See Also: 2011-2012 | 2010-2011 |2009-2010 | 2008-2009 | 2007-2008 | 2006-2007 | 2005-2006 | 2004-2005 | 2003-2004 | 2002-2003
2011-2012 Academic Year
Thursday, April 26
SPEAKER: Frederick Byrd & Yiyang Sun, Honors Undergraduate Students
TIME: 3:35 pm
ROOM: Ayres 405
TITLE: (Frederick Byrd) - Applications of Ultraproducts in Ring Theory
ABSTRACT: The concept of an ultraproduct leads to many interesting results in ring theory. Using this construction, it is possible to create new rings from old ones, which may have entirely different properties than the original rings. Though the component rings of an ultraproduct may have many restrictive characteristics, their ultraproduct typically will not have these characteristics unless it is a field. This can be used to generate counterexamples to many conjectures about ring theory while still retaining the familiarity of well-known rings. Of particular interest is the use of ultraproducts in factorization problems. In this thesis we attempt to use an ultrapower to find an example of an element with non-unique irreducible factorizations whose lengths are unbounded, as well as examine and review many interesting properties of ultrapower rings.
TITLE: (Yiyang Sun) - Mathematical Analysis of Poker Games
ABSTRACT: Poker is an intriguing game which has attracted many scholars' attention throughout History. The study of two-person zero-sum poker models with independent uniform(0,1) hands goes back to Borel and von Neumann. In this presentation we will take a look at three different mathematical poker game models. First we will start with von Neumann's simplified two player poker model where each player antes the same amount to form the initial amount to start the game, then we will change up the betting rules to make it a blind bet poker model. Finally, we will throw in an extra variable to capture the other uncertainties in a poker game.
Friday, April 20
SPEAKER: Dr. Michael Dorff, Brigham Young University
TIME: 3:35-4:25 p.m.
ROOM: Ayres 405
TITLE: "Toy Story 3", the "real" Iron Man Suit, and advising the President of the United States
ABSTRACT: Have you ever been asked "What can you do with a degree in math?" Besides teaching, many people are clueless on what you can do with strong math skills. For the past three years, I have been hosting a "Careers in Mathematics" seminar and inviting mathematicians to talk about how they use math in their careers, from research scientist at Pixar Animation Studios to operations research analyst at the Pentagon in Washington DC. In this talk, we will present some highlights from these mathematicians and their careers.
Thursday, April 12
SPEAKER: Dr. Daniel Fiorilli, Princeton Institute for Advanced Study
TIME: 3:35-4:25 p.m.
ROOM: Ayres 111 -- LOCATION CHANGE
TITLE: Races between prime numbers
ABSTRACT: In a 1853 letter to Fuss, Chebyshev made the remark that there seem to be more primes of the form 4n+3 than of the form 4n+1. If we compute the (normalized) difference between these two kinds of primes up to x, and make a graph of this quantity with x varying, then we get the figure which is shown here. What one remarks is that the difference is almost always positive, confirming Chebyshev's assertion. Interestingly, the statement in Chebyshev's letter is equivalent to the Riemann hypothesis for a certain zeta function; in fact the whole subject of prime number races is highly dependent on the zeros of zeta functions. The goal of the present talk will be to give a taste of the subject, focusing on the amusing facts.
Wednesday, April 4
SPEAKER: Dr. Stephen Shipman, LSU
TIME: 3:35 – 4:25 p.m.
ROOM: Ayres 405
TITLE: Simple models for complex physical phenomena
ABSTRACT: Real physical systems are usually too complicated to describe mathematically in detail. What we try to do instead is to isolate essential features of a system and devise an analytically tractable model system that exhibits those features. I will illustrate this philosophy with a simple chain of beads connected by springs that nicely exhibits the phenomena of wave propagation and inhibition in crystalline materials as well as confinement of energy at defects.
Pizza will be available at 3:15 p.m.
Thursday, March 1
SPEAKER: Dr. Joseph Tien, Ohio State University
TIME: 3:35 – 4:25 p.m.
