| Candidate | Date | Time | Location | Title | Abstract |
|---|---|---|---|---|---|
| Justin Marks | Friday, January 15 | 4:10-5:00 | Herak 237 | Manifold Methods for Averaging Subspaces | For a given data set, a natural task is to find the mean of the set. Given a set of subspaces, perhaps of varying dimension, of a vector space, we discuss the diversity of methods available to compute an average subspace. We present details of an algorithm to compute an average subspace of flexible dimension which is driven by an optimization problem that depends upon principal angles between subspaces (a generalization of the angle between two vectors). We call the resulting mean a flag mean, due to the nested relationship demonstrated by flag means of increasing dimension. To illustrate, we compute the flag mean of subspaces generated from digital photographs of human faces. The primary mathematical tools for the content of this talk are found in undergraduate Calculus and Linear Algebra, and include derivatives, subspaces, eigenvalues and eigenvectors. |
| Diana Davis | Tuesday, January 19 | 4:10-5:00 | Herak 244 | Lines on polygon surfaces | If you glue together opposite edges of a square, you get a torus (the surface of a donut). What happens if you travel forever in a straight line on the square torus surface? How can you keep track of where you've been? I'll discuss this question for the square torus, and for other surfaces made from polygons, and I'll explain the surprising connection to continued fractions. There will be lots of pictures, and a dance video. |
| Andrew Oster | Wednesday, January 20 | 4:10-5:00 | Herak 237 | Calculus, Beer, and the Center of Mass | In this talk, I will introduce the idea of center of mass and derive the formulae for calculating the center of mass. The formulae can be derived as an extension from Archimedes Law of the Lever. These ideas will then be used to solve a fun problem that arises when drinking beer (or root beer) from a can. |
| Joel Pereira | Thursday, January 21 | 5:10-6:00 | Herak 257 | Introduction to Riemannian Geometry | Many of the important facts in geometry concern objects in the plane such as the Pythagorean theorem, law of cosines, etc. A simple question is what happens when we are not in the plane? Do these theorems still hold? Are the modifications that have to be made for an analogous result. A basic topic in Riemannian geometry is the study of curved surfaces. We will introduce some terminology and basic examples. We will define what it means to be a flat surface and a minimal surface. We show that these surfaces come up in real life applications and discuss some consequences. |
| Melody Alsaker | Friday, January 22 | 4:40-5:30 | Herak 237 | The Wonderful Backward World of Inverse Problem | In The Hitchhiker's Guide to the Galaxy, a supercomputer takes 7.5 million years to determine the answer to the Great Question of Life, the Universe, and Everything, which turns out to be 42. However, this was fairly trivial compared to the Inverse Problem: "We know the Answer is 42, but what is the Question?" In mathematics, the general idea of an inverse problem is this: we know the outcome of a process, but can we "go backwards" and determine what caused that outcome? Inverse problems are all around us, and are becoming increasingly important in many fields, including medical imaging, computer vision, geophysics, astronomy, and countless others. This presentation will introduce a few fascinating examples of inverse problems, including the discovery of extra-solar planets, the use of electromagnetic waves to see inside the body, and real-life cloaking devices. We will also delve deeper into the mathematics and theory of inverse problems using a specific example which will be accessible to those with a background in elementary calculus. |
| Jason Lutz | Monday, January 25 | 4:10-5:00 | Herak 237 | Fractals & Mysterious Triangles | Using playing cards, we'll construct mysterious triangles and explore the hidden patterns in the apparent randomness of these triangles. |