California Polytechnic State University

Colloquium Schedule

• The significant-digit phenomenon, or Benford's Law
  Ted Hill, Professor Emeritus of Mathematics, Georgia Tech
  
  Abstract
  A century-old empirical observation now called Benford's Law says that the significant digits of many real datasets are logarithmically distributed, rather than uniformly distributed, as might be expected. This talk will briefly survey some of the colorful history of the problem, including recent proofs and applications to fraud detection, and will then describe new discoveries (joint work with A. Berger and L. Bunimovich, to appear in TransAMS Jan 2005) that Benford sequences are typical in many deterministic sequences such as 1-dimensional dynamical systems and differential equations. For example, the orbit of iterates of almost every rational function obeys Benford's Law - if you iterate x^2, or 3^x, or x^x, or any successive composition of these, the resulting sequence will start with a "1" log 2 ~ 30% of the time, for almost all starting points x_0 > 1. A number of open Benford-related problems in dynamical systems, probability, number theory, and differential equations will be mentioned, and the talk will be aimed for the non-specialist.

 

• Professors' and Students' Reasoning of Mathematical Statements
  Nicole Van Buskirk Lange
  
  Abstract
  College students and professors use a variety of reasoning strategies to convince themselves whether or not a statement is true. This study examined nine people, three college professors and six college students, to explore how each group of people justify mathematical statements. The results of this study suggest that the professors use more formal ways of justification in mathematics than do the students. The study further suggests that the professors use a logical/analytical approach to reasoning through mathematical statements, whereas students' responses vary between analytical and empirical justifications. Other characteristics of the professors and students are also discussed in light of these findings.

 

• Hilbert's Tenth Problem
  Martin Davis
  Visiting Scholar, University of California, Berkeley
  Professor Emeritus, NYU
  
  Abstract
  In the 10th problem in Hilbert's famous list of 1900, he asked for an algorithm to determine whether a given polynomial equation with integer coefficients has integer roots. Work by logicians has shown that no such algorithm exists. In this talk, the ideas leading to this negative solution will be explained, and various applications, extensions, and open problems will be discussed.

 

• On the Interaction of Nearly Parallel Vortex Filaments
  Gustavo Ponce, University of California, Santa Barbara
  
  Abstract
  (A joint work with C. E. Kenig and L. Vega "On the interaction of nearly parallel vortex filaments", Comm. Math. Phys. 243, pp. 471-483, 2003) We study the system for the interaction of N-nearly parallel vortex filaments proposed by Klein, Majda and Damodaran. This model reflects both the self-interaction of each filament and the interaction between them. We deduce explicit solutions of this system, including some in the form of N-helix. Also under appropriate assumptions on the unperturbed configurations and on the initial parameters we prove the global in time stability of some of these special solutions.

• Almost periodic flows and three-manifolds
  Kelly Delp, University of California, Santa Barbara
  
  Abstract
  A flow on a manifold M is a continuous action of R on M. Flows on three-manifolds have connections to many pure and applied areas of mathematics, including hydrodynamics, differential equations, foliations, and knot theory. A manifold which supports a periodic flow is a Seifert fibered space. I will define a notion of almost periodic flow and give conditions under which a manifold supporting an almost periodic flow is a Seifert fibered space. Along the way I will discuss the importance of Seifert fibered spaces in three-manifold topology, as well as presenting some interesting results about flows and foliations.
  
  Kelly Delp is a candidate for a tenure-track position in the Mathematics Department at Cal Poly.

 

• Geometries and covers of 3-manifolds
  Joseph Maher, California Institute of Technology
  
  Abstract
  3-manifolds are analogues of surfaces, but one dimension higher. On a small scale, they look like the space we live in, but on a large scale they may have interesting shapes. We give a brief introduction to geometric structures on 3-manifolds, and various outstanding conjectures, in particular Thurston's conjectures about covers of 3-manifolds. We will also discuss some other topological structures on 3-manifolds, and attempt to relate them to their geometries. We will mention how this relates to my own work, and other areas of math, such as minimal surfaces and discrete groups.
  
  Dr. Maher is a candidate for a tenure-track position in the Mathematics Department at Cal Poly.

