## Professor Kurt M. Bryan

Department of Mathematics

Rose-Hulman Institute of Technology

Terre Haute, IN 47803

Phone: 812-877-8485

E-mail: bryan@rose-hulman.edu

Kurt Bryan (Ph.D., 1990, University of Washington) is Professor of Mathematics at the Rose-Hulman Institute of Technology. He has also held a post-doctoral position at the Institute for Computer Applications in Science and Engineering (ICASE) at NASA's Langley Research Center, worked in industry from 1984 to 1990 as a mathematician and statistician at Blount Industries, and been a visiting faculty member at Rutgers University. His research interests lie mainly in partial differential equations, especially inverse problems related to non-destructive testing. He is particularly interested in teaching applied mathematics to undergraduates, and for the past seven years has directed Rose-Hulman's summer REU program.

**The $25,000,000,000 Eigenvector: The Linear Algebra Behind Google**

When Google went online in the late 1990's, one thing that set it apart
from other search engines was that its search result listings always
seemed to deliver the "good stuff" up front. With other search
engines, one often had to wade through screen after screen of links to
irrelevant web pages that just happened to match the search text. Part
of the magic behind Google is its PageRank algorithm, which quantitatively
rates the importance of each page on the web, allowing Google to rank
the pages and thereby present to the user the more important (and typically
most relevant and helpful) pages first. In this talk, we explore the
wonderful linear algebra that lies behind the traditional PageRank algorithm.

**Inverse Problems in Remote Sensing and Non-destructive Evaluation**

What do medical diagnosis, oil exploration, and airport security all
have in common? Answer: The need to see inside an object---a human body,
the earth, a suitcase. In each case we apply energy (mechanical, electromagnetic,
even thermal) to the object, observe the response, and try to infer internal
structure. We'd prefer to do this without damaging the object! This process
frequently gives rise to mathematical "inverse problems."

Inverse problems are sometimes summarized as the mathematics of "deducing
cause from effect." In this talk I'll discuss the mathematical
formulation of some common inverse problems, and techniques for solving
them, with plenty of examples drawn from applications. This includes
some sophisticated work done by
undergraduates in Rose-Hulman's summer REU program.

**The Mathematics of Cloaking and Invisibility**

Abstract: Cloaking and invisibility are staples of popular fiction, especially science fiction. The pseudo-explanation usually given is that "the selective bending of light rays" (to quote Mr. Spock) around the object to be cloaked can render the object invisible. But with the laws of physics in the real world, is this really possible, even in theory? Scientists and mathematicians have recently found that the answer to this question is a qualified "yes." In this talk I'll give a quantitative, but accessible account of the essential mathematical idea behind one approach to cloaking, in the context of an electromagnetic imaging technique called ``impedance imaging.''

**Picture Perfect: The Mathematics of JPEG Image Compression**

Without effective image compression, the "www" in a web URL would probably stand for "World-Wide Wait." In this talk, I'll discuss the mathematics behind JPEG compression. JPEG compression is a wonderful application of elementary Fourier analysis and linear algebra, which I'll illustrate with plenty of audio and visual demonstrations. I'll then show how some of the shortcomings of traditional JPEG compression are addressed by the new wavelet-based JPEG 2000 standard.

**Making Do With Less: The Mathematics of Compressed Sensing**

Suppose a bag contains 100 marbles, each with mass 10 grams, except for one defective off-mass marble. Given an accurate electronic balance that can accommodate anywhere from one to 100 marbles at a time, how would you find the defective marble with the fewest number of weighings?

You've probably thought about this kind of problem and know the answer. But what if there are two bad marbles, each of unknown mass? Or three or more? An efficient scheme isn't so easy to figure out now, is it? Is there a strategy that's both efficient and generalizable?

The answer is "yes," at least if the number of defective marbles is sufficiently small.

Surprisingly, the procedure involves a strong dose of randomness. It's a nice example of a new and very active topic called "compressed sensing" (CS), that spans mathematics, signal processing, statistics, and computer science. In this talk I'll explain the central ideas, which require nothing more than simple matrix algebra and elementary probability. I'll then show some applications, including how one can use this to build a high-resolution one-pixel camera.