See Also: 2007-2008 | 2006-2007 | 2005-2006 | 2004-2005 | 2003-2004 | 2002-2003
2007-2008 Academic Year
April 11, Dr. Anand Godpole, East Tennessee State University, "Undergraduate Research in Combinatorics".
April 8, Dr. Ivars Peterson, "The Jungles of Randomness", (An award winning journalist on topics in mathematics and computer science. You might have seen his name in "Science News", where he worked for 25 year, on one of his many popular books, such as "The Mathematical Tourist", or as online editor for the MAA's "MathTrek".)
March 25, Dr. Shashikant Mulay, "A quaint question in elementary linear algebra".
February 29, Dr. Art Benjamin, Mathematician, lightening calculator, and magician
February 5, Drs. Doug Birdwell & Tsewei Wang. Drs. Doug Birdwell and Tsewei Wang, from Engineering, work on the national criminal DNA databases. We hear a lot about these on the news these days, so come and hear about the mathematics behind the stories.
January 22, Dr. Chuck Collins, UT - "Mathematics of the Bendy Straw". Bendy straws will be provided.
November 27, Dr. Thomas Papenbrock, Physics, UT, Random matrices and chaos. There is mathematics here that you have probably seen, but it might surprise you to see how it can be used.
October 25, Dr. Robert Lang. "From Flapping Birds to Space Telescopes: The Art and Science of Origami." The free event will take place in the Alumni Memorial Building Cox Auditorium on Oct. 25 at 7 p.m. The discussion will focus on how development mathematics has been applied to the art and technique of origami throughout the past decade. Origami is the ancient Japanese art of folding paper into artistic creations.
October 15, Dr. Steve Weeks, MacArthur Fellow, 'The Shape of Space". When we look out on a clear night, the Universe seems infinite. Yet this infinity might be an illusion. During the first half of the presentation, computer games will introduce the concept of a multiconnected universal Interactive 3D graphics will then take the viewer on a tour of several possible shapes for space. Finally, we'll see how recent satellite data provide tantalizing clues to the true shape of our Universe.
October 2, Dr. Elias Wegert, Germany
September 11, Dr. Remus Nicoara, UT Google's Secret. Everybody knows that Google Inc.'s innovations in search technology made it the No. 1 search engine in the world. Google has recently made public their US patent, which reveals a great deal of how they search and rank web sites. We unveil some of the mathematics behind Google's success: graphs, matrices, eigenvalues and eigenvectors, and deep results such as the Perron-Frobenius theorem.
2006-2007 Academic Year
April 12, Dr. Scott Chapman, Trinity University,What is a Block Monoid?A basic algebra course focuses on the construction of new and different algebraic objects. From our basic set of integers (denoted Z) the cyclic groups Z_n are constructed and eventually endowed with a ring structure. Given a ring R, new extension rings, such as the ring of polynomials over R (denoted R[X]), are developed.
The purpose of this talk is to introduce a new and interesting algebraic structure based on the theory of finite abelian groups. A set M with a binary operation * which satisfies all the group axioms with the exception of element inverses is known as a monoid. If G is a finite abelian group, we shall show how the set of all finite sequences of elements from G which sum to 0 form a monoid. This monoid, denoted B(G), is known as the block monoid over G. The study of block monoids has led to the investigation of two elusive combinatorial constants, known as the Davenport constant and the cross number. We shall discuss these constants and apply them to describe the factorization properties of B(G).
March 27, Dr. Pavlos Tzermias, Things you wanted to know about integrals but were afraid to ask. It is a very difficult problem to decide whether an elementary function has an elementary indefinite integral and, if so, how to compute it explicitly. A precise formulation of this problem was given (in 1833) by Liouville who was the first person to prove that certain functions do not have elementary integrals. In recent decades, the general problem was solved thanks to the combined efforts of several mathematicians. In the process, many unexpected connections between diverse mathematical disciplines were revealed and some popular misconceptions about the nature of the problem were debunked. We will discuss these issues as extensively as time permits.