ROOM: Ayres 405
TITLE: Modeling an autocatalytic enzyme
ABSTRACT: Aspirin works by blocking an enzyme called COX. This enzyme possesses some interesting kinetic features. In particular, it is autocatalytic, meaning that the enzyme can activate itself. I'll discuss a simple model of this enzyme, analyze the model using basic techniques from dynamical systems and multiple scales, and interpret the results biologically. This includes the relevance of COX to cancer research and the pharmaceutical industry.
Pizza will be available at 3:15 p.m.
Thursday, February 23, 2012
SPEAKER: Dave Auckly, Associate Director, Mathematical Sciences Research Institute, UC Berkeley
TIME: 3:35
ROOM: Ayres 405
TITLE: Exploring Scale - A Math Circle Demonstration
ABSTRACT: This will not be a presentation. Instead, it will be a group exploration of some interesting mathematics. We might learn something about hyperplanes, we might learn something about dynamics, we might learn ways in which algebra can be used to solve problems that appear to come from differential equations.
There will be pizza. We will build something. (Since pizza is part of the activity, pizza will be served at 3:35pm.)
Thursday, January 19, 2012
SPEAKER: Dr. Amy Szczepanski, UT, EECS
TIME: 3:35
ROOM: Ayres 405
TITLE: Understanding the Encoding of the JPEG Images
ABSTRACT: JPEG encoding allows large images to be represented as fairly small files. This compression is achieved through clever use of algorithms that take advantage of the fact that pixel intensity and color both tend to change fairly gradually throughout an image.
We'll see how this property, together with well-chosen matrices and smart ways of representing long strings of zeros, allows the JPEG format to encode images efficiently. This talk should be accessible to anyone who is comfortable with matrix multiplication.
Pizza will be available at 3:15 pm.
Thursday, November 17
SPEAKER: Prof. Remus Nicoara
TIME: 3:35
ROOM: Ayres 405
TITLE: Google's Secret
ABSTRACT: Everybody knows that Google Inc.'s innovations in search technology made it the No. 1 search engine in the world. Google 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. We will also discuss other search tools such as Bing, WolframAlpha, and Siri.
Pizza will be available at 3:15 pm.
Thursday, November 10
SPEAKER: Prof. Ben Cooper
TIME: 3:35 – 4:25
ROOM: Ayres 405
TITLE: "Using math to get along with your roommates"
ABSTRACT: We answer the question of whether it is always possible to share fairly with your fellow roommates in the affirmative.
Pizza will be available at 3:15 p.m.
Thursday, October 27
SPEAKER: Professor K. Sundar, LSU
TIME: 3:35 – 4:25 p.m.
ROOM: Ayres B003
TITLE: "Probability and PDEs"
ABSTRACT: The use of probability theory to answer purely deterministic questions would, at first sight, seem paradoxical and surprising.
Starting from the basics, probabilistic reasoning will be employed in this talk to solve the Dirichlet problem and the heat equation. Extensions and applications will be briefly discussed.
Please join us for pizza at 3:15 p.m.
Thursday, September 15
SPEAKER: Professor Jin Feng-University of Kansas
TIME: 3:30 – 5:00pm
ROOM: Ayres 405
TITLE: Entropy, from the point of view of Boltzmann
ABSTRACT: The concept of entropy has been used in many physical contexts. In this talk, I will focus on and explain the original version as was devised by Boltzmann, to describe symmetry of particles in gas kinetics. The whole concept can be derived by using elementary calculations based upon combinatorial counting techniques and an asymptotic formula known as Sterling formula. The modern day version of what we will derive is known as Sanov theorem in the Theory of Large Deviations
Thursday, September 8
SPEAKER: Dr. Bill Holmes, Univ of British Columbia, Alumnus of UT
TIME: 3:35 pm
ROOM: Fourth Floor Colloquium Room, Ayres Hall
TITLE: Foundations of wave propagation in biology and ecology.
2010-2011 Academic Year
Thursday, April 14
SPEAKER: Professor Judy Walker, University of Nebraska
TIME: 1:25 - 2:25 p.m.
ROOM: Ayres 405
TITLE: What color is my hat? And what does that have to do with my ipod?
