Colloquium Schedule
2002-2003 Colloquia
- Some Computational Methods for Hyperbolic
Manifolds
David Reese, Physics and Mathematics (Double Major), Cal Poly, San Luis Obispo
Abstract
We will discuss a possible method for determining if a set of hyperbolic isometries in fact generates a closed hyperbolic manifold. As an example we present a proof of concept program which applies in two dimensions. - Applications of Category Theory to a Few Issues in
Physics
Goro Kato, Mathematics Department, Cal Poly
Abstract
Some notions in category theory will be used to reformulate some fundamental ideas in physics, and also t very small objects in physics. - 2003 Mathematical Contest in Modeling Team Presentation
A Tumor Treatment Model: The Sum of All Spheres
Andre Harmse and Josh Jenkins, Cal Poly
Abstract
Space filling problems can be some of the simplest to state and understand yet hardest to solve. We will be discussing space-filling algorithms as they refer to the gamma-knife brain tumor treatment. Four sizes of spheres will be used to fill an arbitrarily shaped tumor efficiently. Algorithms for filling the shape with non-overlapping spheres will be discussed as well as the analysis of these algorithms and our expectations of their efficiency. We will also describe implementation of one algorithm with a discussion of its abilities and limitations. - 2003 Mathematical Contest in Modeling Team Presentation
Producing Movies with a Real Impact
Nicholas Costa and Nicole Van Buskirk, Cal Poly
Abstract
A scene in an upcoming movie consists of a stunt person who propels himself over an elephant by means of a motorcycle and a ramp. As the stunt coordinators, it is our task to use cardboard boxes to cushion the fall of both the person and the motorcycle. Taking into account the size of the boxes, how many are to be used, and how they are to be stacked, our goal is to achieve the lowest cost of production and to keep the boxes out of sight of the camera. - Approximation Theory and its Role in Science and
Mathematics
John Walkup, Physics Department Cal Poly
Abstract
Problems with summing divergent power series are found in many areas of mathematics and science. Because computers can only add, subtract, divide, and multiply, mathematicians often use power series to represent more complicated functions --- a representation that breaks down if the power series diverges. In chemistry and physics, trouble often appears when evaluating a perturbation series of a quantum-mechanical system at large values of the perturbation parameter. In such cases, the perturbation series diverges and the perturbation theory is then often (wrongly) described as invalid. In many cases such pessimism is unjustified because we can often transform a troublesome power series into an approximant that converges to an accurate result. The success of an approximant at providing a sensible result from a badly behaving power series is often staggering. Unfortunately, many perfectly acceptable analytical approaches to problems are abandoned prematurely because researchers are often unaware that such approximants exist.
In this presentation we will briefly discuss the field of approximation theory. Then we will illustrate a new technique for summing divergent power series by improving the convergence of economized rational approximants and compare its effectiveness with the more common Pade approximants. Finally, we will discuss the role that approximation theory plays in mathematics and science and possible future research in this subject. - Slippery Solutions
Jon Jacobsen, Mathematics Department, Harvey Mudd College
Abstract
"The profound study of nature is the most fertile source of mathematical discoveries" -Joseph Fourier (1768-1830)
Our understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations (PDE's). Examples are the vibrations of solids, the flow of fluids, the diffusion of chemicals, the spread of heat, and the radiation of electromagnetic waves. PDE's also play a central role in modern geometry and mathematical biology, ecology, and physiology. In this talk we will explore solutions to PDE's from a visual point of view. Through demonstrations with soap film we will study minimal surfaces and vibrations of membranes. Undergraduates are encouraged to attend and recipes for soap solutions will be provided! - Fourier Meets Phylogeny
Steven N. Evans, Department of Statistics & Mathematics, U.C. Berkeley
Abstract
We give an overview of phylogenetic invariants: a technique for constructing evolutionary family trees from DNA sequence data. This method is useful in practice and is based on a number of simple ideas from elementary group theory, probability, linear algebra, and commutative algebra. - The
Mechanization of Mathematics
Michael Beeson, Department of Math and Computer Science, San Jose State University
Abstract
The "mechanization of mathematics" refers to the use of computers to find, or to help find, mathematical proofs. Turing showed that a complete reduction of mathematics to computation is not possible, but nevertheless the art and science of automated deduction has made progress. This talk will describe some of the history and survey the state of the art. - On Some Properties of the Numerical Range
Ilya Spitkovsky, Department of Mathematics, College of William and Mary
Abstract
The numerical range of a matrix is the set of values of the associated quadratic form on the unit sphere. In this talk, we will discuss various beautiful properties of the numerical range, starting with the classical ones (such as its convexity and the shape in 2-by-2 case) and ending with rather recent findings, including the discussion of the shape of the numerical ranges of some tridiagonal matrices. - Bounded Composition Operators Induced by a Symbol Function
Having Sup Norm 1 on General Weighted Bergman Spaces
Marian Robbins, Bellarmine University
Marian Robbins is a candidate for a tenure-track position in the Mathematics Department at Cal Poly. - Non-Linear Theory of Generalized Functions and Multiplication
of Schwartz Distributions
Todor Todorov, Mathematics Department, Cal Poly
Abstract
I will present a survey on non-linear theory of generalized functions. This is a branch of functional analysis dealing with commutative differential algebras of generalized functions containing the space of Schwartz distributions. Within an algebra of this type (known as algebras of Colombeau type) the Schwartz distributions can be multiplied (which is impossible in within the space of distributions). A large part of the talk will be devoted on the applications of both linear and non-linear theory of generalized functions to probability theory, Fourier analysis, ordinary and partial differential equations, functional analysis and quantum mechanics. At the end of the talk I will shortly discuss my contribution to the field.
