part of the
The panel “This Could Be YOUR Graduate Research!” is a session of research talks aimed at undergraduate students designed to introduce you to three exciting areas of mathematics which you might not have seen yet. The session will kick off at 1:00 p.m. on Friday, January 7, 2011. The organizers are Ralucca Gera (assistant professor of mathematics at the Naval Postgraduate School and the Editor in Chief of the Young Mathematicians’ Network) and Aaron Luttman (assistant professor of mathematics at Clarkson University, an editor of the Young Mathematicians’ Network, and a member of the MAA Committee on Graduate Students).
ABSTRACT: Calling all undergraduates! Do you think “frames” are what we hang pictures in? Is “network science” what IT people do? What is “numerical” linear algebra? Each of these is a developing and exciting field of mathematics with which you might not yet be familiar, but they are among your myriad options of what research to pursue in graduate school. As an undergraduate, it’s hard to know what research direction you will take, and one primary reason for this is that there are so many areas of mathematics to which you haven’t yet been introduced.
Our speakers are:
Dr. Keri Kornelson of Oklahoma State University
Title: “Loosen up! Frames in Finite Dimensions.”
Dr. Tim Chartier of Davidson College, who will present on
Title: “When Life is Linear: Research in Numerical Linear Algebra.”
Dr. Steven Horton of the United States Military Academy at West Point
In the final presentation, titled “Edge Discovery in a Large Social Network,” (developing field of Network Science and the importance of mathematics and mathematicians to the expansion of the discipline).
Abstracts for the talks above:
1. Title: Loosen up! Frames in finite dimensions.
Abstract: What are frames? In a vector space with an inner product (like R^n with the dot product), an orthonormal basis (ONB) is a collection of vectors which are orthogonal, have length 1, and which form a basis for the space. We learn in linear algebra about the beautiful properties of ONBs - every element can be expressed uniquely as a linear combination of the basis vectors and the coefficients in that linear expansion are computed using inner products. No matrix inversion required! But ONBs are restrictive. The orthogonality and norm constraint mean that they can't always be tailored to fit an application. When we look at collections of vectors with looser conditions than that of ONBs which still retain some of the nice properties, we are studying frames.
Frame vectors need not be orthogonal, linearly independent, or uniform in norm. In fact, in finite dimensions, every spanning set is a frame. We can find frames which satisfy ONB-like properties, however. For example, there exist Parseval frames in which (perhaps non-unique) expansion coefficients are inner products. The array of possibilities introduced by working with frames rather than orthonormal bases has revolutionized mathematical areas such as wavelets and harmonic analysis. The adaptability to existing conditions allows frames to be used in applied settings including signal processing, imaging, sampling, and cryptography.
In this talk, we will discuss a variety of examples of frames and demonstrate some properties. We will also describe some research projects done by undergraduate students in the area of frame theory.
2. Title: "When life is linear: research in numerical linear algebra"
Abstract: Learning to solve a linear system Ax = b is a part of many undergraduates' education. This talk will discuss research directions in linear algebra when the matrix system is solved on a computer numerically. First, researchers explore developing and improving algorithms for solving such linear systems. Complex mathematical models common in modern science lead to matrix systems containing millions or even billions of unknowns. For such problems Gaussian elimination is crippled due to its inefficiency. This talk will discuss how iterative methods attempt to solve Ax = b efficiently and quickly. Second, we will consider areas where numerical linear algebra is useful, particularly, ranking and clustering. While these techniques can be useful for a wide range of applications, we will discuss their use in sports, web search, and social networks like Twitter. Finally, this talk will mention how such research in this area can be conducted both at one's graduate institution and in internships at government national research laboratories.
1. Keri Kornelson received her Ph.D. in 2001 from the University of Colorado, working with Lawrence Baggett in the area of abstract harmonic analysis. She held a VIGRE postdoctoral position at Texas A & M University, where she learned about frames from David Larson and the students in the Matrix and Wavelet Analysis REU every summer. She was an Assistant Professor at Grinnell College from 2004-2008. She moved to the University of Oklahoma in 2008 (solving the infamous 2-body problem). Link to the book:
If you are interested in learning more about frames, or are interested in doing research with students in the area of frames, you might check our our book
Frames for Undergraduates
by Deguang Han, Keri Kornelson, David Larson, and Eric Weber, published in the Student Mathematical Library series from the AMS.