The Rose Mathematics Seminar

Overview

The Rose-Hulman Mathematics Seminar meets on a regular basis throughout the year per the schedule below. The seminar is quite informal and so the topics are likely to vary from week to week .  Anyone --- student, faculty, or visitor --- is encouraged to give a talk or series of talks on any topic of interest.  We're especially happy to have students attend or better yet, give talks! Students can get credit for attending and giving talks, by signing up for MA450. For seminars in past years go to the seminar history page. Here is a printable campus map if needed by our off-campus visitors.

Also of interest: Here is the seminar page for our sister institution Indiana State University.

• Regular Day and Time (Winter 2008-09): Wednesday, period 7 (1:35 P.M.)
• Place: G222
• Organizer: Kurt Bryan bryan@rose-hulman.edu   (812) 877-8485

Current and Upcoming Schedule (latest first)

• Topic:  Modeling Pattern Formation in the Auditory and Visual System
• Speaker:  Kim Montgomery
• Date:  22 Apr 2009
• Abstract:  In the course of biological development interesting hexagonal patterns are formed in both the non-mamalian auditory system and the visual system of the fly's eye. I'll discuss how mathematical models for intercellular signaling and cell motility can be useful in explaining the formation of these patterns.

• Topic:  The Mathematics of Cloaking
• Speaker:  Kurt Bryan
• Date:  18 Mar 2009
• Abstract:  Cloaking and invisibility are old 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 actually 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 an essential mathematical idea behind cloaking, in the context of an electromagnetic imaging technique called "impedance imaging."

• Topic:  Braids, Cables, and Cells: An Interesting Intersection of Mathematics, Computer Science, and Art
• Speaker:  Joshua Holden
• Date:  11 Mar 2009
• Abstract:  The mathematical study of braids combines aspects of topology and group theory to study mathematical representations of one-dimensional strands in three-dimensional space. These strands are also sometimes viewed as representing the movement through a time dimension of points in two-dimensional space. On the other hand, the study of cellular automata usually involves a one- or two-dimensional grid of cells which evolve through a time dimension according to specified rules. This time dimension is often represented as an extra spacial dimension. Therefore, it seems reasonable to ask whether rules for cellular automata can be written in order to produce depictions of braids. The ideas of representing both strands in space and cellular automata have also been explored in many artistic media, including drawing, sculpture, knitting, crochet, and macrame, and we will touch on some of these.

• Topic:  Linear Volterra Inverse Problems - Formulation and Regularization
• Speaker:  Cara Brooks
• Date:  18 Feb 2009
• Abstract:  When solving practical problems, one often tries to gain intuition by first making many assumptions to obtain a simplified model. As the problem becomes better understood, the assumptions can be relaxed and a more complex model can be considered. In this spirit, we will start by examining the problem of differentiating data, then move on to the problem of computerized tomography, demonstrating how a few simplifying assumptions and a lot of calc II lead to solving a linear Volterra integral equation of the first kind. Depending on the function spaces involved, this means solving an ill-posed (inverse) problem. We will then examine regularization techniques for handing some linear Volterra problems and discuss some of the work involved in obtaining ``good'' approximations to the exact solution when using measurement data corrupted with noise.

• Topic:  Nonlinear Design Problems, Model Discrimination, and Impossible Solutions
• Speaker:  Mike DeVasher
• Date:  11 Feb 2009
• Abstract:  This talk will cover three distinct topics. First, an introduction to the fundamental conundrum of optimal design for nonlinear experiments will be discussed. A novel approach in applying Bayesian ideas to the need for prior information will be compared to the historical standard of local optimality. Next, a related nonlinear design problem, that of model discrimination for exponential regression models will be introduced. A brief review of model discrimination techniques for linear models will be offered as well as a discussion of model discrimination techniques particular to nonlinear models. Finally, time permitting, a solution to a variant of Freudenthal’s “Impossible Problem” attributable to Lee Sallows will be discussed. Solutions to the so-called “Superimpossible Problem” will have to wait until a later date.