 

• A new exact solution of a nonlinear model of coastal upwelling
  Paul Choboter, College of Oceanic and Atmospheric Sciences, Oregon State University
  
  Abstract
  In the coastal ocean off the west coast of North America, equatorward along-shore winds drive surface fluid offshore, causing deeper water to be "upwelled" to the surface. The upwelled fluid carries nutrients with it into sunlit waters, thereby driving biological productivity at the base of the food chain. Despite its fundamental importance, wind-driven coastal upwelling remains an incompletely understood process in coastal ocean circulation.
  
  A two-dimensional, frictionless, nonlinear model of coastal upwelling is discussed. The model has been solved previously at steady state and as an initial-value problem; however, the previous solution to the initial-value problem is inconsistent with the steady-state solution. A new solution to the initial-value problem is presented that approaches the existing steady-state solution in the long-time limit. The key to the method of solution is a coordinate transformation that renders the nonlinear coupled set of PDEs linear, and thus provides a systematic way to find the exact closed-form solution to the fully nonlinear system. The transformation itself is not new; our contribution lies in the application of novel boundary conditions in transformed space. The dynamical insights provided by the new solution are discussed.
  
  Paul Choboter is a candidate for a tenure-track position in the Mathematics Department at Cal Poly.

 

• Adaptive wavelet methods for convection-dominated transport equations
  Jiangguo Liu, Department of Mathematics, Texas A&M University
  
  Abstract
  Convection-dominated transport equations arise from petroleum reservoir simulation, groundwater pollution clean-up, and many other interesting applications. These problems usually admit solutions with moving steep fronts and even discontinuities, which present serious numerical difficulties. The capacity of wavelets in capturing singularities in nonsmooth data can be utilized to establish efficient numerical methods for this kind of problem. The multilevel scheme with wavelet compression saves computation but still conserves the total mass. Optimal error estimates and some numerical results will be presented.
  
  Jiangguo Liu is a candidate for a tenure-track position in the Mathematics Department at Cal Poly.

 

• Vanishing of functions on intersections of algebraic varieties
  Sean Sather-Wagstaff, University of Nebraska
  
  Abstract
  
  Sean Sather-Wagstaff is a candidate for a tenure-track position in the Mathematics Department at Cal Poly.

 

• Prolate Spheroidal Wavelets in Computerized Tomography
  Tatiana Soleski, University of Wisconsin, Milwaukee
  
  Abstract
  In computerized tomography, an image must be reconstructed from data given by the Radon transform of the image. This data is usually in the form of sampled values of the transform. In this talk we introduce a method of recovering the image based on the sampling properties of the prolate spheroidal wavelets which are superior to other wavelet systems. It avoids integration and allows the precomputation of certain coefficients. The approximation based on this method is shown to converge to the true image under mild hypotheses. The algorithm is then tested on the standard Shepp-Logan image and is shown to be surprisingly good.
  
  Tatiana Soleski is a candidate for a tenure-track position in the Mathematics Department at Cal Poly.

 

• Counting Macaulay-Style
  Susan Cooper, Queen's University, Kingston, Ontario
  
  Abstract
  One way to obtain information about a set of points in projective space is to look at its Hilbert function. Associated with any set is a collection of special polynomials which pass through the points. If these polynomials are grouped by degree, then the Hilbert function is simply a sequence of numbers which counts the dimensions of these associated polynomials degree by degree. We will see how "Macaulay's Theorem" uses Pascal's Triangle to characterize Hilbert functions of finite sets of distinct points in projective space. We will generalize this method to other "triangles" to classify the Hilbert functions of some very special point sets, namely subsets of complete intersections.
  
  Susan Cooper is a candidate for a tenure-track position in the Mathematics Department at Cal Poly.

   

• Central Methods for Balance Laws
  Stephen Bryson, Stanford University
  
  Abstract
  In this talk we develop numerical schemes for balance laws in two applications. First, a method for solving the two-dimensional shallow water equations with bottom topography is developed for conformal triangular meshes. This work extends a semi-discrete central method recently developed for conservation laws on triangles, as well as a balanced method for shallow water equations with bottom topography on cartesian meshes. It is shown that the existing method on triangles cannot be balanced except in the case of very special meshes. A new second-order method is then developed which is balanced on any conformal triangular mesh. A new second-order interpolant for triangles which minimizes spurious oscillations is also presented. The balance and accuracy of this method is demonstrated with various examples.
  
