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## 2011-2012 Academic Year Colloquium Schedule

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### 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 Citation Click for BibTeX citation

### September 8, 2011

 Title: Ramanujan's Lost Notebook Speaker: Bruce C. Berndt, '61 Professor Department of Mathematics University of Illinois at Urbana-Champaign 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, q-series, 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 Citation Click for BibTeX citation

### 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 (711-1492). 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 Citation Click for BibTeX citation

### 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 Church-sanctioned 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 Citation Click for BibTeX citation

### 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 k-th power be written as a sum of other k-th 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 k-th 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 Citation Click for BibTeX citation

### 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 Citation Click for BibTeX citation

### 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 well-known 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 Citation Click for BibTeX citation

### 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 "phase-transitions." 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 Citation Click for BibTeX citation

### 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 Citation Click for BibTeX citation

### November 10, 2011

 Title: Applications of Dynamic Programming in Macroeconomics Speaker: Kotaro Yoshida Assistant Professor Economics and Management Albion College Albion, Michigan Abstract: Location: Palenske 227 Time: 3:30 PM Citation Click for BibTeX citation

### 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 Citation Click for BibTeX citation

### 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 R2 of degree N. Then either there are at most N+1 possible slopes "at infinity" for the unbounded trajectories, or there are at most N-1 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 Citation Click for BibTeX citation

### 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 site-built construction machines. Location: Palenske 227 Time: 3:30 PM Citation Click for BibTeX citation

### 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 base-free expressions of real numbers and their utility in solving problems such as Pell's equation is well-understood. In particular, if you take the square root of a non-square 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 Citation Click for BibTeX citation

### 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 large-scale 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 symmetry-reduced 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 space-time. 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 space-time 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 three-quarters 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 Citation Click for BibTeX citation

### February 23, 2012

 Title: Careers in Mathematics and Computer Science Speaker: David A. Reimann Associate Professor and Chair Mathematics and Computer Science Albion College Albion, MI, USA Abstract: A degree in mathematics or computer science is excellent preparation for employment in areas such as teaching, actuarial science, software development, engineering, and finance. Come learn about career opportunities awaiting you after graduation. Slides from the talk are available at http://zeta.albion.edu/~dreimann/talks/careers/careers.html. Location: Palenske 227 Time: 3:10 PM Citation Click for BibTeX citation

### 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 in-depth exploration of what it means to be human in a 21st-century digital world. Continuing a line of investigation she began with the 2008 FRONTLINE report Growing Up Online, award-winning 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 long-term effects," says Dretzin. See www.pbs.org/wgbh/pages/frontline/digitalnation/ for more information. Location: Palenske 227 Time: 3:30 PM Citation Click for BibTeX citation

### 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 Citation Click for BibTeX citation

### 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 classically-trained 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 Citation Click for BibTeX citation

### 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 triple-junction 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 Citation Click for BibTeX citation

### 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 10-state 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 Citation Click for BibTeX citation

### 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 Citation Click for BibTeX citation

### 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 Citation Click for BibTeX citation

### 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 Citation Click for BibTeX citation

### 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 who-ha, 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 Citation Click for BibTeX citation