20112012 Academic Year Colloquium Schedule 

September 1, 2011
Title: 
Planning for Graduate Study in Mathematics and Computer Science

Speaker:  David A. Reimann
Associate Professor
Mathematics and Computer Science
Albion College
Albion, Michigan

Abstract: 
A degree in mathematics or computer science is excellent preparation for graduate school in areas such as mathematics, statistics, computer science, engineering, finance, and law. Come learn about graduate school and options you will have to further your education after graduation.

Location: 
Palenske 227

Time: 
3:30

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September 8, 2011
Title: 
Ramanujan's Lost Notebook

Speaker:  Bruce C. Berndt, '61
Professor
Department of Mathematics
University of Illinois at UrbanaChampaign
Urbana, Illinois

Abstract: 
In the spring of 1976, while searching through papers of the late
G. N. Watson at Trinity College, Cambridge, George Andrews found a
sheaf of 138 pages in the handwriting of Srinivasa Ramanujan,
generally regarded as India's greatest mathematician. In view of
the fame of Ramanujan's earlier notebooks, Andrews naturally
called these papers Ramanujan's "lost notebook." This work,
comprising about 650 results with no proofs, arises from the last
year of Ramanujan's life and represents some of his deepest work.
Since many in the audience may not be familiar with Ramanujan, we begin
with a brief biography. Second, we provide a history of the lost
notebook. Third, a general description of the topics found in the lost
notebook will be provided. For some of the topics, in particular,
qseries, theta functions, mock theta functions, continued fractions, partitions,
and infinite series, we offer some details. In the time remaining, the
fourth portion of the lecture will be devoted to a more detailed
discussion of one of the topics prominently addressed in the lost
notebook, namely continued fractions.

Location: 
Palenske 227

Time: 
3:30 PM

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September 15, 2011
Title: 
Symmetry Groups: The mathematical connection between patterns in Moorish architecture and the artwork of M.C. Escher

Speaker:  David A. Reimann
Associate Professor
Mathematics and Computer Science
Albion College
Albion, Michigan

Abstract: 
The mathematical structure of symmetrical patterns can be studied using group theory. The Moors built many magnificent buildings richly decorated with geometric patterns during their rule of the Iberian peninsula (7111492). The graphic artist M.C. Escher visited southern Spain in 1922 and was captivated by the patterns that richly decorate the architecture of the Alhambra, Alcazar, and other Moorish buildings. After a second visit to Spain in 1935, Escher became obsessed with creating patterns of interlocking figures based on these elaborate tiling patterns. While Escher had no formal mathematical training, he used mathematical methods grounded in scientific literature to study these patterns. We will view these patterns through the lens of group theory, one of the great mathematical accomplishments of the 19th century. This talk will be highly visual with many pictures of Escher's works and Moorish architecture.

Location: 
Palenske 227

Time: 
3:30 PM

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September 22, 2011
Title: 
Infinitely Reasonable: Science Revises the Heavens (Program 5 from The Day the Universe Changed)

Speaker:  James Burke (Virtual)
Science Historian
James Burke Institute

Abstract: 
This program explains how from 1550 and forward science began to undermine the Churchsanctioned Aristotelian doctrine of the universe, in which the Sun and all the planets revolved around the Earth. In its place, was established the model to which we adhere today of a clockwork universe, governed by discoverable laws of math and physics.

Location: 
Palenske 227

Time: 
3:30 PM

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September 29, 2011
Title: 
Pythagoras, Fermat and Euler: a progression of Diophantine equations

Speaker:  David C. Murphy
Associate Professor
Department of Mathematics and Computer Science
Hillsdale College
Hillsdale, Michigan

Abstract: 
When can a kth power be written as a sum of other kth powers? Thinking of the Pythagorean Theorem, several examples of squares that are equal to the sum of two other squares will likely come to your mind. For higher powers, however, Pierre de Fermat claimed that it is impossible to write a cube as a sum of two cubes or any power greater than the second as a sum of two others. (This is Fermat's Last Theorem.) While working on Fermat's Last Theorem, Euler conjectured that it is impossible to express a kth power as a sum of fewer than k others, but suggested that it should be possible when you allow k or more summands. If the first part of his conjecture is true, Fermat's Last Theorem would be a special case. In this talk, I will discuss these problems. In particular, I will present both parts of Euler's conjecture, give some answers, and ask more questions.