March 1, Dr. Jochen Denzler, Euler - The man who did what our teachers told us not to do. We'll first have a look at Euler's (`legal') proof, via the geometric series that there are infinitely many primes. Then we study his (as it stands, now `illegal') proof that $\sum 1/k^2 = \pi^2/6$. After a glimpse at how the definition of most calculus notions changed since Euler (and why), we'll have a look how Euler's proof can be revamped to fit today's standards of mathematical rigor.
February 15, Dr. Luis Finotti, Applications of Number Theory in Public Key Cryptography. Encryption is the process of obscuring information to make it unreadable to individuals unfamiliar with its (secret) decoding method. Although encryption has always been important to warfare and security, the advent of the Internet and on-line shopping increases the necessity of an efficient way to share encryption keys. After a very brief introduction to some basics of number theory, we will discuss the RSA cryptosystem in detail, and if time allows, the use of elliptic curves in cryptography.
November 16, Dr. Tim Schulze, Pentominoes. Pentominoes are generalizations of the more familiar dominoes and tetras. In this case, five square tiles are arranged adjacently to form shapes that present the tempting and potentially addictive possibility of fitting them neatly into a rectangular region. Despite the challenge this presents when considered as a sort of jigsaw puzzle, there are a large number of solutions. This talk will discuss an algorithm to determine all of the ways the pentominoes can be fit into a given rectangular region.
October 5, Dr. Michael Frazier, Wavelets: fingerprints, submarines, and car rattles. Abstract: Wavelets were created by pure mathematicians in the 1980s. Wavelets are small waves; that is, wiggly functions like the sine and cosine functions, but which die out instead of going on forever. In wavelet analysis, general functions are broken down into sums of wavelets, much like how general function are broken down into sums of sines and cosines in Fourier analysis.
Since the 1980s, wavelets have found many applications, for example in the FBI's computerization of their fingerprint files. We will explain why wavelets sometimes provide a valuable alternative to traditional Fourier analysis methods in signal analysis. Also, the speaker will describe two personal experiences in wavelet applications: work in 1989-1990 on the detection of Russian submarines, and work in 2002-2003 with Ford Motor company on the analysis of car rattles.
September 21, Dr. S. Mulay, Algebraic Parametrization. In studying geometric objects such as curves, surfaces etc., it is helpful to find ways to parametrize them, if at all it is possible to find a desirable parametrization. In this talk we consider the question of parameterizing curves, surfaces etc., defined by polynomial equations. Our desired parameterizations are by rational or by polynomial functions.
September 7, Dr. Conrad Plaut. The Hadamard Matrix Conjecture. Abstract: Since Fermat's Last Theorem has been proved, the Hadamard Matrix Conjecture may be the oldest and best known unsolved mathematical problem that can be understood with almost no background in mathematics. Aside from being an interesting mathematical problem, the conjecture is also relevant to the modern theory of codes, and so is of interest to computer scientists, algebraists, and others. The conjecture can be explained in terms of pavements made with black and white tiles, but it is useful to reformulate the problem in terms of matrices with 0's and 1's for entries. This talk will present the basic problem, some constructions of Hadamard matrices, and the current status of the conjecture.
2005-2006 Academic Year
March 2, 2006, Title: Quadrilateral Shapes with Round Circles, Speaker: Ken Stephenson, Abstract: Circles are perhaps the most familiar of the "ideal forms" whose study reaches back thousands of years to the ancient Greeks. We will discuss how circles can be used to judge more complicated shapes. In particular, by experimenting with a range of "quadrilateral" shapes --- filling them with patterns of circles --- we will find how to distinguish one from another. Continued fractions, the golden ratio, and some other topics with ancient roots will also make surprise appearances.
January 26, 2006, Speaker: Conrad Plaut, Title: What’s the point of studying surfaces that aren’t smooth? Abstract: The surfaces that you study in calculus are always smooth—except maybe an occasional cone, although you don’t try to do calculus at the point of the cone! But when a computer is processing the image of a surface, even a “smooth” one, the data it receives are finite and describe a surface that is not smooth, but rather has has thousands of vertices joined by tiny edges. The computer has to make sense of this data, sometimes even having to figure out where the surface is within the "cloud" of vertices. Understanding such a surface can be less like calculus and more like combinatorics. In this talk I’ll discuss features of the intrinsic geometry of a surface, smooth or not, including describing some geometric features that have been used in modern image processing problems. I’ll also bring up the old unsolved problem of Alexandrov that I mentioned in a previous JC and wonder aloud if there might be a way to use combinatorics to solve it. Don’t worry if you weren’t at the previous talk—or were there but forgot everything about it. Very little mathematics background is needed—I don’t think I’ll actually compute anything this time. Just draw lots of pictures.
