The Department of Mathematics, Computer Science and Physics Conversation Series offers talks that appeal to a broad range of interests. The Conversation Series includes talks by Roanoke College faculty, students, past graduates and invited speakers from other colleges. We are especially pleased that several of our alumni have given talks on the work they have done since leaving RC.
Dr. Hermann Haberman of the National Research Council presented "The Shifting Role of Statistics in Public Policy." Dr. Haberman has served as Deputy Director of the US Census Bureau and Director of the UN Statistics Division. In a compelling talk, he made the case for the world's need for official statistics that are unbiased, whose underlying process is transparent and has a clear error structure. While social media collects an outrageous amount of data from which statistics may be computed, the need to make money means that their methods will be proprietary, not transparent. Error structure is not being reported to guide us about their statistics' significance or lack thereof, and they have little direct motivation to limit bias. Given the importance of such measures as CPI, GDP, and unemployment rate, we need to protect ourselves by supporting agencies, government or otherwise, that produce reliable official statistics.
Dr. Karin Saoub of Roanoke College presented "Fun With Infinity." After a brief discussion of what an infinite series means, Dr. Saoub presented a seemingly reasonable argument that 1-1+1-1+1-1... should equal to 1/2. If you buy that, a quick slide down a slippery slope leads you to the unfortunate conclusion that 1+2+3+4+5+... equals -1/12. First, how does a sum of integers equal a non-integer? Second, how does a sum of positive integers become negative? It's hard to say which is more disturbing. Nonetheless, Dr. Saoub showed us a "proof" of this result, and its use in a (current) textbook on string theory. The way out of this rabbit hole is to adhere to the careful definition of infinite series given in calculus, from which we must reject 1-1+1-1+1-1... as representing any number. And that removes the bizarre claim that 1+2+3+4+5+... equals -1/12. Now we just need to tell the string theorists.
Dr. Adam Childers of Roanoke College discussed "The Mathematics of Juggling" with the help of expert juggler Chris Taibbi. The connection between mathematics and juggling is that both are all about the creation of patterns. Dr. Childers showed us how to mathematically describe the patterns in juggling. The classic three-ball cascade, for example, is characterized by each ball being thrown every third beat; its pattern is simply "3." In the three-ball shower, there are two types of throws: the high toss (lasting 5 beats) and the short pass (1 beat) from hand to hand; its pattern is "51." The definitions were accompanied by entertaining demonstrations from Mr. Taibbi. We then saw theorems for determining if a sequence of numbers represents a valid juggling pattern, with the unexpected consequence that we can now invent new juggling tricks! The grand finale was a "4-handed" team juggling exhibition!
Dr. Anil Shende of Roanoke College gave a presentation on "How Many Learning Styles Are Adequate?" The first issue in discussing machine learning is to define "learning." Pattern recognition and accurate prediction lead to a usable definition. Dr. Shende explored the basic question of whether a machine could "learn" to identify a function from successive function values. Under certain limitations, he showed how this could be done, but then provided a proof that no machine can ever learn all functions. So, we must allow some mistakes: a machine that can learn all but one function is quite useful. This introduction to machine learning got us all thinking about what it means to learn, and how adept machines may become.
Dr. Rama Balasubramanian of Roanoke College gave a presentation on "Nanotechnology; Why STEM Majors Should Care." They are 20 times stronger than steel, have conductances a million times higher than copper, and are not science fiction; they are carbon nanotubes, and faculty and student researchers are growing them at Roanoke College. Dr. Bala introduced us to the small (1/1000 of a human hair) and fascinating world of nanoscience. Physical properties change at this small size scale: pure gold appears red, catalysts have higher reaction rates, and so on. Dr. Bala and her students have found a new method of making Y-junctions, useful for the fabrication of nanoelectronic devices.
Dr. Richard Grant of Roanoke College gave a presentation on "The Science of Musical Instruments." It is not often that you find trombones, didgeridoos and bagpipes in the same talk! And you need to hear the explanation of why a bagpipe equals three didgeridoos plus one trombone. Demonstrations of wave patterns and software showing each instrument's waveform and spectrum made this a multisensory experience. Demonstrations of different effects on the electric guitar further illustrate the relationships between waveforms and the tones that we hear and enjoy. The kilt goes with the playing of the pipes and honors Dr. Grant's Scottish heritage. And, no, THAT question was not answered.