Location: 
Palenske 227

Time: 
3:30 PM

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October 6, 2011
Title: 
An Introduction to Unit Testing and Test Driven Design

Speaker:  Jeff Jia, `06 and Rob Murdoch, `05
Developers
Menlo Innovations
Ann Arbor, Michigan, USA

Abstract: 
A presentation on the basics of Test Driven Design and Development, with a focus on creating enough curiosity in the subject for further self research.

Location: 
Palenske 227

Time: 
3:30 PM

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October 13, 2011
Title: 
Can You Hear the Harmonic Series Diverging?

Speaker:  Eric Barth
Professor of Mathematics
Mathematics Department
Kalamazoo College
Kalamazoo, Michigan

Abstract: 
The language of mathematics often coincides with that of music. In this talk,I explore the connection suggested by the harmonic series, a mathematical name that has a powerful suggestion of music in it. The harmonic series is wellknown to mathematics students because it provides an interesting example of divergence. Can the musical content of the harmonic series help us understand how this divergence happens? By developing definitions that assign musical sounds to the terms of a series in a natural way, we can produce sonic versions of several convergence theorems. This leads us to the conclusion that yes, you can hear the sound of divergence in the harmonic series if you know what to listen for! Along the way, we will also find satisfying musical examples of convergent geometric series.

Location: 
Palenske 227

Time: 
3:30 PM

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October 27, 2011
Title: 
What if my paper clip was made out of NiTi?

Speaker:  Darren E. Mason
Associate Professor of Mathematics
Mathematics and Computer Science
Albion College
Albion, MI

Abstract: 
Shape memory alloys (SMAs) are fascinating structures in materials science and engineering. Fundamentally, these are materials that are formed at a high temperature and then allowed to cool. Once at a lower temperature (often room temperature), SMAs can be bent and twisted into heavily deformed shapes that appear to be permanently damaged. However, the material has instead reorganized itself crystallography at an atomic level in a reversible fashion. Moreover, upon reheating, the alloy "remembers" its original atomic configuration and can return to it almost instantly. This transformation is viewed by the naked eye as a return to its original formed shape. This microstructural change is an example of a common phenomenon in mechanics and materials science called "phasetransitions." The theoretical study of such material reconfigurations is typically conducted mathematically using the machinery of variational calculus with a view towards material energy minimization.
In this talk we will provide a physical overview of the fundamental behavior of SMAs as well as discuss the basic mathematical framework used to predict such transitions. In the end, it is all about optimization.

Location: 
Palenske 227

Time: 
3:30 PM

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November 3, 2011
Title: 
Dynamic Programming and Contraction Mappings

Speaker:  Daniel Christiansen
Professor of Economics and Management
Economics and Management
Albion College
Albion, Michigan, USA

Abstract: 
This talk provides an introduction to discounted dynamic programming with an infinite time horizon. The role of the optimal return function is discussed, and the contraction mapping theorem is used to provide an existence theorem for an optimal policy or plan.

Location: 
Palenske 227

Time: 
3:30 PM

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November 10, 2011
November 17, 2011
Title: 
On Consistent Bankruptcy Rules

Speaker:  Michael A. Jones
Associate Editor
Mathematical Reviews
American Mathematics Society
Ann Arbor, MI

Abstract: 
The Talmud rule is a method to determine how to allocate an estate
(an amount of money) to two or more individuals who are owed collectively
more than the estate. Using data from the Egyptian Talmud, I will examine
the puzzle of how the data led to the rule and I will give some of the
history of the problem. The Talmud rule is one of a class of consistent
rules. I will demonstrate how such rules (as well as the proportional
rule) can be used to define a dynamic procedure for which the bankruptcy
solution is the unique attractive fixed point.
This is joint work with Jennifer M. Wilson, Eugene Lang College,
The New School for Liberal Arts, New York, NY.

