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Talks and Seminars

A new model for really old thrust faults on Mars: How old are they, how did they form, and why do we care?

Lydia Boroughs,

Friday, November 3, 2006 - 1:30 p.m., Woods Labs 134

Abstract: Thrust faults, which are present throughout the Martian surface, are an enigma due to their pattern concentric to the Tharsis Rise (a large volcanic region). We present a new model, including planetary cooling and the loading of the elastic shell of the planet, which largely explains observed tectonics. Our model strengthens the argument that volcanism began very early in Mars' history, and provides a mechanism which allows volatile (i.e. water) penetration onto the surface of the planet for at least the first billion years.

Imaging Challenges when Placing an Oil Well

Nick Bennett (C' 91), Schlumberger-Doll Research

Friday, October 27, 2006 - 4:00 p.m., Woods Labs 134

Abstract: Placing an oil well in a layer of the Earth containing hydrocarbons can be challenging. One can easily drill through the layer, overcorrect, and end up with an uneconomic well. The proper placement of wells is also very important when preparing to sequester carbon dioxide in the Earth's subsurface and thereby reduce global warming due to the greenhouse effect.

Real-time imaging of the geology around the well, particularly using electromagnetics and acoustics measurements, is critical for decision making while drilling. I will discuss some of the mathematics challenges inherent in this process which is somewhat similar to those found in medical imaging.

After the talk, I will also share some of my own experiences since graduating from Sewanee along the road towards a career in applied mathematics.

The Miller-Selfridge-Rabin Probabilistic Test for Primality

Eric Wilson, Math Major, Class of 2007

Thursday, December 2, 2004 - 4:00 p.m., Woods Labs 230

Abstract: The security of modern encryption standards relies on the inability to factor large numbers in polynomial time. Specifically, the RSA algorithm employs two 154-digit (512 bit) prime numbers in its protocol. Unfortunately, there is no sure-fire method of finding prime numbers, but it is possible to test for primality using probabilistic methods. The Miller-Selfridge-Rabin Strong Pseudoprime Test is a Monte Carlo algorithm used to test whether or not a number is prime. This test is the basis for Mathematica's PrimeQ function. I will discuss the algorithm, its weaknesses and how to improve its error.

Advanced Encryption Standard

David Rose (C' 02),

Friday, October 22, 2004 - 3:30 p.m., Woods Labs 230

Abstract: Since 2000, the Advanced Encryption Standard (AES) has become an industry standard for secure communication, used by governments, banks, and web services alike. This talk gives an overview of AES (a.k.a. "Rijndael"), discussing design, implementation aspects, and security issues. No mathematical or cryptologic knowledge beyond familiarity with finite fields will be assumed.

Reaction-Diffusion Models of Biological Pattern Formation

Selwyn Hollis, Armstrong Atlantic State University

Monday, September 13, 2004 - 3:30 p.m., Room TBA

Abstract: In a 1952 paper, The Chemical Basis of Morphogenesis, Alan Turing explained that spatial patterns of chemical concentration can be generated by simultaneous reaction and diffusion processes, suggesting that this behavior may account for the development of some animal pigmentation patterns. In this talk, we will present an introduction to reaction-diffusion equations and derive conditions for the phenomenon that has become known as Turing (or diffusion-driven) instability, which is the mathematical basis for Turing's theory of pattern formation. The process will be illustrated by several Mathematica-generated animations.

Modeling Bark Beetle Infestations in Pine Forest Ecosystems

Selwyn Hollis, Armstrong Atlantic State University

Monday, September 13, 2004 - 7:30 p.m., Blackman auditorium

Abstract: The southern pine beetle (SPB) is a highly destructive species of bark beetle that inhabits pine forests in the southern United States and parts of Central America. In natural pine forests, the ecological role of the SPB presumably is to hasten the demise of weak or damaged trees, perhaps creating kindling for fires that contribute positively to the long-term health of the forest ecosystem. In commercial forests and developed residential areas, SPBs are responsible for hundreds of millions of dollars in timber losses annually. SPB populations exhibit outbreak behavior similar to other forest insect pests, typically persisting at low endemic levels for a number of years before rapid increase to epidemic levels and subsequent collapse. In this talk we will describe a differential-equation model of the interaction between SPBs and pine trees, in which a crucial role is played by "scramble competition" among SPB larvae. The model successfully mimics outbreak behavior, and numerical experiments indicate that an outbreak may be triggered by a small change in any of several parameters in the model.

