily with singular curves in order to classify the various points within the Monster Tower. In this talk I will discuss my current work with the Monster Tower built from R^{3} instead of R^{2} and discuss some basic classification results that I have been able to find using the singular curve approach. I will also discuss how the singularity theory approach actually fails to give the same results that it did for the R^{2} case and then present a new tool that helps with the classification process, called the isotropy method. I will present the isotropy method and show how one can use it to compute the various orbits at a particular level of the Monster Tower.

May 31, 2012

Bach Flatness and Fischer-Marsden's Conjecture

Wei Yuan 

In 1975, A.Fischer and J.Marsden proposed a conjecture, which conjectured a closed static space can only be either Ricci flat, or the standard sphere. But in fact, the class of the closed static spaces is much larger than this. In this talk, I will give the classification of the closed static spaces providing Bach flatness and also talk about some special noncompact cases.

June 7, 2012

The congruent number problem

Mitchell Owen 

The congruent number problem concerns characterizing which integers are the areas of right triangles with rational sides. This problem was settled (mostly) in 1983 by a theorem of Tunnel; we will discuss a simple way of approaching the problem and present a partial solution, then discuss the connection to elliptic curves which created the almost-complete solution and outline the arguments that contribute to it.