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Schedule - Saturday, March 8, 2008

Funding for SUMS 2008 provided by NSF grant DMS-0536991 through the MAA Regional Undergraduate Mathematics Conferences program and the Clay Mathematics Institute. Additional funding is provided by the Department of Mathematics, the Office of Institutional Diversity, the Division of Applied Mathematics, and the Department of Economics.

Except for the banquet, all SUMS events are in MacMillan Hall. Faculty talks are in room 117 while student talks are in rooms 115 and 117.

List of Faculty Speakers

The following list of speakers is now complete.

• Jonathan Farley, Mathematics, California Institute of Technology
  How to Build the Perfect Terrorist Cell: Or, What Would Osama Do? (Abstract)
  After making assumptions that we hope are not too unrealistic, we attempt to find the structure of the terrorist cell that is least likely to be disrupted upon the capture of a certain number of its members.
• Jerry Hausman, Economics, MIT
  Asymptotic Approximations and Finite Sample Performance in Econometrics (Abstract)

  For approximately 30 years most econometric estimator evaluation was based on first order asymptotic normal approximations. However, in the past 5 years econometricians have realized that these approximations provide a poor guide to actual finite sample estimator performance in the weak instrument/many instrument situation. Higher order approximations of different types have proven useful to analyzing different estimation situations that commonly arise in applied research. My talk will discuss different estimators and how they perform in given applied situations. I will also discuss new estimators that have been invented to treat situations where "optimal" first order estimators do not perform well. I will point to areas of future research where these advanced techniques may be useful.

• Richard Kenyon, Mathematics, Brown University
  Soap bubbles are round, and other consequences of the law of large numbers (Abstract)
  How do large-scale shapes form out microscopic interactions? This is one of the fundamental questions of statistical mechanics. Armed with only few tools, one of which is the so-called ``law of large numbers'', one can make a surprising amount of progress towards understanding these ``limit shapes''. We'll discuss a few other examples, in particular the limit shapes for random partitions in two and three dimensions.
• Donald Saari, Mathematics and Economics, University of California - Irvine
  Mathematics of Voting (Abstract)
  While many people tend to take voting for granted, mathematicians should not. The reason is that it requires the muscle power of mathematics to appreciate and understand what is required to ensure that election outcomes more accurately reflect the views of the voters. In this expository talk, I will outline the kind of mathematics that is needed to understand voting. As for Arrow's Theorem, forget it; after it is introduced, we will show why it does not mean what everyone has thought it meant for the last half century. Also, be prepared to leave the lecture worrying about the legitimacy of the outcome of the last election that was of importance to you.
• Roberto Serrano, Economics, Brown University
  An Economic Index of Riskiness (Abstract)
  Define the riskiness of a gamble as the reciprocal of the absolute risk aversion (ARA) of an individual with constant ARA who is indifferent between taking and not taking that gamble. We characterize this index by axioms, chief among them a "duality" axiom which, roughly speaking, asserts that less risk-averse individuals accept riskier gambles. The index is positively homogeneous, continuous, and subadditive, respects first and second order stochastic dominance, and for normally distributed gambles, is half of variance/mean. Examples are calculated, additional properties derived, and the index is compared with others.