Dr. Roland Minton of Roanoke College discussed "Ranking Systems: Determining #1 in College Football." After honoring his alma mater with more Clemson orange than most wanted to see, Dr. Minton showed how his college football ranking system works, and predicted the games for the upcoming weekend. His system is dependent on the relationships among the teams' schedules. The mathematics involved is primarily the reduction of a matrix. A program written by his son Greg illustrated how highly interconnected the college football world is, including a 34-team path showing Harvey Mudd beating Virginia Tech. This is related to the "small world" phenomenon and the movie trivia game Six Degrees of Kevin Bacon, as illustrated through the web site Oracle of Bacon. This, in turn, is related to PageRank, Google's original system of rating web pages. Mathematics leads to unusual and interesting connections!
Dr. Hannah Robbins of Roanoke College discussed "Crazy Dice" probabilities. The goal of this interesting puzzle is to renumber a pair of six-sided dice (using positive integers) so that the possible outcomes and their likelihoods are the same as for a pair of standard dice. An unexpected connection solves the problem: a die corresponds to a polynomial, and rolling a pair of dice corresponds to multiplying the appropriate polynomials! This connection opens up the possibility of numerous dice puzzles: all we have to do is factor the standard polynomial and re-order. For our problem, it turns out that there is exactly one re-numbering that works. So, don't be worried if Dr. Robbins pulls out a die with an 8 on it for a game of Monopoly.
Dr. Ben Hester of the National Security Agency presented "Everything You Wanted to Know About Mathematics at the NSA But Were Afraid to Ask." Dr. Hester characterized the NSA's two primary charges as "offense" and "defense": protecting our own systems and monitoring the signals of others. The NSA is the world's leading employer of mathematicians. All employees need to know mathematics, programming, and general problem-solving skills. Dr. Hester gave an overview of public key cryptography, with special emphasis on RSA encryption. This form of encryption utilizes some basic facts from number theory, and its security depends on the difficulty of factoring very large numbers into their prime factors. Dr. Hester closed the evening by describing some of the employment opportunities at the NSA, including scholarship programs and summer internships.
Johns Hopkins graduate student Erin Hackett (Roanoke College 2000) gave two presentations relating to her research "Exploring Earth's Oceans." Erin described the role of the oceans on the climate and cooling of the land we inhabit. Studies of this kind involve the unification of several sub-disciplines within physics like fluid dynamics, kinetic theory of gases, and analog electronics. She also gave a thorough discussion of ocean wave formation, variance of tides, and catastrophic events like tsunamis. Erin focused a portion of the discussion on her own research, which relates to the interchange of temperature and nutrients between the floor and surface of coastal regions. Erin's research combines both observational monitoring as well as computer modeling.
Clemson graduate student Matt Troutman (Roanoke College 2005) presented "What We Can Learn From Recently Discovered Exoplanets." Matt discussed the observational techniques for planet discovery, including some basic insights about a few specific exo-solar planets. For example, several new exo-solar planets are observed with molecular absorption lines like carbon dioxide and methane. These signatures are thought to indicate the existence of planetary atmospheres composed of these molecules. Although the newly discovered planets are not orbiting within the "habitable zone'' as understood for carbon-based lifeforms, the prospect of molecular atmosphere discovery encourages planetary astronomers. Observational and theoretical research in this realm of astrophysics aids in the understanding of our own solar system formation and evolution.
Dr. Jeff Spielman of Roanoke College presented "The Statistics of the New Hampshire Primary." With a little help from Stephen Colbert, Dr. Spielman resolved the apparent paradox of political polls with a 4-5 percent error being off by the 15 percent that the New Hampshire pre-primary polls were. The talk discussed the pervasiveness of statistics in modern political campaigns, through public opinion polls and exit interview breakdowns of voting demographics. Dr. Spielman used resampling simulations to demonstrate how unlikely the New Hampshire would have been if the pre-primary polls were accurate.
Dr. Adrienne Bloss of Roanoke College discussed "The Future of Computing." A brief history of computing was given to contrast traditional computing with the possibilities of quantum computing. The audience was introduced to Schrodinger's cat and invited to feel dizzy contemplating the strange world of quantum mechanics. Factorization of integers was presented as an example of the potential power of quantum computing. The difficulty of factorization lies at the heart of RSA and other encryption systems, but even though quantum computers render such systems obsolete, quantum cryptography promises truly unbreakable computer security. Depending, of course, on whether that cat is alive, dead or both.