  The second application to solving a balance law is the study of shocks propagating upward in flux tubes in the solar atmosphere. This phenomena is modeled by the one-dimensional Euler equations in gravity. In this case the required balance is the initial hydrostatic equilibrium, which is obtained via a special choice of variables. The simulation is performed with existing second and third order semi-discrete central methods, though new interpolants for irregular computational meshes are developed. These simulations use various observational data for initial and boundary conditions. The results of these simulations show a surprising (in view of the simplicity of the model) match with observed phenomena in the solar atmosphere.
  
  Stephen Bryson is a candidate for a tenure-track position in the Mathematics Department at Cal Poly.

 

• Prospective Teachers' Understanding of Mathematical Objects in the Context of Transformational Geometry: Implications for Proof
  Todd Grundmeier, Mathematics Department, Cal Poly
  
  Abstract
  The talk will focus on research that investigated prospective teachers' views of mathematical objects through the implementation of a curriculum module that highlighted the relationships between transformational geometry and linear algebra. The prospective teachers were all mathematics education majors enrolled in a one semester geometry course that is geared towards the mathematics education major. Data analysis suggests that these prospective teachers viewed isometries as processes and viewed mathematical objects (triangle, etc.) as "perceived". Results also suggest that these two views influenced students' abilities to construct geometric proofs in transformational geometry. The talk will provide an overview of the project, a discussion of the curriculum developed, and results of the associated research.

• Combinatorial Probability of Algebraic Objects
  Kent Morrison, Chair, Mathematics Department, Cal Poly
  
  Abstract
  This expository talk is a survey of questions and results about the combinatorics and probability of various types of algebraic objects. Qquestions include:
  • What is the probability a permutation moves everything?
  • What is the probability that two random integers are relatively prime?
  • What is the probability that two random elements generate the full symmetric group?
  • What is the probability that a linear map over a finite field moves every non-zero vector?

   

• Statistics Colloquium: An Introduction to Prototype Patterns
  Katherine Tranbarger, Statistics Department, University of California, Los Angeles
  
  Abstract
  Data from fields as diverse as neurology, seismology, economics, astronomy, forestry, and epidemiology can take the form of point patterns. This talk discusses point process data and considers the problem of how to measure the distance between two point process realizations. Several properties of a distance metric proposed by Victor and Purpura for examining neuronal impulse spike trains are examined and shown to aid in the measurement algorithm. Given a dataset consisting of many realizations of a point process, description of a typical point pattern can be accomplished by using the distance metric to define a prototype pattern such that the distance between the prototype and all observed point patterns in the data set is minimized. Examples of this measurement approach and prototype extension applied for earthquake aftershock analysis and a problem in computer network security are discussed to illustrate the potential this type of analysis has for use in a variety of fields. In the earthquake context, for instance, prototypes are used to characterize the typical aftershock behavior expected following an earthquake of magnitude between 7.5 and 8.0.
  About the Speaker
  Alumna Katherine Tranbarger graduated from the Cal Poly Statistics Department in the Spring of 2001 before beginning graduate studies at the University of California, Los Angeles. While at UCLA, her research has focused on applications and extensions of the spike time distance metric for analysis of point process data. In 2003 she was selected as a UCLA Collegium of University Teaching Fellow, an honor that enabled her to develop and teach her own course: Making Sense of Lies, Damned Lies, and Statistics. Completion of her Ph.D. is expected this June. This Fall 2005 she will be joining the faculty of Amherst College as the sole statistician in their Department of Mathematics and Computer Science.

 

• Differentiable Dedekind Sums in Integer Linear Programming
  Sinai Robins, Temple University
  
  Abstract
  We define a new class of differentiable Dedekind sums, associated to rational polytopes in R^n, that allow us to locate integer points in rational polytopes (as opposed to just straightforward enumeration). This approach answers the 'feasibility question' in computer science rather quickly, at least in low dimensions. This is joint work with Helaman Ferguson. The talk should be accessible to undergraduates with a little analysis.
  
  Sinai Robins was a receipient of the NSA young investigator's award and is the co-author of an upcoming Springer-Verlag UTM series book.

 

• Statistics Colloquium: Bayesians, Frequentists, and Modern Science
  Brad Efron, Stanford University
  
  Abstract
  Broadly speaking, 19th century statistics was Bayesian while the 20th century was frequentist, at least from the point of view of most scientific practitioners. Here in the 21st century scientists are bringing statisticians much bigger problems to solve, often comprising millions of data points and thousands of parameters. Which statistical philosophy will dominate practice? Professor Efron¹s guess, which he will try to back up with some recent examples, is that a combination of Bayesian and frequentist ideas will be needed to deal with our increasingly intense scientific environment.
  