ABSTRACT: As each of three people enter a room, either an orange hat or a white hat (with the color chosen randomly and independently) is placed on his head. Each person can see the other hats but not his own. They can discuss strategy before they enter the room, but after they've entered no communication is allowed. Once they've looked at the other hats, the players must simultaneously guess their own hat colors or pass. The group shares a prize if at least one person guesses correctly and no one guesses incorrectly. The "obvious" strategy (one person guesses "Orange" no matter what and the other two pass) yields a 50% success rate. Is there a better strategy? What if there are more than three players? We will use the theory of error-correcting codes to find the optimal strategy for this game in many situations.
Thursday, March 31
SPEAKER: Professor Suzanne Lenhart
TIME: 3:35 - 4:25 p.m.
ROOM: Ayres 405
TITLE: "Optimal Harvesting in Fishery Models"
ABSTRACT: We discuss two types of partial differential equation models of fishery harvesting problems. We consider steady state spatial models and diffusive spatial-temporal models. We characterize the distribution of harvest effort which maximizes the harvest yield, and in the steady state case, also minimizes the cost of the effort. We show numerical results to illustrate various cases. The results inform ongoing debate about the use of reserves (regions where fishing is not allowed), and are increasingly relevant as technology enables enforcement of spatially structured harvest constraints.
Thursday, March 24
SPEAKER: Prof. Mike Langston, EECS, Un of Tennessee
TIME: 3:35 - 4:25
p.m.
ROOM: Ayres 405
TITLE: Mathematical and Computational Advances in High-Throughput Biological Data Analysis
Abstract: We will discuss the utility of innovative graph algorithms in the analysis of high-throughput biological data. Using gene co-expression network analysis as a central theme, we will describe the use of novel tools for handling noisy data, and the role of model organisms in successful applications to human health. We will also discuss the utility of powerful mathematical methods and computational platforms. Algorithms designed for shared memory machines can be difficult to translate effectively to distributed memory models. Moreover, problems are frequently of an enumerative flavor, and thus output bound. The situation is confounded by FPGAs, multi-core processors, GPUs, green computing and other so-called disruptive technologies.
Efficient combinatorial search remains a core concern. As time permits, we will also touch on the quest for biomarkers and machine learning.
Thursday, March 10
SPEAKER: Prof. Tim Schulze
TIME: 3:35 – 4:25 p.m.
ROOM: Ayres 405
TITLE: "The Mathematics of Poker and Bluffing"
ABSTRACT: In this talk we take a look at the game of poker. We quickly find that a complete analysis is not practical. Multiple players, multiple rounds of betting, multiple raises, and variable bet sizes combine to make the game intractable. We are able to make progress, however, if we restrict our attention to heads-up play with some simplified betting rules. We expand upon this basic situation to include some elements of the popular game of Texas Hold 'Em. One focus will be identifying the sets of hands we should bluff with.
Thursday, February 24
SPEAKER: Professor Andrew Miller, Belmont University
TIME: 3:35 – 4:25 p.m.
ROOM: Ayres 405 (Colloquium Room)
TITLE: The Trouble with Tribbles (for Hairdressers)
ABSTRACT: In a classic episode of the original Star Trek TV series, the Enterprise is overrun by adorable, spherical, fur-covered creatures called tribbles. Suppose you are a Federation stylist and are tasked with taming a tribble's tresses. Could you comb all of its fur flat? A cylindrical or donut-shaped tribble would provide no trouble (can you imagine how it could be done?), but spherical tribbles present more of a challenge. We will tackle the mathematical version of this question, encountering such ideas as vector fields, singularities, and the Euler characteristic. The answer to the tribble challenge provides an example of the many fascinating interactions between global and local behavior on surfaces.
Thursday, February 10, 2011
SPEAKER: Mr. Hai Nguyen
TIME: 3:35 p.m.
ROOM: Ayres 405
TITLE: "Analysis of the Expected Industry Price"
ABSTRACT: We explore an intuitive microeconomic question of the behavior of the expected industry price, and see how we can apply basic concepts of Calculus to arrive at the answer. The question here is: given firms in a certain industry that are constrained by some production capacity K, what will happen to the overall price when we exogenously raise K?