Todor Todorov is a candidate for a tenure-track position in the Mathematics Department at Cal Poly. He will be available to visit with faculty on Thursday from 11:10 - 12:00 in the Math Department Conference Room, 25-208B. - Pre-service Teachers' Problem Posing and Beliefs About the
Relationship Between Problem Posing and School Mathematics
Todd Grundmeier, University of New Hampshire
Abstract
Literature in mathematics education has suggested the inclusion of problem posing at all levels of mathematics instruction. The research discussed in this talk incorporated problem posing into a mathematics content course for pre-service elementary and middle school teachers. During the semester participants posed mathematics problems as re-formulations of problems that they had previously solved and from sets of given information. Participants problems posed as problem re-formulation where characterized by seven re-formulation techniques and participants problems posed as problem generation were coded based on their plausibility, sufficiency of information, and complexity. Participants also responded to nine prompted journal entries related to class, their future teaching, and their beliefs about problem posing. During the semester participants became more efficient problem posers, articulated possible benefits of student problem posing for learning, and suggested ways to incorporate problem posing in their future classrooms. These results, other characteristics of participants' problem posing, and participants' beliefs about the relationship between problem posing and school mathematics will be the focus of this talk.
Todd Grundmeier is a candidate for a tenure-track position in Mathematics Education at Cal Poly. He will be available to visit with faculty on Friday from 11:10 - 12:00 and 2:30 - 4:00 p.m. in the Math Department Conference Room, 25-208B. - An Investigation of Prospective Secondary Mathematics
Teachers' Conceptions of Proof and Refutations
Kate Riley, Montana State University
Abstract
A quantitative, descriptive research study was conducted to investigate prospective secondary mathematics teachers' conceptions of proof and refutations as they were near completion of their preparation program. To research the primary question of the study, the researcher addressed two components of participants' conceptions of proof--1) understanding of the logical underpinnings of proof, and 2) ability to complete mathematical proofs. Both components focused on direct proof, indirect proof, and refutations. These components are common proof themes emphasized by the MAA (1998) and the NCTM (2000) Standards.
Kate Riley is a candidate for a tenure-track position in Mathematics Education at Cal Poly. She will be available to visit with faculty on Friday from 2:00 - 3:30 p.m. in the Math Department Conference Room, 25-208. - Unpacking Mathematical Content through Problem
Solving
Elaine Young, Department of Mathematics, University of Oklahoma
Abstract
Pre-service teachers bring 15+ years of mathematics learning to the college classroom. However, this knowledge is generally faded, fragmented, and not available in useful form for new problems. Using problem solving tasks, pre-service teachers can "unpack" this previous knowledge, revisit, reconstruct, and make connections.
Elaine Young is a candidate for a tenure-track position in Mathematics Education at Cal Poly. She will be available to visit with faculty on Wednesday from 10:00 - 11:00 and 3:00 - 4:00 in the Math Department Conference Room, 25-208B. - Dynamics of Multivalued
Functions
Jim Wiseman, Department of Mathematics and Statistics, Swarthmore College
Abstract
A multivalued function is a function under which the image of a point is not another point, but a set. These arise naturally in economics, numerical simulations, game theory, and even high school math. I'll talk about where multivalued functions come from and how people study them (you can think of normal, single-valued functions as a special case, so there's a lot to study). In particular, I'll discuss the dynamics, that is, the long-term behavior of these functions, and how we can combine them with computer approximations to detect chaos in physical systems.