Location: 
Palenske 227

Time: 
3:30 PM

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January 26, 2012
Title: 
Differential Equations and Projective Geometry

Speaker:  Robert R. Bruner
Professor
Mathematics
Wayne State University
Detroit, Michigan

Abstract: 
After a quick introduction to the projective plane, we show
that extending a differential equation to the projective plane
is a quick and effective way to study the asymptotic behavior
of its solutions. The simplest approach leads to equations
with ugly singularities, so we also show how to use the method
of rescaling time to desingularize them.
The main theorem is as follows:
Theorem: Let x' = f(x,y), y' = g(x,y) be a polynomial differential
equation in R^{2} of degree N. Then either
 there are at most N+1 possible slopes "at infinity" for the unbounded trajectories, or
 there are at most N1 slopes which are omitted: all other slopes "at infinity" actually occur.
The theorem is proved by exhibiting the polynomial whose roots are the
possible slopes in case (1), or the possibly omitted slopes, in case (2).
It is simple to apply and gives information that would otherwise be difficult
to extract.
The only real prerequisite will be differentiation and integration
of functions of one variable. The approach to differential equations
will be qualitative and intuitive, so the talk could also serve as a
good introduction to the geometric point of view on differential equations.
The audience will be left with an open ended list of examples and applications
to explore.

Location: 
Palenske 227

Time: 
3:30 PM

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February 2, 2012
Title: 
Brunelleschi's Dome: A Room With A View

Speaker:  Daniel E. Bollman
Architect and Owner
east arbor architecture
East Lansing, Michigan

Abstract: 
In August 1418, the City of Florence, Italy commissioned a competition design of the dome of the
Florence cathedral. Although construction on the cathedral began in 1296, the dome's design  which
was the largest span of its time  was not proposed until 1367. Even then, its construction details
remained to be determined. The competition's entrants were instructed to establish the means of
construction, as well as the method of raising tons of brick, stone and marble, 180' vertically to the drum
where the dome would be built.
The competition's eventual winner, architect Filippo Brunelleschi, confronted these obstacles, employing
his ingenuity and skill to construct the first dome built since the time of the Romans.
The talk will feature a brief historical overview, followed by mathematical issues that address the
problems of construction and logistics. The discussion includes structural analyses, the geometric
properties of Brunelleschi's dome, external forces acting on the dome and the forces internal to
Brunelleschi's sitebuilt construction machines.

Location: 
Palenske 227

Time: 
3:30 PM

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February 9, 2012
Title: 
Sum of Continued Fraction Expansions

Speaker:  Brad Emmons, '97
Associate Professor
Mathematics
Utica College
Utica, New York

Abstract: 
Continued fractions can be thought of as basefree expressions of real numbers and their utility in solving problems such as Pell's equation is wellunderstood. In particular, if you take the square root of a nonsquare integer $D$, the continued fraction expression of the $\sqrt{D}$ is known to be periodic. Yet the length of the
period is slightly mysterious, and much of the available literature explores the length of this period. However, there is no literature on the sums of the terms in the period.
Related to the continued fraction expansion of $\sqrt{D}$ is the quadratic form $Q(x,y) = x^2  Dy^2$. This can be seen by factoring the form as $x^2  Dy^2 = (x + \sqrt{D}y)(x  \sqrt{D}y)$. A particularly useful technique for investigating this quadratic form is the topograph. The topograph of this particular form is
known to be periodic, and this period is somewhat mysterious as well. However, this period is not the period of the continued fraction representation of $\sqrt{D}$, but rather it is related to the sum of the terms of the continued fraction representation.
In this talk, we will introduce continued fractions, topographs of quadratic forms, and show how topographs can be used to help find an upper bound for the growth of the period sequence. This talk will be accessible to a general audience.

Location: 
Palenske 227

Time: 
3:30 PM

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February 16, 2012
Title: 
Choosing the right quantum cosmology

Speaker:  Rachel Maitra
Visiting Assistant Professor of Physics
Department of Physics
Albion College
Albion, MI

Abstract: 
In order to model the largescale evolution of the universe, it is necessary to make simplifying assumptions as to its approximate shape. One typically assumes the universe is highly symmetric, then imposes these symmetries within Einstein's theory of general relativity to solve for a complete history starting with the Big Bang. Such a symmetryreduced model of the universe is known as a cosmology.
However, the Big Bang is also a point of singularity where general relativity breaks down, yielding an infinity when we attempt to compute the curvature of spacetime. To obtain a coherent account of the universe's origin, we must construct a quantum version of general relativity able to address the microscopic geometry of spacetime and hence to reveal the state of the universe when it was very tiny. Quantizing general relativity is a formidable endeavor which has been underway for the past threequarters of a century.
Quantizing a cosmological model, on the other hand, is relatively straightforward. The difficulty arises when we ask whether this model is accurate to the history of the universe which would be predicted by a full quantization of general relativity.
In this talk, I will discuss my ongoing work with collaborators using results from a computer simulation to choose promising quantum cosmologies. We parametrize a family of quantizations and construct a series solution to a differential equation to assess each quantization's prediction about the infancy of our universe.