To answer this question, we consider a simplified market in which demand is represented by a simple linear function, and supply is made available by only two identical firms. Furthermore, capacities of the firms are bounded by K. We then introduce some game theoretic notions to derive firms' strategies in setting prices, given K. With appropriate assumptions, the prices that firms set will be identical, and given in the form of a probability distribution over a set of prices. This probability distribution, as expected, is a function of K. We use this information, together with the expected market share of each firm, to arrive at a formula for the overall expected industry price. From there, we examine the derivative of a double integral to obtain our primary result.
Intuitively, when firms are able to produce more, the competition in the market becomes more intense, and thus the expected overall price will drop. We will see that, indeed, this is the result that we get from our calculations.
Thursday, November 18, 2010 (last talk for the fall semester)
SPEAKER: Dr. Joan Lind
TIME: 3:35
ROOM:
HBB 102
TITLE: Random Walks
ABSTRACT: Suppose you decide to go for a walk, and each time you reach a corner, you randomly choose the direction to take for the next block. Your path will be an example of a simple random walk. In this talk, we will take a look at this and other random walks, and we will discuss Schramm-Loewner Evolution, an exciting new random process which is related to these random walks. There will be plenty of pictures and simulations along the way.
Thursday, November 11, 2010
SPEAKER: Dr. Sara DelValle, Los Alamos National Laboratory
TIME: 3:35 – 4:25 pm
ROOM: HBB 102
TITLE: "Great Careers in the Mathematical Sciences"
ABSTRACT: Applied mathematics and computational science have become essential tools in the development of advances in science and technology. In this talk, I will discuss different mathematical applications used in transportation, infectious diseases, and economic modeling. I will also describe some universal distinguishing traits of good career choices that can guide your decisions and increase your chances of having a great career.
Thursday, November 4, 2010
SPEAKER: Dr. Paul Thurston
TIME: 3:35 - 4:25
ROOM: HBB 102
TITLE: An Introduction to Fractal Geometry with Applications in Engineering, Biology and Finance
ABSTRACT: In an undergraduate calculus course, we usually study geometric objects that are, in some sense, regular. Examples of such 'regular' geometric objects include circles, spheres, and polyhedra. Experiencing the natural world around us, we often find that the most common objects in nature, such as trees, mountains, and even the circulatory systems of animals, are much more 'jagged', or irregular. The term 'fractal' is often used to describe irregular objects such as these examples found in nature. Fractal Geometry provides a mathematical tool with which to study such irregular geometric objects, and hopefully, gain more insight into naturally occurring phenomena.
Popularized by Benoit Mandelbrot, many of us have seen the exotic artwork inspired by fractals adorning paintings, posters and clothing. Recently, fractals have found many practical applications in Engineering, Biology and Finance. (For example, every modern smart phone nowadays comes equipped with a fractal, and these smart phones wouldn't work very well without the fractal.) In this talk, I'll describe the basic elements of fractal geometry and present numerous practical examples of fractals in Engineering, Biology and Finance.
Students with any background in undergraduate calculus should find the presentation very accessible.
Thursday, October 14, 2010
SPEAKER: Professor David Anderson
TIME: 3:35 – 4:25
p.m.
ROOM: HBB 102
TITLE: "The Quaternions"
ABSTRACT: 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 Brougham Bridge. The man was Sir William Rowan Hamilton, and these equations describe what is now called the division ring of quaternions, a number system like the complex numbers except that multiplication is not commutative. In this talk, we will discuss the quaternions, their role in the development of algebra, and the number of solutions of a quadratic equation. No specific mathematics is needed other than a knowledge of the complex numbers.
Thursday, September 16, 2010
SPEAKER: Professor David Dobbs
TIME: 3:35 – 4:25 pm
ROOM: HBB 102
TITLE: "The Probability of a Statistical Oddity in Baseball"
ABSTRACT: 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.
2009-2010 Academic Year
Thursday, April 22
TIME: 3:35 – 4:25 p.m.
ROOM: HBB 102
SPEAKER: Professor Lou Gross, Director, National Institute for Mathematical and Biological Synthesis, James R. Cox Distinguished Professor of Ecology and Evolutionary Biology and Mathematics
TITLE: “What's math got to do with it? Connections between math and biology at NIMBioS”
Friday, April 9
TIME: 3:35 p.m.