September 29, 2005, Speaker: Conrad Plaut (UT), Title: A pair of geometric inequalities. We will consider two questions: (1) What is the maximum area that can be bounded by a curve of length 1 in the plane, and which curves, if any, realize this maximum? (2) What is the maximum area of a convex surface in 3-space having diameter 1, and which surfaces, if any, realize that maximum? (I will explain what a convex surface is—the concept is very simple and intuitive).The answer to the first question was known to the Greeks (1/4pi, realized only by a perfect circle), although it wasn’t until the latter half of the 19th century that the statement was really proved, by Weierstrass. The answer to the second question is still unknown. In 1955 A.D. Alexandrov conjectured that the area is bounded above by pi/2, and he exhibited a very simple surface that realizes this area. But this surface, unlike the circle, is not smooth, and not even really a convex surface. At the present time it is not even known whether there is a maximum area, or, if there is one, if it is realized by a smooth surface. After 50 years this very interesting question is still waiting to be solved.
October 20, 2005, Speaker: James Conant, Title: Euler's formula and the classification of polyhedra. Abstract: In any convex polyhedron, the number of vertices, edges and faces satisfy a certain linear equation. (Can you figure out what it is?) We will prove this equation (Euler's formula) and use it to show that there are at most eight convex polyhedra with equilateral faces. The audience will be invited to construct these eight "deltahedra" using plastic triangles which I will bring.
November 3, 2005, Speaker: Jochen Denzler, Title: The abyss of the continuum and the predictability of unpredictabilty. Abstract: It has been said that the continuum (the set of real numbers) is a well-defined collection of mostly undefinable objects. Buried in this vastness is the paradox that a phenomenon can be completely random and completely deterministic at the same time. In seemingly artificial examples, this coexistence is blatantly obvious, rather than paradoxical. However it became clear in the 2nd half of the 20th century that these `artificial' examples can faithfully represent a subset of behaviors of simple mechanical systems. One one hand, this observation has indeed torn to shreds the idea of viewing the world like a clockwork, even within the confines of Newtonian mechanics. On the other hand, this genuine scientific progress got camouflaged by unwarranted popular generalizations, marketed in a way that has earned some disrepute. We'll get some glimpses of the genuine mathematics behind the now fading pop-sci fluff on dynamical systems.
November 17, 2005, Speaker: Don Hinton, UT, Title: The Brachistochrone Problem. Abstract: In 1696 Johann Bernoulli issued the following challenge to the world's mathematicians. "Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time." We will examine Bernoulli's solution which is based on Fermat's principle of least time. Other problems stated by Bernoulli will be considered as well as other least time problems. As Bernoulli pointed out, his solution was the same as the tautochrone which is the curve of constant time investigated by Huygens. (Huygens was interesting in building a perfect clock, i.e., a pendulum clock whose period is independent of its amplitude). In conclusion we will connect Bernoulli's differential equation with the Euler-Lagrange equation developed some time later.
2004-2005 Academic Year
04/14/05 David E. Dobbs (UT) The probability of a statistical oddity in baseball. We begin by comparing the batting performances of four baseball players. Tables include each player's batting averages for the first half of the season, the second half of the season, and the entire season. It is noticed that Player A has "lost paradoxically" to Player D in the following sense: Player A had a higher batting average than Player D in both halves of the season, but Player D had a higher batting average than Player A for the entire season. The main part of the talk determines the probability that such a "paradoxical loss" could befall Player A. The analysis makes certain realistic assumptions. A theorem is proved involving a branch of a rectangular hyperbola, and we also make use of the SOLVER feature of a graphing calculator. The discussion depends only on Precalculus and Calculus II (the latter is needed only because some definite integrals arise in the calculations) and thus provides an introduction to the calculation of probability for continuous (as opposed to discrete) uniform probability density functions. We determine the probability that Player A could lose paradoxically to a player with a lower first-half batting average, as well as the probability that Player A could lose (not necessarily paradoxically) to such a player. We close with some comments about more realistic models, some philosophic musing about paradoxes, and a theorem that reinforces one's intuition about fractions.