  Bradley Efron is the Max Stein Professor of Statistics and Biostatistics at Stanford University. He is immediate past president of the American Statistical Association and a recipient of a MacArthur genius grant. Among Professor Efron¹s many statistical accomplishments, he is best known for developing the foundations of bootstrapping along with a great deal of bootstrap methodology.

 

• Complex Symmetric Operators
  Stephan Garcia, Department of Mathematics, University of California, Santa Barbara
  
  Abstract
  Roughly stated, a complex symmetric operator is one which can be represented by a symmetric matrix with complex entries. This surprisingly large class includes all normal operators, compressed Toeplitz operators (including finite Toeplitz matrices and the "compressed shift"), all Hankel operators, as well as many standard integral and differential operators. We will discuss several general structure theorems for this class, illustrated with several examples. This talk should be accessible to advanced undergraduates who have taken upper-division linear algebra.

 

• Higher-Dimensional Group Theory
  Alissa Crans, Loyola Marymount University
  
  Abstract
  Group theory plays a prominent role in many branches of science where symmetries appear. In many contexts where we are tempted to use groups, however, it is actually more natural to use a richer sort of structure, that of a higher-dimensional group, or '2-group.' A 2-group blends together the notion of a group with that of a category. Thus, in addition to group elements describing symmetries, a 2-group also has isomorphisms between these, describing symmetries between symmetries. This talk consists of an introduction to higher-dimensional group theory in which we will examine examples of 2-groups and address their contexts and motivation.

 

• The Art of Dihedral Groups
  Gwen Fisher, Mathematics Department, Cal Poly
  
  Abstract
  The symmetry of regular polygons is described mathematically by way of dihedral groups. Accordingly, one can easily transform a regular polygon into a piece of artwork with dihedral symmetry. I will discuss the construction of regular polygons and the structure of dihedral groups. Numerous examples of art that exhibits dihedral symmetry and the structure of dihedral groups will be presented. Pieces include artwork made by Cal Poly students and myself, as well as from folk art of different cultures. This talk should be accessible to everyone.

 

• The Booths of Hazard
  André Harmse, Mathematics Undergraduate, Cal Poly
  
  Abstract
  Toll booths are a common feature of the modern American highway system, which makes the analysis of tollbooths an interesting and relevant topic. One of the most fundamental aspects of any toll plaza is the number of toll booths that are open at any given time. The problem at hand is to determine what number of booths is optimal with any given traffic flow and any number of incoming lanes. In formulating parts of a model we use some ideas from queuing theory and network theory. The final result is a specification of the number of booths to operate at various times of the day for a toll plaza with a certain expected daily traffic distribution over a fixed number of lanes. An alternative approach to the design of toll plazas which may be able to significantly increase efficiency is also introduced. This talk is largely conceptual and thus accessible to any student.
  
  This talk is based on a paper by Cal Poly math undergraduates Tyler Barry, Nicholas Walton, and Andre Harmse. Dr. Jonathan Shapiro was the faculty advisor.

 

• Four-Handed Shuffles and Affine Groups
  André Harmse, Mathematics Undergraduate, Cal Poly
  
  Abstract
  Shuffling cards gives rise to an interesting way to visualize certain groups. The common method of shuffling cards, where a deck is broken in two and interleaved back together, is called a two-handed shuffle. If each shuffle is viewed as a permutation on the arrangement of cards, then all the two handed shuffles on a deck forms a group. The group formed by two-handed shuffles is completely understood. The talk will introduce some infinite families of groups to which certain families of four-handed shuffles are equivalent. This talk will be accessible to any student familiar with the basic concepts of linear algebra and group theory. This talk is based on a paper by Dr. Kent Morrison, Sarah Wright, Amanda Cohen, and Andre Harmse.

 

• Composition Operators and Hypergeometric Series
  Dylan Retsek, Mathematics Department, Cal Poly
  
  Abstract

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Last Update: mm/dd/yyyy

Department of Mathematics
California Polytechnic State University
San Luis Obispo, CA 93407-0403
805.756.2206
math@calpoly.edu