ROOM: HBB 102
SPEAKER: Dr. Aparna Higgins, University of Dayton
TITLE: “Pebbling and defense strategy”
ABSTRACT: We describe an application of pebbling on graphs to defense strategy. When deploying troops, it is important to ensure that the force that arrives at a trouble spot is large enough to deal with the trouble effectively. Each move made by troops has a cost associated with it. It is desirable to Move as many troops as necessary while keeping the costs as low as possible. Viewing regions of the globe as nodes of a graph, where proximate regions are joined by an edge, we can move pebbles (representing troops) between regions according to certain rules, assessing a cost for each move made. We compare certain common strategies, such as defense-in-depth and deterrence, and provide a hypothetical explanation for the fall of the Roman Empire.
Thursday, March 25
TIME: 3:35 – 4:25 p.m.
ROOM: HBB 102
SPEAKER: Professor Jennifer Quinn, University of Washington, Tacoma
TITLE: “Mathematics to DIE For: A Battle Between Counting and Matching”
ABSTRACT: Positive sums count. Alternating sums match. So which is "easier" to
consider mathematically? From the analysis of infinite series, we know that if a positive sum converges, then its alternating sum must also converge but the converse is not true. From linear algebra, we know that the permanent of an n x n matrix is usually hard to calculate, whereas its alternating sum, the determinant, can be computed efficiently and it has many nice theoretical properties.
In this talk, we will visit a variety of positive and alternating sums as two mathematical techniques (direct counting versus matching) compete one-on-one for the title of "Most Superior." You will be the judge and jury. I ask you to consider how the terms in each sum are concretely interpreted. What is being counted? What is being matched? Which leads to simpler results? Which is most elegant? The outcome is not predetermined. You decide!
Thursday, March 18
Undergraduate Research Colloquium ((MURC))
TIME: 3:35 – 4:25 p.m.
ROOM: HBB 102
SPEAKERS: Mr. Michael Roberts and Mr. Michael Horning
TITLE: “From Knoxville, Tennessee to Dresden, Germany”
It turns out that our research results are pretty good and we are going to present them this May at the AIMS 8th International Conference, Dresden, Germany.
Did we do this research in order to travel to Germany for free? Did we do this research to be able to go to a better graduate school? And finally how did we succeed to obtain interesting results which are worth a presentation to a regular session at a very prestigious International Conference.
You are welcome to come and to find out the answers to all these questions.
Thursday, February 25
TIME: 3:35 – 4:25 p.m.
ROOM: HBB 102
SPEAKER: Professor Grozdena Todorova
TITLE: “Blow-up vs global existence”
ABSTRACT: We will talk about:
What is a partial differential equation (PDE)?
Where do PDE's come from?
What does it mean to ``solve'' a given PDE?
There is a rich variety of physical phenomena which can be modeled by PDE's. Special attention will be paid to nonlinear Schrodinger equations which are modeling processes in nonlinear optics and laser physics.
We will discuss phenomena related to the so called "Blow-up'' of solutions, the rate of the blow up, how the blow-up can be arrested, singularity formations and instability.
Thursday, November 12
TIME: 3:35 – 4:35 p.m.
ROOM: HBB 102
SPEAKER: Professor Conrad Plaut
TITLE: “A pair of geometric inequalities”
ABSTRACT: We will consider two questions: (1) What is the maximum area that can be bounded by a closed 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, 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 almost 55 years this very interesting question is still waiting to be solved.
Thursday, October 29
TIME: 3:35 – 4:35 p.m.
ROOM: HBB 102
SPEAKER: Professor Suzanne Lenhart
TITLE: “The power of optimal control: from confining rabies to improving CPR”
ABSTRACT: This talk will present optimal control of two examples which are discrete in time. The first example involves difference equations that model cardiopulmonary resuscitation. The goal is to design an external chest and abdomen pressure pattern to improve the blood flow in the heart in standard CPR procedure. The second example is an epidemic model for rabies in raccoons on a spatial grid. The goal is to find the optimal distribution pattern for vaccine baits to slow the spread of the disease.