List of Student Speakers

• Matt Buchta and Yubin Wang, Mathematics, Western Connecticut State University
  Wavelet Based Non-Parametric Regression Model for Stock Price (Abstract)
  The wavelet transform is an effective tool used for signal processing. Unlike the Fourier transform, a feature of this method allows us to analyze the time and frequency information of a signal concurrently. In this research we are going to create a new wavelet based time series method for forecasting stock market prices. We assume stock price is a function of various factors, such as previous day(s) stock price, Federal interest rate, dividend yield, etc. We apply the wavelet transform to each factor, breaking it up into approximation and detail components. We can then use multivariable linear regression on the combination of approximation and detail for each factor, allowing for more accurate next time period predictions than current financial methods allow.
• Abdulmajed Dakkak, Mathematics, University of Toledo
  Solving Partial Differential Equations with Dirichlet Boundary Conditions on the Disk and Finding their Bifurcation Points (Abstract)
  In the following talk we present our research where we solved a certain class of Partial Differential Equations (PDE) under the Dirichlet boundary condition on the disk region. We present relevant concepts from differential equations, algebra, and graph theory to find solutions to equations of the form , where is the Laplacian operator, effectively. Two main algorithms used to find the bifurcation points will be presented: tangent General Newton Galerkin Algorithm (GNGA) and cylindrical GNGA, along with the significance of these bifurcation points.
• Jeffrey Hatley, Mathematics, The College of New Jersey
  The Probability of Relatively Prime Polynomial (Abstract)
  Let be a finite field of elements. In a recent paper, Arthur Benjamin and Curtis Bennett used an elegant argument involving the Euclidean algorithm to show that, given two randomly chosen polynomials and in , the probability that they are relatively prime is . We will present this result and then discuss a generalization of the theorem to polynomials from where is a prime and . As grows larger, the differences between this problem and the original problem quickly become magnified. This talk will focus specifically on the case when .
• Xiang He, Mathematics, Williams College
  Markov Chain Convergence and Applications (Abstract)
  Markov Chain Monte Carlo (MCMC) is a seminal simulation technique widely adopted in diverse disciplines. In physics, it plays an important role in modeling phase transitions. In computer science, it is used in probabilistic approximation algorithms as well as artificial intelligence. We will first present the basic theory behind Markov Chain convergence, namely conditions of irreducibility, aperiodicity, and reversibility. An MCMC solution to the NP-complete traveling salesman problem will be presented along with other colorful examples.
• Cihan Karabulut, Mathematics, Montclair State University
  Solving Diophantine Equations Using Invariant Theory (Abstract)
  We study polynomial solutions to certain Diophantine equations. Our focus is on equations of the form , where are positive integers and are univariate coprime polynomials over the complex field . We describe an algorithm that produces polynomial solutions to such Diophantine equations.
• Brendan Kelly, Mathematics, The College of New Jersey
  Investigating the Algebraic Structure of Zero-Divisor Graphs (Abstract)
  The zero-divisor graph of a commutative ring with unity is given by a vertex set which contains all non-zero zero divisors and an edge set which contains if . This graph displays information about the multiplicative structure of the zero divisors. The talk will address the algebraic information that is embedded into these graphs.
• Scott Kominers, Mathematics, Harvard University
  Configurations of Extremal Even Unimodular Lattices (Abstract)
  We introduce the notion of an extremal even unimodular lattice and extend the results of Ozeki on the configurations of extremal even unimodular lattices. In particular, we have the following configuration results: If is an extremal even unimodular lattice of rank , , or , then is generated by its minimal-norm vectors and if is such a lattice of rank (), then is generated by its vectors of lengths and .
• Jasmine Nirody, Mathematics, New York University
  Snakes on a Plane: Mathematical Models of Limbless Locomotion (Abstract)
  Snake locomotion has always been regarded as mysterious and foreign, and has been the subject of much research. Previous studies regarding snake motion, however, were mostly constrained to the biological sciences, regarding the muscular or neurological mechanisms that control it. This study concerns the propulsion of limbless creatures that rely on solid-solid friction rather than hydrodynamic forces for forward motion. Limbless creatures exhibit a variety of terrain-specific modes of locomotion. These include lateral and vertical undulation, sidewinding, concertina, and rectilinear, all of which will be discussed briefly. Lateral undulation, the "typical" serpentine motion, and rectilinear progression, the gait used primarily by the "giants", will be analyzed more deeply. In addition, I will also discuss the limitations a living snake faces that are not considered in most mathematical models, including flexibility of the vertebrae and the wear rate of snake skin.
• Ajay Shenoy, Mathematics, University of Connecticut
  Math vs. Cancer: Mathematically Modeling Leukemia and its Treatment (Abstract)
  Averse as it is to being dissected and analyzed, the living human body is understandably unwilling to surrender the secrets of its inner workings. Through the use of mathematical modeling and nonlinear dynamics, however, doctors and mathematicians can join forces to at last gain insight into these inner workings to find the optimal means of destroying something that threatens them: leukemia. After introducing the basics of dynamical modeling with ordinary differential equations in mathematical biology, I will present original research conducted during the summer of 2007 directed at modeling and treating B-Cell Chronic Lymphocytic Leukemia, the most common form of leukemia in the Western world. This research included a model of the untreated disease, modeling of treatments, and ideal treatment regimes for combating the disease.
• Adam Smith, Mathematics, Emmanuel College
  A Mathematical Model to Reduce Electricity Consumption (Abstract)
  Rising electricity prices mean that consumers must spend more just to use the same amount of energy. Two possible ways to solve this problem are by conserving energy or generating electricity at home. We present a quadratic programming model that can be used to decide which actions to take to minimize a home's electricity costs. During the presentation, we will describe the model, show the user interface, and look at results of an example run of the model.
• Hiro Tanaka, Mathematics, Brown University
  Constructibility on Surfaces of Constant Curvature (Abstract)
  The question of constructible lengths and shapes using straightedge and compass has been answered to a satisfatory degree on the plane. This talk will explore the constructibility of lengths and shapes on two other surfaces of constant curvature: the sphere and the Poincare Disc. The only prerequisite knowledge is basic differential geometry of surfaces and standard results from Galois theory.
• Jordan Tirrell, Mathematics, Lafayette College
  Nested Chain Partitions of Normalized Matching Posets (Abstract)
  In what ways can we partition a partially ordered set (poset) into linearly ordered subsets (chains)? We will report on recent progress made by our Claremont REU team on a thirty year old conjecture. In particular, two chains and in a finite ranked poset (a finite poset is ranked if all maximal chains have the same size) are said to be nested if implies that the levels occurring in are a subset of the levels occurring in . A thirty-year old conjecture of Griggs gives a sufficient condition--the so-called normalized matching condition, also known as the LYM property--for guaranteeing a decomposition of a poset into pairwise nested chains. In this talk, we will present our results in support of the conjecture. As a consequence of our main theorem, the conjecture is true for rank 3 posets of width (size of the largest collection of incomparable elements) less than 12.
• Pam Welch, Mathematics, Nazareth College of Rochester
  Investigating the Structure of a Double Bubble Cookie (Abstract)
  When two drop sugar cookies are baking, occasionally they will run together forming what we refer as a double bubble cookie. I will be presenting conjectures and results regarding the properties of the curve along which the two cookies meet. In addition, I will discuss some general characteristics of the double bubble cookie as well as the effects of this curve on its anatomy.
• David White, Mathematics, Bowdoin College
  An Investigation into the Structure of Digroups (Abstract)
  Digroups are algebraic structures with two associative binary operations allowing multiple identities that satisfy certain relations. They were created in order to provide a possible solution to the generalization of s third theorem for Leibniz Algebras. In this presentation I will present research investigating the algebraic structure of digroups, including definitions and results regarding subdigroups, the commutant, trivial digroups, and the idempotency class. Also, I will prove a Lagrange-style correspondence between digroups and subdigroups, show how to construct a digroup containing any given number of identities whose order is any multiple of that number, classify all digroups with a prime number of inverses, and discuss various structural propositions as well as directions for further research.