04/07/05 Paul J. Nahin (University of New Hampshire): An elevator stopping problem. Each morning a man gets into an elevator on the garage level of a building with 11 office floors, along with n other riders (n=>0). The man's office is on the 9th floor. If it is assumed that each of the n riders is as likely to exit the elevator on one office floor as on any other office floor, and if we further assume, since it is early in the morning, that there are no people on any of the floors waiting to enter the elevator, then what is the average number of elevator stops (including his stop) that the man experiences in getting to his office? The answer for the n=0 case is of course trivially obvious (one stop, always). Can you work out the answers for the n=1 and n=2 cases? Almost surely you can (if you're careful!) Can you work out the answer for the n=17 case? 'Probably' not (this means the speaker couldn't do it.) The problem is easy to simulate on a computer for any value of n, however, and the theoretical calculations for the n=1 and n=2 cases will be done to allow partial validation of a MATLAB simulation. As a special treat, copyright-free listings of the computer program will be handed-out for as long as they last (come early and be sure to get yours along with the pizza!) This talk is based on a work-in-progress, the sequel to the speaker's book DUELLING IDIOTS AND OTHER PROBABILITY PUZZLERS, Princeton 2000.
03/17/05 Tim Schulze (UT): Pentominoes. Pentominoes are generalizations of the more familiar dominoes and tetras. In this case, five square tiles are arranged adjacently to form shapes that present the tempting and potentially addictive possibility of fitting them neatly into a rectangular region. Despite the challenge this presents when considered as a sort of jigsaw puzzle, there are a large number of solutions. This talk will discuss an algorithm to determine all of the ways the pentominoes can be fit into a given rectangular region.
03/10/05 Carl Wagner (UT): Fibonacci, Lucus, and the Menage Problem.
02/17/05 Wojbor A. Woyczynski (Case Western Reserve): Physical and mathematical traditions in probability theory The physical and mathematical communities often approach the same problems in a strikingly different fashion. The simple example of tossing a coin repeatedly makes statisticians think about sequences of independent Bernoulli random variables but for a physicist it is just a simple dynamical system at work. What is the connection between those two world views? I will discuss several examples of historical and current interest of similar dillemas that the two communities have to face over the years.
02/10/05 Grozdena Todorova (UT): Development of SingularitiesExplosions, Black Holes, Shock Waves and Self Focusing of Laser Beams Nonlinear wave equations arise naturally in physical, engineering, biological sciences and many others. The basic question is when a solution can develop a singularity and what is its nature. For example, in general relativity black holes are singular solutions of Einstein's equations. The formation of singularities at some time T starting with very smooth initial data is a purely nonlinear phenomenon. The smoothness and the smallness of initial data can't arrest in general the development of singularities for nonlinear wave equations. One type of singularity occurs when a solution or some of its derivatives become infinite in finite time. This phenomenon, called finite time blow-up, has physical examples such as the self-focusing of a laser beam which is modeled by the nonlinear Schrodinger equation. We will present several interesting phenomena of the instability for nonlinear wave equations.
11/18/04 Boris Yordanov (UT): Differential Equations as Mathematical Models Differential equations are a powerful tool for modeling various real world processes and predicting certain outcomes. We will show for example that differential equations could help explain or prevent some of the greatest engineering disasters such as the famous collapse of Tacoma Narrows Bridge, airplane wing shear and catastrophic structure failure during earthquakes. We will also explain why some skydivers survive when their parachute fails to open. Any quantity that changes over time may potentially be described using differential equations.