Cell Metabolism, Mathematics, and Public Health

Mike Reed, Duke University

Monday, April 19, 2004 - 7:30 p.m., Blackman Auditorium

Abstract: Population studies over the past 50 years have shown statistical associations between dietary habits and certain cancers. Similarly, certain gene variants appear to be linked to increased risks of particular cancers. Mathematical models can help us to understand the actual mechanisms by which these genes or dietary factors contribute to a sequence of cellular events that result in cancer. This will be illustrated by recent work on folate and methionine metabolism. Understanding the mechanisms is important, both for making wise public health decisions and designing excellent intervention strategies in patients with cancer.

Probability Theory and Neurobiology

Mike Reed, Duke University

Monday, April 19, 2004 - 3:30 p.m., Woods Labs 230

Abstract: A fundamental question in neurobiology is to understand how sensory information is sharpened as it proceeds through the brainstem to the midbrain and cortex. For example, neurons in the auditory nerve, which runs from the ear to the brainstem, are inherently sloppy timing devices, since their firing times under repeated trials with the same sound have standard deviations of approximately one millesecond. Yet, neurons further up in the brainstem show exceptionally precise timing (standard deviations of 70-100 microseconds) even though they get all their information from these sloppy auditory nerve neurons. And, it is known that a variety of mammalian behaviors depend on analysis of auditory timing differences at the microsecond level. Thus, the brainstem must extract precise timing information from the group of auditory nerve neurons. Investigation of how this occurs leads naturally to questions in probability theory.

Diffie-Hellman Key Exchange and Quantum Cryptography

Laurence Alvarez, University of the South

Friday, April 2, 2004 - 2:30 p.m., Woods Labs 230

Abstract: The ability to exchange encrypted messages is necessary to maintain the confidential nature of transactions in today's electronic exchange of funds and information. To exchange encrypted information participants must agree on a key for the encryption and decryption. This talk will explain how it is possible for two parties to have a public conversation and agree on a key to use for encryption without any observers being able to learn what the key is. It will also explore how quantum cryptography could be used to establish a key. This talk covers part of the speaker's work during his sabbatical leave this year.

The Generalized Riemann Integral

Hannah Johnson, Math Major, Class of 2004

Friday, February 20, 2004 - 1:30 p.m., Woods Labs 230

Abstract: In 1854 Riemann defined an integral that most of us learn in an introductory Calculus classroom. In 1902, Lebesgue defined an integral that could integrate more discontinuous functions than Riemann. Then in the 1960's, Ralph Henstock and Jaroslav Kurzweil developed the generalized Riemann integral (also may be called the Henstock integral) that could not only integrate everything that Lebesgue and Riemann could and get the same value, but could integrate every derivative regardless of whether it is bounded or not. The use of the generalized Riemann integral allows the Fundamental Theorem of Calculus to be stated without assuming integrability of the derivative. By only assuming that F:[a,b] --> R is differentiable and setting f(x) = F'(x), the generalized Riemann integral exists and the integral from a to b of f is equal to F(b) - F(a). Surprisingly, the proof of the fundamental theorem in this setting avoids the use of the mean-value theorem. Moreover, with the aid of the generalized Riemann integral many improper integrals become ordinary integrals and the class of integrable functions is extended even beyond those of Lebesgue.

