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Schedule - Saturday, March 3, 2007

Funding for SUMS 2007 provided by NSF grant DMS-536991 through the MAA Regional Undergraduate Mathematics Conferences program. Additional funding provided by the Brown University Lecture Board and the Brown University Division of Applied Mathematics.

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 is the complete list of faculty speakers from SUMS 2007.

• John H. Conway, Mathematics, Princeton University
  The Symmetries of Things
• David Dumas, Mathematics, Brown University
  Shapes of Polygons (Abstract)
  Two polygons have the same shape if they are related by a Euclidean similarity. We will discuss the (surprisingly rich) geometry of spaces of shapes of polygons, focusing on some explicit examples such as pentagons and hexagons with fixed angles or side lengths.
• Noam D. Elkies, Mathematics, Harvard University
  Canonical forms: A mathematician's view of musical canons (Abstract)
  Musical canons, from simple rounds like "Three Blind Mice" to the compendium of canons Bach compiled in his Musical Offering, have a history almost as long as that of Western music itself, and continue to fascinate musical composers, performers and listeners. In a canon the same melody is played or sung in two or more parts at once; this melody must therefore make musical sense both as a tune and in harmony with a delayed or otherwise modified copy of itself. How does one go about constructing such a melody? This challenge has a mathematical flavor. It turns out that some kinds of canons are so easy to create that they can be improvised in real time, while other kinds are more demanding, and in some cases only a handful of examples are known. The talk will be illustrated with both abstract diagrams and specific musical examples, and may also digress into generalizations of canons (the forms known collectively as "invertible counterpoint") and the reasons -- besides showing off -- that so many composers incorporate canons into their music.
• Anna Nagurney, Operations Research/Management Science , University of Massachusetts
  Operations Research and the Captivating Study of Networks and Complex Decision-Making (Abstract)

  In this talk, I will overview some of the major early and recent contributions to the formal mathematical study of networks and associated decision-making, from the perspective of an operations researcher. I will highlight novel mathematical tools, such as nonlinear optimization, game theory, variational inequalities, and projected dynamical systems, that have been utilized for the rigorous formulation of numerous network-based problems, and their effective and efficient solution. Some of the operation-research applications that I will discuss are: congested transportation networks and the Internet, including the Braess paradox (with fixed and time-varying demands), supply chains, financial and social networks, and energy/environmental networks.
  
  The mathematical network-based discoveries continue to impact numerous disciplines, including: engineering, computer science, physics, economics, and biology, where the formalism of networks brings new, refreshing, and unifying insights. Interestingly, and, not surprisingly, several of the fundamental discoveries in the network-application domain have been made by Brown faculty and Brown graduates!

• Meinolf Sellmann, Combinatorial Optimization, Brown University
  Structure and Symmetry in Constraint Programming (Abstract)
  Constraint Programming is one of the key techniques to solve real-world applications. Many problems exhibit a lot of symmetry which complicates the solution process considerably. Consequently, symmetry breaking was found to be an important method to speed up the search in constraint satisfaction problems that contain symmetry. When breaking symmetry by dominance detection, a computationally efficient symmetry breaking scheme can be achieved if we can solve the dominance detection problem in polynomial time. We study the complexity of dominance detection when value and variable symmetry appear simultaneously in constraint satisfaction problems (CSPs). Particularly, we devise an efficient dominance detection algorithm for CSPs with interchangeable variables and values. Our method yields symmetry-free search trees and is based on the abstraction to the actual, intuitive structure of a symmetric CSP.

List of Student Speakers

The following is the complete list of student speakers from SUMS 2007.

