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Schedule - Saturday, March 7, 2009

Except for the banquet, all SUMS activites are in the Barus & Holley building. Faculty talks are in room 166 while student talks are in rooms 166 and 168. The poster session is in the lobby.

List of Faculty Speakers

• Colin Adams, Mathematics, Williams College
  Blown Away: What Knot to Do When Sailing (Abstract)
  by Sir Randolph Bacon III, cousin-in-law to Colin Adams
  
  Being a tale of adventure on the high seas involving great risk to the tale teller, and how an understanding of the mathematical theory of knots saved his bacon. No nautical or mathematical background assumed.
• Robert Devaney, Mathematics, Boston University
  The Fractal Geometry of the Mandelbrot Set (Abstract)
  In this lecture we describe several folk theorems concerning the Mandelbrot set. While this set is extremely complicated from a geometric point of view, we will show that, as long as you know how to add and how to count, you can understand this geometry completely. We will encounter many famous mathematical objects in the Mandelbrot set, like the Farey tree and the Fibonacci sequence. And we will find many soon-to-be-famous objects as well, like the "Devaney" sequence. There might even be a joke or two in the talk.
• Govind Menon, Applied Mathematics, Brown University
  Mathematics and Turbulence (Abstract)
  Turbulence is often considered the grand unsolved problem of classical physics. Somewhat remarkably for a physics problem, many of the obstructions are mathematical. This talk will be a gentle overview of this relationship. Little knowledge of fluids or differential equations will be presumed.
• Joseph Silverman, Mathematics, Brown University
  Dynamical Systems from a Number Theoretic Perspective (Abstract)
    Dynamics is the study of iteration of functions, while number theorists often study integer and rational solutions to equations. The new field of arithmetic dynamics involves number theoretic questions that arise when polynomial or rational maps are iterated. Here are two typical problems:
    (1) If f(z) is a rational function with rational coefficients and c is an initial rational number, under what circumstances can the set of iterates {c, f(c), f(f(c)), f(f(f(c))), ...} contain infinitely many integers?
    (2) For a given rational function f(z) with rational coefficients, how many initial rational values c have a finite set of iterates?
    In this talk I will discuss what is known and what is conjectured about these and other problems in arithmetic dynamics. The talk will require no background in either number theory or dynamics.
• Suzanne Sindi, Applied Mathematics, Brown University
  Chaos in Biology (Abstract)
  Mathematical models have long been used to study biological systems. Earlier work using linear models to study population growth predicted and verified observed data. It was long believed that chaotic dynamics simply did not occur in natural populations. I will discuss the mathematical modeling and experimental work in studying the population dynamics flour beetles. Researchers were able to predictably alter population parameters to drive the dynamics through period doubling bifurcations and produced one of the first examples demonstrating chaos in ecology. No prior knowledge about dynamics (or flour beetles) is assumed.