List of Student Posters

Registration for undergraduate posters is still open. If you are interested in presenting a poster, please register today.

• Voula Collins, Mathematics, Wellesley College
  Ordering Using the Magnus Transformation (Abstract)
  This poster will discuss a particular ordering on the Baumslag-Solitar group using the Magnus transformation. The Baumslag-Solitar groups are given by the presentation and while this ordering is effective on and it is still unclear for values where . The Magnus transformation sends the algebraic term to and I will define an order on this rather than on the original group. I will then prove the Magnus transformation to be injective to show that this order can indeed be applied to the Baumslag-Solitar groups.
• David Hansen, Mathematics, Brown University
  The Sato-Tate conjecture on average (Abstract)
  The Sato-Tate law is a remarkable conjecture governing the number of points on a smooth genus one curve over as varies. Richard Taylor and his collaborators recently proved this conjecture, using a great deal of heavy machinery. I will discuss how to obtain a very strong version of it on average, using comparatively simple techniques from the theory of modular forms.
• Theodora Hinkle, Computer Science, Brown University
  E-Cash and Cryptography for Peer-to-Peer File Sharing (Abstract)
  E-cash is a scheme for using cryptography to give users coins they can spend without the possibility that their purchase can be linked back to them, while also giving merchants the opportunity to find out from a central bank if a user has tried to spend a coin twice. The first characteristic is desirable in many online applications so that users can make transactions without giving away any personal information. The second is necessary to ensure that no one will cheat and duplicate their money. Thus it provides a level of security to users and merchants that is useful in an online world. This summer a group of undergrads, graduate students, and professors at Brown will be designing and implementing a cryptographic e-cash library and a peer-to-peer file sharing system that uses it. E-cash depends on zero-knowledge proofs for these traits.
• Alex Kruckman, Mathematics, Brown University
  Chains of Probability Distributions and Benford's Law (Abstract)
  Benford's law states that for certain data sets in base B, the probability of having a mantissa of at most is . Alex Ely Kossovsky recently conjectured that the distribution of leading digits of chains of probability distributions converges to Benford's law as the number of chains grows. We prove his conjecture in many cases, and provide an interpretation in terms of products of independent random variables and a central limit theorem.
• Aaron Mazel-Gee, Mathematics, Brown University
  A Combinatorial LSB Theorem on the d-Cube (Abstract)
  The classical Lusternik-Schnirelman-Borsuk (LSB) theorem states that if a -sphere is covered by closed sets, then at least one of the sets must contain a pair of antipodal points. We prove a combinatorial version of this theorem for hypercubes. While the LSB theorem applies to cover sets of codimension one, we consider cover sets of codimension two. For each hypercube, we find minimum-dimensional faces of which at least one cover set must contain an antipodal pair. We then show that for all dimensions except five, this minimum is sharp. The sharpness of our minimal result on the 5-cube remains open.
• David White, Mathematics, Bowdoin College
  An Investigation into the Structure of Digroups (Abstract)
  Digroups are algebraic structures with two associative binary operations allowing multiple identities that satisfy certain relations. They were created in order to provide a possible solution to the generalization of s third theorem for Leibniz Algebras. In this presentation I will present research investigating the algebraic structure of digroups, including definitions and results regarding subdigroups, the commutant, trivial digroups, and the idempotency class. Also, I will prove a Lagrange-style correspondence between digroups and subdigroups, show how to construct a digroup containing any given number of identities whose order is any multiple of that number, classify all digroups with a prime number of inverses, and discuss various structural propositions as well as directions for further research.