• Topic:  Inverse Problems on Resistor Networks
• Speaker:  Kurt Bryan
• Date:  04 Feb 2009
• Abstract:  Suppose we have a rectangular grid (of finite extent) of resistors in the 2D plane. The “interior” resistors are not accessible to us and have unknown resistance. However, we have access to the resistors on the boundary of the grid, to which we can apply voltages and measure the resulting currents. Can we use this kind of information to determine the resistance of the inaccessible interior resistors? What if we have access to only part of the boundary? This kind of problem is a natural discrete analogue to certain problems in nondestructive testing and “impedance imaging”, but easier to analyze---all we need is linear algebra. I’ll show some results obtained by students in the mathematics REU last summer at Rose-Hulman.

• Topic:  Solutions to the Pure Parsimony Problem
• Speaker:  Joshua Burbrink, Nicole Fehribach, Tony Ferrell, Fred Freers, Casimir Ksiazek, Jason Sauppe, Jeremy Schendel and Al Holder
• Date:  28 Jan 2009
• Abstract:  The students in MATH 444 addressed the problem of finding the least amount of genetic diversity needed to describe a population, which is known as the Pure Parsimony Problem. This talk will start with a succinct introduction to the problem and then will proceed to a discussion of the proposed solution methods. In particular, local search methods and their efficiency will be discussed. The talk will end with a mathematical result that identifies a sub-class of these APX-Hard problems that can be solved in polynomial time.

• Topic:  The role of beauty in the search for world-record cages
• Speaker:  Robert Jajcay, Indiana State University
• Date:  21 Jan 2009
• Abstract:  A (k,g)-cage is a very neat and efficient mathematical creature; a k-regular graph of girth g that has the smallest number of vertices possible. As finding the (absolutely) smallest cage is extremely hard, researchers often settle for finding a graph that is smaller than anyone else's in the world -- the world record cage. This gives rise to a curious area of mathematics, an area where tables of current record holders are constantly being updated and closely watched, and every new entry gives rise to frantic attempts at beating it. It is also an area where everybody has an equal chance (well, not really, being smart helps quite a bit), and even newcomers may get their chance for their 15 minutes of fame (or how long it takes until someone else beats their record and erases their name from the tables). In our talk we intend to introduce some order into the competition by looking into the relation between beauty and efficiency. We make a very non-mathematical claim that beautiful (i.e., highly symmetric) structures have the best chance for being the world records, and we support this claim with a little bit of evidence and a lot of speculation. We will take great care not to start fights with proof-demanding mathematicians, but cannot make any promises.

• Topic:  Differentiating the QZ Algorithm with Application to Gradient Based Output Feedback Optimization
• Speaker:  Brad Burchett
• Date:  17 Dec 2008
• Abstract:  Special time 10'th Hour PARTA: The QZ algorithm gives a robust way of computing solutions to the generalized eigenvalue problem. The generalized eigenvalue problem is used in linear control theory to find solutions to Ricatti equations, as well as to determine system transmission zeros. In state space linear system analysis, the system poles and transmission zeros are particularly important for determining system time and frequency response. Here we embed calculation of the eigenvalue derivatives in the QZ algorithm such that the derivatives of system poles and transmission zeros are computed simultaneously with the poles and zeros themselves. The resulting method is further exercised in finding generalized eigenvalues and their sensitivities required for finding the derivatives of system residues. This technique should openthe door to solutions of problems of interest by unconstrained gradient based methods. Typical numerical results are presented. PART B: A new method for gradient based determination of H2 optimal output feedback gains is presented. Constraints representing the dynamics of a linear time invariant system are substituted into the quadratic cost function. Sylvester’s expansion is used to write the matrix exponential in a form which can then be integrated closed-form. The cost function and its derivatives can then be written as algebraic expressions in terms of the system eigenvalues. PART C (time permitting): A numerical model of the Ares I upper stage main propulsion system is ormulated based on first principles. Equations are written as non-linear ordinary differential equations. The GASP Fortran code is used to compute thermophysical properties of the working fluids. Complicated algebraic constraints are numerically solved. The model is implemented in Simulink and provides a rudimentary simulation of the time history of important pressures and temperatures during re-pressurization, boost and upper stage firing. The model is validated against an existing reliable code, and typical results are shown.