Dr. David Taylor of Roanoke College discussed "Monty Hall's Sudoku Problems." The first part of the talk introduced us to the wacky world of Let's Make a Deal and the famous Monty Hall Problem. There are three doors with a car behind one door and goats behind the others. You choose a door, are shown a goat behind one of the other doors, and are given the option of switching doors. Should you? Does it matter? The second part of the talk detailed some of the interesting questions about Sudoku, especially how many different puzzles there are, and what is the minimum (or maximum) number of clues. Dr. Taylor then solved a different puzzle -- why are these two problems in the same talk? The answer gets at the heart of mathematics problems. Hypotheses are critical, and the difference between an easily solved and an impossible problem can be very subtle, but fun to discover!
Dr. Jane Ingram of Roanoke College discussed "Math + Computer Science = Pretty Pictures." The talk is designed to make us see mathematics in a new way, and showed us several ways that fractals have helped mathematicians see properties of functions in new ways. Newton's method for approximating solutions of equations produces many spectacular images, even for simple polynomials. The general idea of function iteration is behind the varied and beautiful images of the Mandelbrot set and associated Julia sets. Iterated function systems use repeated matrix multiplications and additions to create detailed and realistic images like Barnsley's fern.
Dr. Bud Brown of Virginia Tech discussed "Chocolate Key Cryptography." Bud introduced us to some of the basics of cryptography, quickly introducing the key concepts of a one-way function and a public key. A clever demonstration of public keys involved audience participation, Bud's briefcase, and a message from a student to Bud about his bow-tie being crooked! Chocolate entered the talk with another clever demo about how to publicly but securely negotiate a common key. M&M's of various colors filled the role of the key, and the student volunteers were able to find a common key without knowing each other's private key. They then, of course, ate the evidence!
Clemson graduate student Hampton Smith (Roanoke College 2007) discussed "When Skynet Has Math Homework: Computers Writing Proofs." Hampton introduced us to his graduate research on the RESOLVE proof verification system. After a brief discussion of what a proof is (and the significance of Godel's Incompleteness Theorem), Hampton showed us popular proof-checkers Isabelle and Coq, and discussed the ins and outs of using such tools. Improvements being implemented in his work were discussed. The reality of where computer proofs are currently at was contrasted with the Hollywood fantasy of Skynet.
Dr. Chris Lee of Roanoke College discussed "Evolutionary Algorithms." These are new mathematical solution techniques based on the way nature solves problems. For a basic genetic algorithm, potential solutions are modeled as a sequence (analogous to a strand of DNA) and then given rules to evolve (analogous to sexual reproduction and mutation) to a better solution. Particle swarms simulate large populations of potential solutions in which solutions move in the direction of better solutions and (ideally) swarm to the optimal solution. These techniques have proved to be very powerful, finding solutions of complex problems that humans have found intractable.
Dr. Bill Franz of Randolph-Macon College discussed "Officiating Games That We Watch and Play." Given that reaction times for most humans are in the 0.2 second range, the use of clocks in football and other sports that go into the tenths or hundredths of a second may be misguided. The audience acted out a basketball play, and the five student timers recorded clock times from 0.2 to 1.8 seconds! The play illustrated Dr. Franz's point that precision is different from accuracy. The talk then examined the offside call in soccer, in which an official is asked to look in two different directions at the exact moment the ball is kicked. The limitations of vision in humans under stress combine with the fact that the ball is in contact with the foot for a significant length of time to make this call impossible.
Dr. Matt Fleenor of Roanoke College presented "Counting Planets: How Astronomers Discover Extrasolar Planets and What Makes Them Count." The talk focused on a number of recent and dramatic events in astronomy. The demotion of Pluto from planet status has received the most media attention. The discovery of a "Goldilocks" planet orbiting in just the right place for life is also noteworthy. To find planets that cannot actually be seen, astronomers infer their existence by different techniques, one measuring star "wobble" and another measuring decrease in "brightness" as the planet passes in front of its star. Dr. Fleenor expects rapid progress in finding planets as more space telescopes are deployed.
Dr. Durell Bouchard and senior Timmy Balint of Roanoke College presented "Microsoft Kinect and Motion Capture." A demonstration of Kinect illustrated its abilities and limitations to mimic the movements of a person. Dr. Bouchard then discussed the basics of how Kinect works, including its method of distinguishing one joint from another. The method is analogous to a game of twenty questions, but the human and computer labor needed to identify the best questions is intense. Timmy discussed some of his intense labor in the CAVE, training a neural network to recognize faces. Identifying the hue of the face can then help find hands and other flesh. The process of then tracking the movements of hands and face is complicated by various factors, including hands crossing the face and becoming obscured.