Fractional Calculus

Elizabeth Shroyer, Math Major, Class of 2004

Tuesday, December 9, 2003 - 3:00 p.m., Woods Labs 216

Abstract: Did you every wonder what is the half derivative of x²? Surprisingly, the fractional derivative has been proposed for over three hundred years. In this talk, I will give a brief history of fractional calculus, define the definition of the derivative, and look at a few applications including the Tautochrone problem.

Calculated Bets: Computers, Gambling, and Mathematical Modeling to Win

Steven Skiena, SUNY Stony Brook

Wednesday, November 5, 2003 - 7:30 p.m., Blackman Auditorium

Abstract: This talk describes a system that we have developed using computer simulations and mathematical modeling techniques to predict the outcome of jai-alai matches to bet on them successfully. It is based on my book of the same title, recently published by Cambridge University Press.

Jai-alai is a Basque variation on handball, and it is legal to bet on in Connecticut, Florida, and Rhode Island. The "Spectacular Seven" scoring system used in the United States, creates strong inherent biases in the probability of different outcomes, even if all players are equally skillful.

By coupling Monte Carlo analysis with statistical analysis of player skills and the public's betting patterns, we built a system which increased our initial stake by 544 percent over a four month period of actual betting.

In the talk, I will introduce the sport of a jai-alai, explain the scoring system, describe the theory underlying our system, and finally report on our experiences putting it to the test.

"Lions and Lambs," Understanding the Variability of Wintertime Weather

Edwin Gerber (C' 00), Princeton University

Friday, October 10, 2003 - 4:00, Woods Labs 230

Abstract: As of late, large scale variability of the earth's climate on monthly to seasonal timescales has generated much interest in the research community. Our ability to predict events that evolve over such time periods is limited by the chaotic nature of the atmosphere, which makes weather predictions past ten to fifteen days impossible. On the other hand, it is of vital importance to understand the natural vacillations of the climate system if we hope to be able to distinguish potential changes of anthropogenic origin, such as global warming. El Nino is the most famous of these "low frequency" weather patterns, but another in our region is the North Atlantic Oscillation, which can dramatically affect the wintertime weather in both North America and Europe. I will present simple mathematical models that help us understand the cause of "NAO," hoping to explain to you why some winters roar through like a lion while others slip by gently as a lamb.

The Symbiotic Relationship Between Undergraduate Research and Faculty Research

Anant Godbole, East Tennessee State University

Thursday, April 10, 2003 - 7:30 p.m., Blackman Auditorium

Abstract: I have run a successful Research Experiences for Undergraduates (REU) program since 1991 - first from Michigan Tech University and, since 2000, from East Tennessee State University. I am often asked how I find suitable problems for my talented summer research apprentices. Equally often, I am asked how I can find the time to conduct this program -- time that must surely be spent at the expense of my own research agenda. My answers to these questions are invariably the same: the problems I have undergraduates work on are precisely the ones I would otherwise work on myself.

I will provide a variety of examples* that drive home the fact that my undergraduates have made me what I am, particularly during the early '90s, when I was frenetically seeking advancement to the rank of this-or-that, and now during the early '00s, when I find myself with administrative duties that leave me craving for June 15, when my new REU crew arrives on campus.

* Including the stellar role played by Sewanee's Daphne Eudora Skipper, a protege © of Professor Sherwood Ebey, and a recent Ph.D. from the University of Kentucky.

Probabilistic Versions of Seymour's Distance Two Conjecture

Anant Godbole, East Tennessee State University

Friday, April 11, 2003 - 7:30 p.m., Blackman Auditorium

Abstract: Paul Seymour's distance two conjecture states that in any digraph there exists a vertex (a "Seymour vertex") that has at least as many neighbors at distance two as it does at distance one. Seymour's conjecture is known to be true for all (round-robin) tournaments. We explore the validity of probabilistic versions of this conjecture that focus on (i) the existence of Seymour vertices; (ii) the number of Seymour vertices; and (iii) almost everywhere questions in random tournaments and general infinite random digraphs. We are thus inexorably led to statements such as "Seymour's conjecture is true "with probability one".

