Graduate Student Advisory Committee (GSAC) Colloquium Schedule:

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On relations between geometry and topology of hyperbolic 3-manifolds

Check department seminar listing for abstract.

Monopoles and Maxwell

Sung to the tune of "Monorail" from the Simpsons:
What if they're not here at all?
Then the Standard Model falls.

Why are they so hard to find?
That's beyond my meager mind.

Why believe this incarnation?
It would explain charge quantization.

To bring Maxwell's symmetries forth,
we need a South without a North!

Monopole! (What's that word?)
Monopole! (That name again?)
MONOPOLE!

In this talk I'll give a quick and dirty derivation of Maxwell's equations, and discuss how they're almost symmetrical. I'll then mention how this "almost" problem can be fixed, and then derive an amazing result. Namely, that if there exists one magnetic monopole, anywhere in the universe, then it implies charge quantization!

Department Colloquium

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Department Collowuium

Check department seminar listing for abstract.

Flat Maps of a Round World: An Exploration of Map Projections

It has long been known that the world is round. For almost as long, it has been known that making a perfect map of a round earth on a flat sheet of paper is impossible. All map projections preserve some properties at the expense of others. In this talk, I will explore some of the characteristics of various map projections as well as some specific impossibilities in cartography.

The dark side of the Universe

About 95% of the stuff that composes our Universe is yet to be discovered and appears in the form of Dark Matter and Dark Energy. I will focus on the theoretical and observational motivations that led astronomers to postulate these forms of energy, and some current theories that attempt to explain these mysteries.

A Minimum Variational Principle for the Complex Helmholtz Equation

The solution of Laplace‚s equation by minimization methods is a classical example of how the Calculus of Variations can be used to solve a PDE by minimization. However, for the Helmholtz equation, the applicability of the Calculus of Variations is not nearly as clear. I will review how this works for Laplace's equation and then present the work of Milton, Seppecher, and Bouchitt´e as it applies to deriving a minimization variational principle for the solution to the complex Helmholtz equation.

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History of Geometry - CANCELLED

Geometry was born around 5000 years ago in Babylonia. Classic geometry was focused on ruler and compass constructions, while in modern times geometric concepts have been generalized to a higher level of abstraction. We will briefly explore the history of geometry and some of the topics in modern geometry.

Clifford Algebras and Spinor Neurons

A generalization of the complex numbers and the quaternions, Clifford algebras provide a powerful tool for a variety of fields--like geometric computing, physics, computer vision, and robotics. Given a quadratic space over a particular field, I will discuss the defining universal property for Clifford algebras and demonstrate how to construct such an algebra from that quadratic space.

Clifford algebras have a useful application to neural networks and have inspired the spinor neuron model--which allows orthogonal transformations to be computed in a manner faster and more robust than real neurons. In fact, some transformations (like Moebius transformations) not learnable by real neurons can be learned by Clifford neurons. During the talk, I'll cover this application in more detail as motivation for the "abstract nonsense."

This talk is intended for a general math audience, and I will define several basic algebraic concepts which will be used again in the next two GSAC talks.

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