List of Student Speakers

• Sarah Anderson and Jennifer Diemunsch, Presbyterian College and University of Dayton
  On the diameter of the Unidirectional Hyper-Stars (Abstract)
  Star graphs were introduced as a competitive model to the hypercubes. Recently, hyper-stars were introduced to be a competitive model to both hypercubes and star graphs. The vertex set of the HS(n,k) is the set of all {0,1}-strings of length n with exactly k 1's, and two vertices are adjacent if and only if one can be obtained by exchanging the first symbol with a different symbol (1 with 0, or 0 with 1) in another position. These graphs have nice connectivity and structural properties, and their edges can be oriented to obtain unidirectional hyper-stars UHS(n,k). In this talk, we will present computational results on finding the directed path between two vertices in UHS(n,k), and prove an upper bound on its diameter.
• Voula Collins, Wellesley College
  Subgroup Lattices as Graph Theoretic Objects: A Study of Chromatic Number (Abstract)
  In this talk we will consider subgroup lattices as graphs, where the subgroups are the vertices and the containment lines are the edges. We will examine the chromatic number of a selection of infinite and finite subgroup lattices, observe some general results, and discuss where research in this area is headed.
• Rachel Hudson, Williams College
  The Spiral Index of Knots (Abstract)
  Closely related to some classical invariants like the bridge index of a knot, the spiral index of a knot is a new invariant that captures the minimum number of maxima over all knot projections that don't contain inflection points. I will introduce this new invariant and some of its implications. Work of the 2008 SMALL Knot Theory Group.
• Caitlin Leverson, Wellesley College
  Delta Sets of Numerical Monoids (Abstract)
  Suppose S = (n_1, n_2, n_3) is a three-generated numerical monoid, that is, an additive submonoid of N_0 with the property that for all m in S there exist a_1, a_2, a_3 in N such that m = a_1n_1+ a_2n_2+ a_3n_3. For an element m in S, a_1n_1+ a_2n_2+ a_3n_3 is a factorization with length a_1+ a_2+ a_3. Let L(m) = {m_1, ... , m_t} be the set of all lengths of factorizations of m. Then the delta set of m is Delta(m)={m_{i+1}- m_1:0 < i < t}, and the delta set of S is the union of the delta sets of all elements of S. We will explore the delta sets of specific types of three-generated numerical monoids. These include supersymmetric monoids, which are of the form (ab, ac, bc) with gcd(a,b) = gcd(a,c) = gcd(b,c) = 1, supple monoids, which are of the form (n, cb, n+b) or (n, n+b, cb) with gcd(n,b) = 1, and monoids with three as the first generator. Work of the 2008 NSF Trinity REU Algebra Group.
• Gregory Loftus, University of Massacusetts Boston
  Competitively Coupled Map Lattices (Abstract)
  A coupled map lattice (CML) is a set of identical maps on the nodes/sites of an n-dimensional lattice, with coupling between sites. CMLs were introduced by Kunihiko Kaneko as a technique for studying spatiotemporal chaos without the use of PDEs. CMLs are often coupled diffusively. The patterns that arise over such CMLs are generated by the chaotic dynamics at the individual lattice sites (chaotic maps). By considering a system where the coupling is resource dependent, we introduce a dynamics over the lattice that is largely independent of the individual site dynamics. Thus even when the individual site dynamics is trivial (fixed points, 2-cycles), these "competitively coupled" map lattices (CCML) can demonstrate emergent patterns (such as frustration) more complicated than those at individual sites.
• Josh Mollner, University of Notre Dame
  The Group Law for Cubic Curves (Abstract)
  Elliptic curves are an incredibly important area of study in mathematics, with far-reaching implications in a wide variety of fields including Number Theory, Complex Analysis, and Algebraic Geometry. They were used by Andrew Wiles in his famous proof of Fermat's Last Theorem and have applications in cryptography and the sphere-packing problem. Consequently, it is an interesting fact that it is possible to define a binary operation on the set of points of an elliptic curve such that they form an abelian group under this operation. This talk will describe this operation and prove that it gives the points of an elliptic the structure of a group.
• Sylvia Naples, Bard College
  An upper bound on the number of graceful labelings for a path with n edges (Abstract)
  The concern of this talk is to provide an effective way to measure the rate of growth for the number of graceful labelings of a path graph with n edges, as n increases. We introduce the graceful labeling diagram, which we use to systematically construct graceful labelings, and develop analytical tools that exploit the structure of the diagram to compute an upper bound on the number of graceful labelings of a path. We conjecture that a path with n edges has order of sqrt(n!) graceful labelings.
• Si Pan, Brandeis University
  Autocorrelation Measurements of Green Picosecond Pulses Based on the Two-photon-induced Photocurrent in a Photodiode (Abstract)
  At Cornell University, the on-going Energy Recovery Linac (ERL) project requires the laser illuminating the photoinjector to produce ultra-short pulses in the range of picoseconds. These pulses are difficult to measure due to their ultra-short time duration. One suitable method is the autocorrelation measurement based on two-photon absorption in a photodiode. Two commercial photodiodes are tested for suitability in the summer of 2008. Two major types of error associated with autocorrelation measurement based on two-photon induced photocurrent were identified. A mathematical model was constructed to investigate the errors. In my presentation I will briefly describe the ERL project at Cornell, which leads to the motivation of my project, and the results of my work in the past summer at the Laboratory for Elementary Particle Phyiscs at Cornell.
• Hung Tran, Berea College
  The Isoperimetric Problem in the Plane with Density r^p (Abstract)
  The least-perimeter way to enclose a given amount of area in the plane with density r^p, surprisingly, depends on the sign of p. I report on our efforts to find the answer. Work of the 2008 NSF SMALL undergraduate research Geometry Group.
• Ken Zhao, New York University
  Barotropic Instability of Interacting Planetary Waves (Abstract)
  Zonal jets are observed in the gas planets as well as Earth's atmosphere and oceans, but some aspects of their development and evolution are not well understood. In general, rotating, stratified planetary fluids exhibit Rossby waves, which exist due to the meridional gradient of the Coriolis parameter, Beta. We use the barotropic vorticity equation, an idealized model of planetary fluids exhibiting both Rossby wave and zonal jet solutions, which provides two possible mechanisms for jet formation in the presence of Rossby waves: 1)The barotropic instability of zonal Rossby waves in the presence of small amplitude perturbations, which results directly in the growth of zonal jet modes and 2)The Rhines effect. In 1), we compute the analytic solution to the linear barotoropic instability problem and find the wavenumber of maximum growth as a function of Beta. The results indicate that as Beta increases, the horizonatal scale of the fastest growing mode also increases. In 2), turbulent interactions cascade energy to large scale where it interacts with Rossby waves, resulting in turbulently driven zonal jets. However in this case, as Beta increases, the scale of the jets are expected to decrease. To analyze the two jet formation mechanisms, we run a series of non-linear numerical simulations of the barotropic vorticity equation, initialized with a zonal Rossby wave with varying Beta. The results show that before the dominant, turbulent inverse cascade towards the Rhines scale takes place, the interactions of the barotropic instability cause a predictable pattern of peak wavenumber growths proportional to the Beta parameter. Thus, both effects are present, but in the case of strong turbulent interactions, the Rhines effect ultimately determines the jet scale.

List of Student Posters

• David Hansen, Brown University
  Equidistribution problems for quadratic forms
• Vivian Healey, University of Notre Dame
  A Modification of the Penrose Aperiodic Tiles (Abstract)
  The Penrose aperiodic protoset, a set of two triangular tiles with accompanying adjacency rules, admits only non-periodic tilings of the plane, meaning that any tiling of the plane constructed from these two tiles will possess translational symmetry in at most one direction. These aperiodic tiles are useful in modeling quasicrystals with icosahedral symmetry (5-fold symmetry is not permitted under the classical theory of crystals). A modification of these tiles, discovered by Robert Ammann, contains three tiles with no adjacency rules. My research explores this interesting set of tiles and its relationship to the algebraic structure of Penrose tilings. (Research conducted at Canisius College REU, summer 2008.)
• Anna Soybel, Mathematics, Williams College
  Isoperimetric Sequences (Abstract)
  The isoperimetric theorem states that the least-perimeter way to enclose a given volume is with a sphere. I will discuss how to transition the isoperimetric problem to numbers by examining sequences and the Cartesian plane.