This is joint work with undergraduates Zach(ary) Cohn from the University of Chicago and E(liz)abeth Wright of Tulane University.

Fables, Tales, Dilemmas: Randomized Game-Theoretic Solutions to Conflict Situations

Raghav Virmani, Math Major, Class of 2003

Tuesday, March 25, 2003 - 4:00, Woods Labs 230

Abstract: In their groundbreaking work, "Theory of Games and Economic Behavior," Princeton scholars John von Neumann and Oskar Morgenstern presented a game-theoretic solution to a dilemma seen by the latter in Sir Arthur Conan Doyle's classic, "The Adventures of Sherlock Holmes." Starting with an overview of game theory as developed by von Neumann and Morgenstern, this talk discusses von Neumann's solution of the Holmes-Moriarty dilemma, using his seminal "randomization" strategy, as presented in James Case's article, "Paradoxes Involving Conflicts of Interest," in the American Mathematical Monthly. The talk also discusses concepts such as strictly and non-strictly determined games, convex combinations, and dominance relations, along with a proof of the famous "minimax" theorem.

Parrondo's Paradox

Tina Hill, Math Major, Class of 2003

Tuesday, March 25, 2003 - 3:00, Woods Labs 230

Abstract: Have you ever wondered how in a game of chess, pieces are sometimes sacrificed in order to win the overall game? Or in engineering how two unstable systems can be combined to become stable? In these scenarios losing strategies are combined to have a winning outcome. This scenario works in game theory as well. There is now hope for all of the losers out there! Two losing games can be played together to become winning. This surprising result was discovered by Dr. Juan Parrondo.

A demonstration of Parrondo's paradox will be given by playing three games. Game One and Game Two both have losing probabilities. However, even though Game Three is some random combination of the losing Games One and Two, it ends up being a winning game. Markov processes are key in proving this phenomena.

Now everyone don't run out and catch the next flight to Las Vegas. Parrondo's paradox sadly does not apply to casinos. Also, it only works under certain conditions, so we will study the parameters under which this miraculous paradox exists.

How do you measure a cloud?

Jennifer Sinclair, Math Major, Class of 2003

Thursday, March 6, 2003 - 4:00, Woods Labs 230

Abstract: Historically, mathematicians were merely concerned with smooth or regular functions and sets. Recently, mathematicians have realized that "non-smooth" sets are not only of pure mathematical interest; they also provide improved representations of nature in comparison with our "classic geometry." The geometry of fractals is a relatively new area of interest for mathematicians and physicists. The underlying mathematics covers many areas of study including analysis, probability, number theory, and more. Many of the results are somewhat intuitive and others are surprising. I will introduce two different measurements of fractals, Hausdorff and Box-Counting dimensions. I will need to explain what it means to be a strictly self-similar set described by iterated function schemes, using examples and diagrams for clarity. Under specified conditions, a fractal has equal Hausdorff and Box-counting dimensions. This result applies to fractal sets such as the Cantor set, the Sierpinski gasket, and others. I will provide a proof of this result using necessary principles. Fractals can be randomized by flipping a coin or using a random number generator to decide where to add or eliminate a segment of the fractal construction. Random fractals provide a model closer to the real world, and thus model the distance around a cloud or the length of a coastline. I will be available for further discussion during the reception following the lecture.

A presentation of the prime number theorem by statistical methods

Tamara Bailey, Math Major, Class of 2003

Tuesday, March 4, 2003 - 2:00, Woods Labs 230

Abstract: In a search for a law governing the distribution of the primes, the decisive step was taken when mathematicians gave up futile attempts to find a simple mathematical formula yielding all the primes or giving the exact number of primes contained among the first n integers, and sought instead for information concerning the average distribution of the primes among the integers.

This talk gives a brief history of the prime number theorem, presents a heuristic discussion of the theorem, and describes some implications of this theorem.