• Monique Ethier, Mathematics, University of Connecticut
  Fractals and fixed points (Abstract)
  A fixed point of a function is a solution of the equation . For example, the function has one fixed point, approximately .73908, which can be found graphically by intersecting the graphs of and ; at the intersection point we have . The fixed point of cosine can also be found numerically by hitting the cosine button (in radians, please) on your calculator repeatedly starting from any initial value you wish: will always tend to the fixed point .73908... (try it!). We will indicate why the concept of a fixed point is important in mathematics, and in particular see how a fractal like the Sierpinski gasket is a "fixed point" which can be approximated by iteration starting from any initial set in the plane.
• David Hansen, Mathematics, Brown University
  The Birch and Swinnerton-Dyer Conjecture (Abstract)
  In the early 1960s, Bryan Birch and Peter Swinnerton-Dyer conjectured a relation between the group of rational points on an elliptic curve and a special value of the Hasse-Weil L-function attached to the curve. I will present an exposition of their conjecture, some numerical evidence in support of it, and an overview of progress made in the last 40 years. No prior knowledge of elliptic curves required.
• Joseph Hirsh, Mathematics, City University of New York, Queens College
  The Halting Problem and Computability Theory (Abstract)
  A brief introduction to computability theory via "The Halting Problem." Discussion will include the relationship between diagonalization and the noncomputable nature of both the halting problem and the set of functions whose domain is all the natural numbers. This will segue to a brief introduction to the Turing Degrees, via oracles and a partial order "<" on noncomputable sets, where A < B means A is "easier" to compute than B.
• Sam Lewallen, Mathematics, Harvard University
  Knots, Polynomials, and Khovanov Homology (Abstract)
  My goal is to give an introduction to Khovanov's homology theory of knots and links, as a categorification of the Jones polynomial. This new knot invariant has connections to many older invariants, including of course the Jones polynomial, as well as the knot determinant. I will discuss these connections, make some sample calculations, and describe some conjectured combinatorial patterns. If there's time, I'll discuss a newly-proved result relating Khovanov homology to maps from the knot group into SU2. Note: No prior knowledge of knot theory required.
• Aaron Mazel-Gee and Nick Haber, Mathematics, Brown University
  Generalized Isoperimetric Problems for Space Polygons (Abstract)
  It is well-known that for a quadrilateral of given edge lengths, the largest area is achieved when the vertices lie on a circle. What if we twist the quadrilateral into three-space to form the vertices of a tetrahedron? Which such tetrahedron will have the largest volume? More generally, what is the largest hypervolume in n-space of the smallest convex set determined by the vertices of a k-gon with given edge lengths (the convex hull of the polygon)? We solve the problem for an equilateral pentagon in 4-space and show its relationship with equiangular pentagons investigated by others. We solve the problem for the equilateral n-gon in (n-1)-space and the equilateral pentagon in 3-space.
• Adam Smith, Mathematics and Economics, Emmanuel College
  What Are the Determinants of Free Agents' Salaries in Major League Baseball? (Abstract)
  This paper uses ordinary least squares to determine which statistics are most important in determining free agent's salary in Major League Baseball. Offense, defense, experience, and position variables are all examined. The primary determinants of a free agent's salary are his offensive statistics. Teams are more interested in attracting good offensive players and fans are more interested in watching offensive ball games.
• Josiah Sugarman, Mathematics, City University of New York, Queens College
  Measures of Growth (Abstract)
  A thorough discussion of orderings for functions is presented with particular emphasis on two of them. A proof that smooth functions are dense in C(R, R) is given and then used to prove the existence of large smooth functions. Integrals of certain large functions are shown to be less than the initial function, but not significantly less. A generalization of these orderings is given and then used for a more refined discussion of the system of derivatives at a point.
• Jennifer Tam, Mathematics, Tufts University
  Variational Principle for Locating Periodic Orbits of Dynamical Systems (Abstract)
  I will introduce a variational principle for finding periodic orbits of dynamical systems. The method may be applied to both discrete-time and continuous-time dynamical systems, and may be used to locate either stable or unstable periodic orbits. I will describe the variational principle, and present examples of its application, including the determination of various unstable periodic orbits of the Lorenz attractor.
• Elena Udovina, Mathematics, Harvard University
  Kummer and Fermat's Last Theorem (Abstract)
  I will outline a 19th century partial proof of Fermat's Last Theorem in the case where the exponents are regular primes. The formulation of these ideas is due to Kummer. The talk will touch on the theory of cyclotomic fields and Bernoulli numbers; it may hint at class field theory as well. Disclaimer: these ideas have little to do with the work of Andrew Wiles, and in no way suggest a complete proof of Fermat's Last Theorem.