Location: 
Palenske 227

Time: 
3:30 PM

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February 23, 2012
March 1, 2012
Title: 
Digital Nation: Life on the Virtual Frontier

Speaker:  PBS FRONTLINE Video

Abstract: 
Within a single generation, digital media and the World Wide Web have transformed virtually every aspect of modern culture, from the way we learn and work to the ways in which we socialize and even conduct war. But is the technology moving faster than we can adapt to it? And is our 24/7 wired world causing us to lose as much as we've gained?
In Digital Nation: Life on the Virtual Frontier, FRONTLINE presents an indepth exploration of what it means to be human in a 21stcentury digital world. Continuing a line of investigation she began with the 2008 FRONTLINE report Growing Up Online, awardwinning producer Rachel Dretzin embarks on a journey to understand the implications of living in a world consumed by technology and the impact that this constant connectivity may have on future generations. "I'm amazed at the things my kids are able to do online, but I'm also a little bit panicked when I realize that no one seems to know where all this technology is taking us, or its longterm effects," says Dretzin.
See www.pbs.org/wgbh/pages/frontline/digitalnation/ for more information.

Location: 
Palenske 227

Time: 
3:30 PM

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March 22, 2012
Title: 
The Separability Problem in Referendum Elections: Some Recent Developments

Speaker:  Jonathan Hodge
Associate Professor and Assistant Chair
Mathematics
Grand Valley State University
Allendale, MI

Abstract: 
In referendum elections, voters are often required to register simultaneous votes on multiple proposals. The separability problem, first identified in the late 1990s, occurs when a voter's preferences on one or more proposals depend on the known or predicted outcomes of other proposals. In this talk, we will survey several recent developments pertaining to the separability problem, including: (1) structural properties of interdependent preferences; (2) the impact of separability on election outcomes; (3) causes and models of nonseparability; and (4) the potential of iterative voting to solve the separability problem. This talk should be accessible to most undergraduates; in fact, most of the results in it were discovered and/or proved by undergraduates!

Location: 
Palenske 227

Time: 
3:30 PM

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March 29, 2012
Title: 
One! ... No, Zero! ... No, Both!

Speaker:  Tim Rambo, `09
PhD Candidate
Electrical Engineering and Computer Science
McCormick School of Engineering, Northwestern University
Evanston, Illinois

Abstract: 
Traditionally, bits are constrained to have a single value, either 0 or 1. Interestingly, if bits are treated as quantum mechanical objects, they are allowed to be a combination of both 0 and 1. Using this and other effects of quantum mechanics, it is possible to perform certain tasks better than the best known methods which rely on traditional (classical) bits. One example is the quantum algorithm for factoring large numbers, which is exponentially faster than the best known classical technique. Another example is that quantum bits can be used to distribute unconditionally secure cryptographic keys to geographically separated parties. Unfortunately, there are limited known useful computational applications for quantum computing, possibly because the language for describing quantum computation is complex and opaque to our classicallytrained minds. In light of this, much work has gone into finding more intuitive models for quantum computers. In this talk, I will present the
mathematical description of quantum bits and operations. I will also discuss in detail two example uses of quantum information: quantum key distribution and quantum search. Finally, I will conclude by showing a new (hopefully) more intuitive model for quantum computation and our work in designing a proof of principle device to demonstrate the feasibility of this model.

Location: 
Palenske 227

Time: 
3:30 PM

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April 5, 2012
Title: 
Surface Energy, Magnesia, and the Kaczmarz Algorithm

Speaker:  Darren E. Mason
Associate Professor of Mathematics
Mathematics and Computer Science
Albion College
Albion, MI

Abstract: 
Surface energy associated with a free plane of atoms in a crystalline solid is manifestly important for processing and predicting the behavior of polycrystalline materials. Minimization of the surface energy at a triplejunction separating adjacent grains requires satisfaction of a fundamental partial differential equation first posed by Conyers Herring in 1952. Using magnesia (MgO) as a test material, atomic force microscopy is used to gather generally noisy experimental data from equilibrated thermal grooves circumscribing island grains. This dataset is then required to satisfy the Herring equation at many material locations along the thermal grooves, leading to a large and overdetermined system of linear equations. The corresponding inverse problem is then solved using a novel technique that is statistical in nature, multiscale in implementation, and draws on the iterative projection algorithm due to S. Kaczmarz in 1937. The resulting discrete solution represents a st
atistically significant representation of the surface energy of MgO as a function of surface orientation.