The Mathematics Behind the RSA Encryption / Decryption Algorithm

Greg Garland, Math Major, Class of 2003

Monday, February 24, 2003 - 4:00, Woods Labs 230

Abstract: The RSA encryption and decryption algorithm allows for present day communication using secret code. The algorithm itself relies mainly on Euler's generalization of Fermat's Little Theorem, which is hundreds of years old. The RSA algorithm provides assurance that only in the rarest of circumstances can encrypted messages be decrypted by anyone other than the intended recipient. Come find out what it takes for the RSA encryption and decryption algorithm to work, as well as the possibility of cracking the code!

The Most Flexible Owl in the World: an introduction to circle packing

Matt Cathey (C' 98), University of Tennessee

Friday, October 18, 2002 - 4:00, Woods Labs 136

Abstract: A circle packing is a configuration of circles such that the circles share tangencies according to a prescribed pattern. Circle packing came to prominence in a 1985 talk by William Thurston. Thurston asserted that circle packings could be used to approximate classical analytic maps from the unit disk to any simply connected domain in the plane. The existence of these maps was shown by Riemann, but his proof gave no method of actually finding them. This talk will give an overview of the basics of circle packing, and will exhibit some of its applications, including Riemann maps and current research in brain mapping. The talk will be accessible to undergraduates and will have plenty of illustrations.

Accounting for Uncertainty in the Solution of a Simple Seismic Travel Time Inversion Problem

Nick Bennett (C' 91), Schlumberger-Doll Research

Monday, September 30, 2002 - 4:00, Woods Labs 230

Abstract: Solving a geophysical inverse problem means determining the parameters of an earth model given a set of measurements. In solving many practical inverse problems, accounting for the uncertainty of the solution is very important to aid in decision-making. In this work, we address the problem of determining the posterior uncertainty of the solution of a simple one dimensional seismic travel time inversion problem. Further, we shall indicate how solving this inversion problem can play an important role in the process of safely drilling an oil well.

One inversion methodology is to pick a one dimensional layered model, prepare a prior mean and covariance matrix for the model parameters, compute a posterior mean and covariance, and then to sample from this posterior distribution. We also sample different choices of layered models in proportion to their posterior probability. These samples span the uncertainty of the inverse problem solution, accounting for both the uncertainty in the choice of model and of the model parameters.

As time allows, we will also discuss the solution of the same one dimensional seismic travel time inversion problem for models that arise from decimated wavelet bases. The same inversion methodology can be applied where one samples different choices of basis decimation and wavelet model parameters.

This talk makes prominent use of Bayes' Rule (which will be reviewed) and provides some interesting examples of normal random variables. "Data Analysis: A Bayesian Tutorial" (D.S. Sivia) is a good reference for this talk.

The History of the Liberal Arts

Hardy Grant, York University, Toronto, Canada

Wednesday, September 18, 2002 - 4:30, Convocation Hall

Abstract: Until his recent retirement Professor Grant specialized in an undergraduate course in the humanities that focussed upon the role of mathematics in the history of western culture.

The liberal-arts tradition can be traced back continuously for 2500 years, to ancient Greece. Vigorous debate over the goals and methods of teaching led in time to the concept of a number of subjects comprising an enkuklios paideia, a "well-rounded education"; and the Romans, taking over the idea, spoke of such disciplines (artes) as appropriate for the training of a free (liber) citizen of the Republic.

Only in the "Dark" Ages did the number and identity of these special subjects reach their classic forms -- four of them (the quadrivium) mathematical, three others (the trivium) verbal. In Christian contexts, including the first universities, these disciplines were seen as preparation for higher studies in philosophy and theology. The unique place of the "seven sisters" in European culture and education was much weakened by the vast expansion of intellectual horizons after the 12th century, and of course the term "liberal arts" eventually ceased to refer to a specific set or number of subjects; but the ideals cherished in the ancient tradition live on in our time.

Professor Grant will sketch the history of that legacy in a talk sponsored by the University Lecture Series. The public is cordially invited to attend.