Thursday, October 22
TIME: 3:35 – 4:35 p.m.
ROOM: HBB 102
SPEAKER: Dr. Sharon Bewick, NIMBioS
TITLE: Modeling Local Community Responses to Climate Change
ABSTRACT: I will be talking about the development of a mechanistic mathematical framework that models both competitive and mutualistic interspecific interactions with the goal of interpreting community dynamics and altered community structure under a warming regime. In particular, I will focus on climatic change as it affects ant communities in the temperate forests of eastern North America. To that end, the primary interspecific interactions that I will be discussing will relate to competition between ants for food resources.
Previous models have considered competitive interactions between ants in terms of dominance-discovery tradeoffs. Certainly, global climate change may perturb both the dominance relationships between species and/or the discovery abilities of individual species, and this may have predictable consequences on community composition. More recently, however, several empirical studies have suggested that a dominance-thermal tolerance tradeoff may be more important than a dominance-discovery tradeoff, at least in the temperate forests of eastern North America. With this tradeoff, the impact of global climate change is even more obvious. I will therefore discuss the development of mechanistic mathematical models that capture the features of dominance-thermal tolerance tradeoffs and the possibility of using these models to predict community composition, both under current climatic conditions and under a warming regime. Finally, I will briefly touch on aspects related to modeling the impact that the ant community has on the plant community through ant-plant seed dispersal mutualisms.
Thursday, October 8
TIME: 3:35-4:35 p.m.
ROOM: Haslam Business Building, 102
SPEAKER: Assistant Professor Fernando Schwartz
TITLE: A taste of differential geometry: The Gauss-Bonnet theorem
ABSTRACT: Will give a proof of one of the most beautiful theorems in differential geometry. Vector calculus is the only prerequisite for this talk.
Thursday, September 17
TIME: 3:35 -- 4:35 p.m.
ROOM: HBB 102
SPEAKER: Dr. Blair D. Sullivan, Oak Ridge National Laboratory
TITLE: "Why Graph Theory is Strongly-Connected"
ABSTRACT: Worried about the job market, but still want to be a mathematician? Perhaps graph theory is just what you need. Come hear how it prepares you for a diverse set of careers - including matchmaking, mapmaking, and truck driving! On a more serious note, this talk will provide a motivated introduction to graph theory, highlighting several active research areas, connections to other branches of mathematics, and real-world applications (including Facebook, as illustrated above).
Thursday, September 10
TIME: 3:35 – 4:35 p.m.
ROOM: HBB 102
SPEAKER: Professor Mark Meerschaert, Michigan State University
TITLE: “The Fractal Calculus Project”
ABSTRACT: Fractional derivatives are almost as old as their integer-order cousins. Recently, fractional derivatives have found new applications in engineering, physics, finance, and hydrology. In physics, fractional derivatives are used to model anomalous diffusion, where a cloud of particles spreads ...expanded abstract
The entire talk will be accessible to advanced undergraduate students.
2008-2009 Academic Year
Dr. Chaim Goodman-Strauss, University of Arkansas
Thursday, April 2, 2009, 3:35p.m., HBB 102
TITLE: “Puzzles, Games and Undecidabilility”
Dr. Amy Szczepanski
Thursday, Feb. 26, 2009, 3:35 p.m., HBB 102
TITLE: “Untangled Threads: Mathematical Knitting"
Dr. Keith Devlin, Stanford University, "The Math Guy"
Wednesday, Feb. 11, 2009, 3:35 p.m., HBB 102
TITLE: "Why do golf balls have dimples? and other mysteries of everyday life (that can be explained only with mathematics)."
ABSTRACT: Among the other mysteries of life that Devlin will address are: What makes airplanes fly, given that the explanation using Bernoulli's equation that you find in many books is flat wrong -- as you realize when you observe that airplanes can fly upside down. Which way do you have to turn the handlebars to make your bicycle turn to the right?