11/04/04 Conrad Plaut (UT): Don't ask Marilyn! Marilyn vos Savant claims the highest IQ ever recorded. This distinction has earned her a weekly column in Parade Magazine called "Ask Marilyn" in which she answers questions from readers, usually puzzles involving elementary mathematics. Shortly after Wiles's proof of Fermat's Last Theorem was announced, Marilyn declared in her column that hyperbolic geometry is "invalid" and therefore Wiles's proof, which uses hyperbolic geometry, is incorrect. In this talk I will remind you of what Fermat's Last Theorem is, consider Marilyn's argument and give a construction of hyperbolic geometry that uses only elementary calculus. As ususal, basic calculus is all you need to know to understand this talk. 09/30/04 Suzanne Lenhart (UT): Can you parallel park your car with Lie brackets? This talk will introduce the idea of controllability for a certain type of ordinary differential equations. The idea of Lie brackets can give conditions to guarantee such controllability. Parallel parking works because of the non-commutativity of the operations involved; Lie brackets measure this non-commutativity.
09/16/04 David F. Anderson (UT): The quaternions; what are they? On the evening of October 16, 1843, a man and his wife were strolling along the Royal Canal in Dublin. He had a flash of genius, took out his penknife, and carved the equations i2 = j2 = k2 = ijk = -1 on a stone on the Brougham Bridge. The man was Sir William Rowan Hamilton, and these equations define what is now called the division ring or quaternions, a number system which behaves like the real and complex numbers, but multiplication is not commutative. In this talk, we will discuss the quaternions and their role in the development of modern algebra and physics. No specific mathematics is needed other than a basic knowledge of the complex numbers.
09/02/04 Ken Stephenson (UT): Circle packing: first encounter. Circle packings are configurations of circles having specified patterns of tangency which are finding applications in areas as diverse as complex analysis and brain imaging. I will begin this talk with a visit to a menagerie of concrete examples. Then I can ask you, the audience, to start sorting out what the mathematical issues might be. We share a certain affinity with circles which traces back to the ancient Greeks and beyond, so after I have thrown in a little hyperbolic geometry, you will know almost everything you need to jump right in. At the end of the talk I will give a little overview of what lies ahead if you get serious with the topic.
2003-2004 Academic Year
04/15/04 David Linwood (UT): Gravity Simplified. The physicist George Gamow outlined a simple view of a particle theory of gravitation in 1956. He said that if one accepts the existence of a universal field of very high speed and very tiny particles, and a simple scattering law for them as they interact with matter, then one can deduce Newton's inverse-square radius law of attraction between masses, by using elementary calculus. He gave no mathematical details. We accept the challenge, for your entertainment and amusement, and, perhaps, astonishment. Anyhow, I hope the pizza is ok.
04/01/04 Davar Khoshnevisan (University of Utah): Random Thoughts (In Two Acts). Randomness affects all of us every day. So perhaps we should ask ourselves, "What does it mean to be random?" I will describe two short stories that might convince you that the honest answer is, "No one really knows." The only formal prerequisite to this talk is a term of freshman calculus. Act 1: It may be chaotic but it's surely not random! Act 2: Is that a normal-number in your hands?
03/12/04 Pavlos Tzermias (UT): The transcendental side of mundane numbers. We will explore the arithmetic nature (irrationality and transcendence) of some familiar numbers. Questions of this type have fascinated professional and amateur mathematicians throughout history. We will discuss concrete examples leading to the theorems of Lindemann and Gelfond-Schneider. The talk will be accessible to students who have mastered Math 142.
02/26/04 Amy Szczepanski (UT): Computer Typesetting and the Letter S. There are several standards for encoding fonts on a computer. To efficiently display typefaces in a variety of sizes and weights, the font is described in terms of the equations of curves in the plane. Defining an aesthetically pleasing letter S becomes an issue of finding an ellipse which passes through specified points and whose tangent line has a given slope.
11/20/03: Jim Conant (UT): Mathematics without numbers: Probing the secrets of DNA with the mathematical theory of knots. The DNA inside a cell nucleus is a complicated, tangled mess. In order for processes such as replication, transcription and recombination to take place, the DNA needs to be untangled by enzymes. D.W. Sumners and others have analyzed the behavior of some of these enzymes using tools from a beautiful area of mathematics called knot theory. In this talk we will look at some basic ideas from knot theory, and see how they were used to discover the effect of the enzyme Tn3 resolvase.