• Topic: 
• Speaker:  S. Allen Broughton, Voronoi Tesselations, Delaunay Tesselations and Flat Surfaces
• Date:  10 Dec 2008
• Abstract:  Voronoi tessellations are all about us. In crystallography, the can be used to define a unit cell. In coding theory the can be used measure effectiveness of detection and correction of errors in transmission. The sizes of the cells can give us information about uniform placement of points on a sphere such as satellites in the sky. Delaunay tessellations are dual to Voronoi tessellations and have their own uses. In the first part of this talk we will give some examples of the tessellations and discuss algorithms for determining them. In the second part of the talk we will look at how these tessellations can be used to understand the geometry of flat surfaces, such as a cube or icosahedron. This talk is the second of two sabbatical report talks from Professor Broughton's sabbatical at Indiana University last spring. The first talk "Billiards and Flat Surfaces" was a motivational introduction to flat surfaces intended for a general audience. This second talk, will discusses additional geometrical concepts and problems about flat surfaces suitable for undergraduate research.

• Topic:  (Almost) The Poincare Conjecture or “What’s the difference between a ball and a doughnut?”
• Speaker:  Bill Butske
• Date:  29 Oct 2008
• Abstract:  In this talk I’ll discuss how to (mathematically) answer one of the burning questions of our time and indeed of this election cycle. Namely, what is the difference between a doughnut and a ball (there’s a giant hole in one of them) and how it relates to the recently proven Poincare Conjecture (which has to do with holes in things). This talk is intended for a general audience (this means you M. Fouts) though you’ll learn lots of fancy words that you can use to impress the public at large.

• Topic:  Algebraic Tori and Their Applications
• Speaker:  Arnold Yim, Rose Student
• Date:  22 Oct 2008
• Abstract:  In this talk, we will discuss the structure of algebraic tori. In particular, we will go over what it takes for an algebraic variety to be rational. We will then look at an example of a small torus. Finally, I will describe the applications of algebraic tori in public key cryptosystems. Most of the mathematics involved in this talk should be accessible to anyone, however, familiarity with finite fields and basic algebraic ideas will help.

• Topic:  Billiards and Flat Surfaces
• Speaker:  Allen Broughton
• Date:  01 Oct 2008
• Abstract:  What do flat surfaces like a cube or icosahedron have to do with billiards? The billiard question is simply: If you hit a billiard on a polygonally shaped billiard table and it continues indefinitely, will it eventually get near to every point on the table? The answer is fairly easy for rectangular shaped tables but more complicated for other shapes. In this talk we will discuss how flat surfaces arise from the discussion of billiards and look at some of the properties of flat surfaces, including a suitable interpretation of Euler's formula. This talk is the first of two sabbatical report talks from Professor Broughton's sabbatical at Indiana University last spring. The first talk is a motivational introduction to flat surfaces and is intended for a general audience of Rose faculty and students. The second talk, to be given later in the year, will discuss additional concepts and problems about flat surfaces suitable for undergraduate research topics.

• Topic:  Mathematical Programming, Systems Biology, and Undergraduate Research
• Speaker:  Allen Holder
• Date:  24 Sep 2008
• Abstract:  Several recent advances in biology, medicine and health care are due to computational efforts that rely on new mathematical models. The underlying mathematics largely lies within discrete mathematics, statistics & probability, and optimization, which are combined with savvy computational tools and an understanding of cellular biology to advance our biological insights. One of the most significant areas of growth is in the field of systems biology, where we are using information from high-throughput computing to construct models that describe larger entities. We will introduce the overriding goal of systems biology and will highlight the role of mathematical programming. In particular, places for undergraduate research will be discussed.

• Topic:  My Summer as an Actuarial Intern
• Speaker:  Casimir G. Ksiazek III, Rose Student
• Date:  17 Sep 2008
• Abstract:  In this talk, I will describe the work I did as an actuarial intern this summer at Allstate Insurance in Northbrook, IL. The talk will be informal, with discussion and questions encouraged. I STRONGLY recommend that anyone even remotely interested in actuarial science attend. Most of the mathematics involved should be accessible to anyone, though a familiarity with regression will help.