Finite Fields: An Application in the Internet

Stephen Marlowe, Math/CS Major, Class of 2002

Friday, April 19, 2002 - 2:00, Woods Labs 230

Abstract: Distributing a large data set to many users across the network is a common task, but currently it is one that is not extremely efficient. A new protocol to address some of the current issues with this type of bulk data distribution will be presented. In addition to the distribution algorithm, the underlying abstract algebra, on which the system is based will be discussed.

Drag Force on a Sphere

Troy Eichenberger, Math Major, Class of 2002

Wednesday, April 17, 2002 - 4:00, Woods Labs 230

Abstract: Drag force is the resistance on an object moving through a fluid. An exploration of the drag a sphere moving through a fluid will be presented. Dimensional analysis is applied to reduce the number of variables in the experiments. Experimental results are derived to find a relationship between the drag coefficient and the Reynolds number. A resulting log-log graph is then used to develop two models for the drag force. The models are subsequently used to derive differential equations for spheres falling through fluids. Practical applications of the concept will be offered by examples.

The Uniqueness Of The Infinite Open Cluster For 2-Dimensional Bond Percolation

John Sears, Math Major, Class of 2002

Friday, April 12, 2002 - 2:00, Woods Labs 230

Abstract: The proof, which I presented last semester, that the critical probability for bond percolation on the 2-dimensional integer lattice is 1/2 relied heavily on the fact that the infinite open cluster, if it exists, is unique. Harris first obtained the result in 1960 as a by-product of showing that the percolation probability is zero for an edge probability of 1/2. Harris's work relied on a great deal of geometric complexity, and we will thus follow the refined work of Burton and Keane to show that the infinite open cluster is unique (1989).

Colley's Matrix Method

Devin Delaughter, Math Major, Class of 2002

Monday, April 1, 2002 - 4:30, Woods Labs 230

Abstract: Wes Colley's Matrix Method of ranking college football teams has been included as one of the deciding polls to determine college football's national champion as part of the Bowl Championship Series. The BCS as it is commonly referred, is a system that is used to determine the #1 and #2 teams in college football given the fact that it is impossible for all 117 Division I-A college football teams to play each other in one season. The BCS combines "human" polls with various computer-generated polls to solve this problem.

College football became more complicated in the sense of determining who is #1 as interest and participation grew in the 1900's. Using various mathematical tools, Professor Wes Colley of Princeton University has developed a simple method of ranking college football teams. Expanding upon method earlier used in rating chess players as well as other arenas of competition, Colley uses the mathematical method introduced by Pierre-Simon Laplace to develop an initial ranking of each team. Initially considering a flat distribution of expected values of outcomes, Colley alters the initial rankings using an iterative scheme to adjust for strength of schedule. Using the iterative scheme he shows that each teams ranking converge to a final or true ranking. As Colley continues in this method he recognizes that simple linear equations can be formed and can be put into matrix form. Using simple manipulations of these matrices will generate the same final results as the iterative scheme. Colley goes on to prove that the iterative method will always converge for each team, as well as proving the conservation of a 1/2 average ranking of the teams. Finally he goes on to prove that the Colley Matrix is positive definite thus will always produce a final ranking for the teams.

The Shooting Method

Bryan Daniele, Math Major, Class of 2002

Tuesday, March 12, 2002 - 1:30, Woods Labs 230

Abstract: Everyone that has dealt with math seriously has had to deal with numerical approximations. As one digs deeper into the realm of numerical approximation one might stumble upon the problem of approximating a non-linear differential equation. While there are numerous methods to approximate linear differential equations, the choices become quickly scarce when one considers the idea of non-linear systems. The Shooting Method is one of the few numerical approximation techniques that can be used efficiently for both non-linear and linear differential equations. One of the niceties of the method is the simplicity of application. Another strong benefit is that when it converges, it is usually more efficient than other numerical methods. The Shooters Method's technique will be presented with examples and compared to other well known numerical methods.