11/06/03 Bob Daverman (UT): Using Sperner's Lemma to Find Fixed Points Abstract. A function f: X --> X has a fixed point if there exists a point x* in X for which f(x*) = x* --in other words, the equation f(x) = x has a solution in X-and the space X has the fixed point property if every continuous function f: X --> X has a fixed point. The Intermediate Value Theorem from calculus assures that the unit interval has the fixed point property. This talk will exploit Sperner's Lemma, a remarkable combinatorial device, to establish a famous theorem of L.E.J. Brouwer (1881-1966) that the unit square, the unit cube, the unit 4-dimensional hypercube...all have the fixed point property.
10/23/03 James Serrin (University of Minnesota): What do Tornadoes Look Like? We discuss the dynamical interaction of a central vortex flow with a plane surface, a situation which occurs when a tornado makes contact with the ground. This interaction leads to a secondary swirling flow within the vortex, which can be of several different types depending on the coherence of the vortex. Among the possible kinds of induced secondary flows superimposed on the main vortex, there are those with central updrafts (this type presumably carried Dorothy to the Land of Oz). More surprisingly, however, there are others which have a central downdraft, these being associated with a cascade or fountain effect at the point of contact. The latter motions are of particular interest in explaining the cascade behavior observed in actual tornadoes, as seen in photographs and video recordings taken by eyewitnesses. Some of these remarkable videos will be presented during the lecture.
10/09/03 Conrad Plaut (UT): The Hadamard Matrix Conjecture Since Fermat's Last Theorem has been proved, the Hadamard Conjecture may be the oldest and best known unsolved mathematical problem that can be understood with almost no background in mathematics. Aside from being an interesting mathematical problem, the conjecture is also relevant to the modern theory of codes, and so is of interest to computer scientists, algebraists, and others. I will explain the conjecture and present some basic constructions of Hadamard matrices.
2002-3 Academic Year
04/23/03 Jochen Denzler (UT): Imaginary numbers are for real. Imaginary numbers often enter into the mathematical career of high school students as solutions to quadratic equations that really have no real solutions. The relation between students and imaginary numbers will not recover from this mystifying encounter for many years of undergrad' study. I'll try to dispel the apprehensions against complex numbers by showing how they blend naturally into what you are learning in your undergraduate courses, in particular in the cases where they are conspicuous by absence. You will encounter complex numbers in precalculus algebra, in first year calculus and in your apartment, and you will encounter a number of famous mathematicians who contributed to making imaginary numbers a reality.
04/03/03 David Anderson (UT): The quaternions, what are they? On the evening of October 16, 1843, a man and his wife were strolling along the Royal Canal in Dublin. He had a flash of genius, took out his penknife, and carved the equations i^2 = j^2 = k^2 = ijk = -1 on a stone on the Brougham Bridge. The man was Sir William Rowan Hamilton, and these equations define what is now called the division ring of quaterions, a number system which behaves like the real and complex numbers, but multiplication is not commutative. In this talk, we will discuss the quaternions and their role in the development of modern algebra and physics. No specific mathematics is needed other than basic knowledge of what a complex number is.
03/06/03 Lou Gross (UT) An overview of mathematical ecology: past, present, and future.
02/20/03 Jim Dudziak (Michigan State) How Many Archimedean Solids Are There? The Platonic and Archimedean solids form the building blocks of three-dimensional space and are central to architecture, chemistry, and atomic physics. Most people are familiar with some of the Platonic solids: the tetrahedron, the cube, and the octahedron. Fewer are also familiar with the other two Platonic solids: the dodecahedron and icosahedron. Still fewer are familiar with the Archimedean solids, a family of polyhedra less symmetric than the Platonics but just as beautiful. We will generate members of both families, with physical models available for your tactile pleasure, and then show that we do indeed have all of them.
11/18/02 John Conway (UT) What are the regular polyhedra? You all know what a regular polygon is in the plane. A regular poyhedron is the analogous geometric object in three dimensional space: a solid with equal edges, angles, and with faces made up of the same size regular polygons. The cube is the easiest example. What are some others? This talk will illustrate an ideal of pure mathematics: the complete classification of a set of natural objects. Moreover this will be done by applying some simple algebra, thus giving an example of one area of mathematics answering questions in another.
11/11/02 Grozdena Todorova (UT)