Location: 
Palenske 227

Time: 
3:30 PM

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April 12, 2012
Title: 
Chutes and Ladders for the Impatient

Speaker:  Michael A. Jones
Associate Editor
Mathematical Reviews
Ann Arbor, MI, USA

Abstract: 
I will review the rules and game board for Chutes and Ladders, define a Markov chain to model the game, and describe how properties of Markov chains can be used to determine the expected length of the game. Because the resulting Markov chain has 101 states, the analysis is first done for a 10state variation in which the board has a single chute and a single ladder. The approach is used to determine the optimal spinner range to minimize the expected number of turns for a player to complete the game. This allows one to modify the game so it takes less time to play—perfect for the impatient player!

Location: 
Palenske 227

Time: 
3:30 PM

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April 26, 2012
Title: 
Uses of mathematics in Geographical Information Systems

Speaker:  Alexander Jasinowski, `12
Computer Science Major

Abstract: 
Geographical Information Systems (GIS) and Remote Sensing (RS) allow us to map and study our world. they utilize some of the latest technology as well as some of the most basic. There are many applications of mathematics used in GIS and RS to derive an incredible quantity of information about our world from massive sets of satellite imagery and data. In my presentation I will talk about two small areas: Spatial Autocorrelation and Normalized Difference Vegetation Index (NDVI). Spatial Autocorrelation deals with how data is related to itself in an XY plane, and NDVI analysis shows us the relative health of vegetation.

Location: 
Palenske 227

Time: 
3:30 PM

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April 26, 2012
Title: 
Math in the Madness

Speaker:  Luke Walker, `12
Mathematics Major

Abstract: 
Learning math can be unexciting for the average teenager. To spark an interest in this amazing subject, teachers must find ways to connect math to the interests of the students. My passion in math and basketball, so I combined these two into a lesson on the statistics behind the NCAA Basketball Tournament in March. Students will explore the field of the tournament to determine their winner based on the calculation they perform each round. Through all the fun, students practice calculating percentage, average, mean, median, and mode while also learning to think outside of the norm.

Location: 
Palenske 227

Time: 
3:50 PM

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April 26, 2012
Title: 
Mathematics used in Magnetic Resonance Imaging.

Speaker:  Will Spencer, `13
Mathematics Major

Abstract: 
MRI is an indispensable diagnostic tool in the medical field which may be used to
identify certain abnormalities in an individual in a noninvasive way. The mathematics behind
conversion of the data collected lays the foundation for making such data interpretable by
those who need to use it. I will briefly elaborate on the physics underlying nuclear
magnetic resonance to build an understanding of the concept, coupled with a very brief
description of what NMR is used for in the medical field. Then, I plan to discuss how the signals
collected are converted into images using mathematics.

Location: 
Palenske 227

Time: 
4:10 PM

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April 26, 2012
Title: 
Fractals: Modeling Nature with Beauty

Speaker:  Holly Williams, `12
Mathematics Major

Abstract: 
Fractals bridge the connection between mathematics and art as we know it. Fractals
can be mathematically represented as a set of numbers with complex ratios but, more
importantly, fractals can be characterized by their "self similar" tendency. When a fractal
image is zoomed in, the new zoomed view looks very similar to the original picture.
Examples of self similar, fractal like objects in nature are abundant and include trees,
mountains, and veins. Benoit Mandelbrot coined the term "fractal" and realized the
importance of this newfound mathematics branch in the 1950's. Despite Mandelbrot's
important realization, members of the mathematical community had a hard time accepting
Mandelbrot's work. Math was supposed use the language of math, which according to
Galileo was triangles, not fractals. But, after much whoha, Mandelbrot's work was finally
accepted. Mandelbrot's famous set has inspired many works of art, and changed the
computer graphics industry forever.

Location: 
Palenske 227

Time: 
4:30 PM

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