Professor Ken Stephenson, UT
Thursday, Feb. 5, 2009, 3:30 p.m., HBB 102
TITLE: “Circle Packing: Games and More”
ABSTRACT: Circle packings are configurations of circles with specified patterns of tangency. They link combinatorics and geometry in ways that stretch from elementary to profound --- and are typically surprising. We'll generate and manipulate patterns in real time and talk about their uses, from number theory to brain mapping. One thing you can depend on with circle packing, the pictures are always wonderful! Come and enjoy.
Professor Michael Frazier, UT
Thursday, Jan. 22, 2009 3:40 p.m., AC 113A
TITLE: “Applications of Wavelets: Fingerprints, Submarines, and Car Rattles”
ABSTRACT: One of the remarkable things about mathematics is how an idea developed in one context can have applications far outside the scope of its origination. Early in my career as a pure
mathematician, I collaborated in some work that was a precursor to the theory of "wavelets."
Wavelets are functions that look like small waves; they are localized in space but they oscillate
like a sine or cosine function. It turns out that all reasonable functions can be constructed as a superposition of wavelets. I will discuss some applications of wavelets, including the computerization of the FBI fingerprint files, and two projects that I contributed to: One to track enemy submarines in the ocean, and one to automate the diagnosis of car problems based on the sound the car makes.
Mr. Derek Rose, Elec Engr & Comp Science, UTK
Thursday, November 20, 3:40, Ayres 214
TITLE: "A Quick Introduction to Reinforcement Learning"
ABSTRACT: Reinforcement learning (RL) is an exciting and relatively new machine learning discipline,
which corresponds to a broad class of methods that allow a system to learn how to behave in
stochastic environments based on reward signals. A key concept in RL is that the intelligent agent
learns by itself, based on acquired experience, rather than by being explicitly instructed or
supervised. The talk will cover the basic concept pertaining to the theory and practice of RL.
Professor Conrad Plaut, UT
Thursday, October 2, 3:35 p.m., Ayres 214
TITLE: "Don't Ask Marilyn!"
ABSTRACT: 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 tell (or remind) you what Fermat's Last Theorem is, discuss Marilyn's argument, and give a construction of hyperbolic geometry that uses only elementary calculus.
Basic calculus is all you need to understand this talk.
Professor Bob Compton, UT Chemistry/Physics
Thursday, September 18, 2008, 3:40 p.m., Ayres 214
TITLE: “The title is "Fullerenes: A new allotrope of carbon”
ABSTRACT: In the early 1980’s the C_60 ^+ molecular ion curiously appeared in the mass spectra of graphite at the Exxon Laboratories. The same type of experiment in the Smalley lab at Rice enhanced the C_60 ^+ ion, leading his group to surmise that its structure was that of a truncated icosohedron. They dubbed it La@C_60 ^+ .
We now know that C_60 is one of an infinite number of closed cage carbon clusters called fullerenes after the American architect Buckminster Fuller. The dodecahedral C_20 is the smallest, but there is no limit to the size of larger fullerenes, which include the class of so-called carbon nanotubes. Fullerenes are made of interconnected five and six member carbon structures. From Euler’s theorem for closed polyhedra (vertices + faces = edges + 2) it is easy to show that all C_n fullerenes consist of 12 five member rings and n/2 – 10 six member rings. The most famous fullerene, C_60 , has 12 five member rings and 20 six member rings and resembles most soccer balls.
Professor Morwen Thistlethwaite, UT
Thursday, September 4, 2008, 3:40 p.m., Ayres 214
TITLE: “Classification and symmetries of knots”
ABSTRACT: For the topologist, a (classical) knot is a smooth simple closed curve in 3-dimensional space. Two knots are equivalent if one can continuously deform one to the other: stretching and bending are allowed but cutting is prohibited. We can picture a knot as a length of rope with its ends glued together, and with a small amount of practice we can draw pictures of knots, seemingly making the subject approachable. However, knot theory is teeming with problems that are easy to ask and very hard to solve. Even the problem of deciding whether two given knots are equivalent can be tricky, and has caught people out over the years. A very useful approach is to transform a knot theoretical problem into a (hopefully) easier problem in algebra. Some of the simpler algebraic invariants will be presented; the audience is encouraged to bring paper so that they can draw some knots and perform some knot theoretic calculations.
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)