Task Planning for Intelligent Mobile Robots

Gabriel J. Ferrer, University of Virginia (Mr. Ferrer is a candidate for the CS position)

Monday, February 25, 2002 - 4:30, Woods Labs 230

Abstract: The task planning problem involves devising a sequence of actions that enables a robot to achieve formally stated goals given a particular world description. In general, planning is a very hard problem. Many interesting planning problems are NP-Complete, PSPACE-Complete, or undecidable.

Furthermore, in the context of a planner being used to control a robot, events beyond the control of the robot may occur that cause its plan to fail. When a plan fails, it is necessary to devise a new plan that accommodates the exogenous event or events that caused the old plan to fail. This new plan must be devised in a timely manner.

Anytime planners seek to allow the system to exchange execution time for solution quality. I have analyzed how local subplan replacement could be utilized to implement an anytime planning system that devises new plans to accommodate plan failures.

My results demonstrate that, in a number of interesting scenarios, this approach enables better overall system performance than global replanning in a dynamic environment. I also describe some theoretical and practical limitations of this approach and suggest directions for further research.

The Transendence of e

Naylene Orr, Math Major, Class of 2002

Friday, January 18, 2002 - 1:30, Woods Labs 230

Abstract: The number e is a relatively new comer on the mathematical scene, especially in comparison to pi's immense background. However, there have already been many advances in the understanding of e. In 1873, Charles Hermite established one very memorable piece of history for this number. He discovered and proved that e is a transcendental number. The proof itself is very intricate and detailed, and a simplified version provided by Hilbert shows to be very challenging as well. This revelation led the way for many other mathematicians to investigate other aspects of e, and thus contributing to a better understanding of this number.

Grappling With Intractability: Beyond NP-Completeness

William E. Duncan (C' 00), University of Tennessee PhD Candidate

October, 2001, Woods Labs 230

Abstract: We define the notion of fixed-parameter tractability and review a hierarchy of complexity classes. Several fundamental notions are highlighted, including the apparent fixed-parameter intractability of some classes of problems.

We then describe a basic tool for proving inapproximability and show the relationship between inapproximability and fixed-parameter tractability.

We conclude this talk with a brief discussion of immersion order problem: results, ongoing work and open problems.

The Critical Probability for Bond Percolation on the Two Dimensional Integer Lattice

John Sears, Math Major, Class of 2002

Friday, November 16, 2001 - 1:00, Woods Labs 230

Abstract: Fundamental to percolation theory is the concept of a critical probability. For the probability of a given edge being open less than the critical probability, there exists zero probability that a given vertex lies in an infinite cluster of open edges. For the probability of a given edge being open greater than the critical probability, there exists a strictly positive probability that a given vertex lies in such a cluster of open edges. A brief introduction of bond percolation will be offered, followed by establishing nontrivial bounds of the critical probability for bond percolation in two dimensions, and finally a short proof of the critical probability on the two dimensional lattice.

"Quite apart from the fact that percolation theory has its origin in an honest applied problem, it is a source of fascinating problems of the best kind for which a mathematician can wish: problems which are easy to state with a minimum of preparation, but whose solutions are apparently difficult and require new methods. At the same time, many of the problems are of interest to or proposed by statistical physicists and not dreamed up merely to demonstrate ingenuity." -- Geoffrey Grimmett

Mathematical Modeling for Genetic Trends

Anjali B. Patel, Math Major, Class of 2002

Tuesday, October 2, 2001 - 4:00, Woods Labs 230

Abstract: There seems to be a growing number of problems in genetics which can be modeled through mathematical interpretation. An in-depth look at population genetics with the application of discrete mathematics, calculus, and numerical analysis will help to establish criteria for the stability of genetic equilibrium and reproductive capacity of a biological species. A simulation of a growing population reveals a strong dependence upon a minimum slope for convergence through the use of a recursive relationship and the mean value theorem.