Jim Wiseman is a candidate for a tenure-track position in the Mathematics Department at Cal Poly. She will be available to visit with faculty on Friday from 10:00 - 12:00 and from 2:00 - 3:00 in the Math Department Conference Room, 25-208B. - Rook Theory, Stirling Numbers, and Minding
Your P's and Q's
Karen Briggs, Department of Mathematics, University of California, San Diego
Karen Briggs is a candidate for a tenure-track position in the Mathematics Department at Cal Poly. - Bounding Graded Betti Numbers and the Lex Plus Powers
Conjecture
Ben Richert, Department of Mathematics, University of Michigan
Abstract
Doubtless we all have (at some time or another) added, subtracted, multiplied, differentiated, solved, and graphed polynomials. That is, polynomials seem to be important mathematical objects. It is sometimes important to classify certain collections of polynomials, which we can do in various ways. One of the most important methods is to consider how many polynomials it takes to generate our collection, calculate the number of relations on these generators (this is a measure of how "close together" our generators are; for instance, xy is closer to x^2 than y^2 is because xy and x^2 share a common factor, while y^2 and x^2 do not), calculate the number of relations on the relations, and so on. The numbers we compute are called Betti numbers, and we next consider whether these integers are bounded. In particular, we consider a conjecture of Eisenbud, Green, and Harris concerning a sharp upper bound for the graded Betti numbers of all ideals attaining a given Hilbert function and containing a regular sequence whose terms lie in certain degrees.
Ben Richert is a candidate for a tenure-track position in the Mathematics Department at Cal Poly. He will be available to visit with faculty on Thursday from 9:30 - 11:00 a.m. and from 2:00 - 3:00 p.m. in the Math Department Conference Room, 25-208B. - Finite Split Extensions of
Coxeter Groups
Anton Kaul, Tufts University
Abstract
An underlying theme in Geometric Group Theory is the notion that a great deal of information can be extracted about a given group by observing its action on a topological space. The class of groups known as Coxeter groups fit nicely into this picture: they are precisely the groups that act "by reflections" on topological spaces. Some of the first groups encountered by an undergraduate algebra student (eg., the dihedral and symmetric groups) are, in fact, Coxeter groups and play an important role in the theory.
A group G is a CAT(0) group if it acts "geometrically" on a CAT(0) space X (the CAT(0) axiom may be viewed a global non-positive curvature condition). Such groups have many nice properties and are of particular interest to geometric group theorists. A result of G.
Moussong states that every Coxeter group is a CAT(0) group. In recent work done jointly with M. Gutierrez and K. Ruane we consider the following question: Given a Coxeter group W and a finite split extension
1 --> W --> G --> K --> 1,
is G a CAT(0) group? Using a construction due to Serre, we provide an affirmative answer in the case that W is "right-angled."
Anton Kaul is a candidate for a tenure-track position in the Mathematics Department at Cal Poly. - Wonderful World of Composition
Dylan Retsek, Department of Mathematics, CSU Fresno
Abstract
The notion of the composition of two functions is familiar from calculus, where we learn that many important features of functions are preserved by this operation. Among these are continuity, differentiability (the Chain Rule!), and in some cases integrability (u-substitution). What other features of functions are preserved under composition? In what sense is the act of composing two functions continuous? Come and find out!
Dylan Retsek is a candidate for a tenure-track position in the Mathematics Department at Cal Poly. - A Taste of Variation
Daniel Canada, Mathematics Department, Portland State University
Abstract
This talk will illustrate recent research on the understanding of variation. Sample tasks and responses will be shared from ongoing work being done with middle, high school, and college students.
Daniel Canada is a candidate for a tenure-track position in Mathematics Education at Cal Poly. - Groups of Arithmetical Functions
Jim Delany, Mathematics Department, Cal Poly
Abstract
An arithmetical function is a mapping from the positive integers to the real or complex numbers. These form a commutative ring with unity under ordinary addition and the Dirichlet product, or convolution. Examples of beautiful formulas in number theory will be derived using simple algebraic calculations in this ring. A complete group-theoretic description of the group of units will be presented. It will be shown that the functions for which f(1) = 1 may be regarded as a vector space over the rationals, with the multiplicative functions forming an important subspace. Linear independence will also be discussed.
This talk will be accessible to undergraduates. - Elections! What a Chaotic Mess!
Donald Saari, Department of Mathematics, U.C. Irvine
Abstract
From the perspective of mathematics, elections seem to be trivial - just by knowing how to count we can tally ballots. But, in this expository lecture that will be accessible to undergraduates, it is shown that elections can be highly "chaotic;" indeed, mathematical chaos is even involved in understanding what can happen. In fact, there is so much that can go wrong with elections that you may leave this lecture with a skeptical attitude about whether the outcome of your organization's last election was "fair." - Proofs That Really Count
Jennifer Quinn, Mathematics Department, Occidental College
Abstract
Counting leads to beautiful, often elementary, and very concrete proofs. As human beings we learn to count from a very early age. A typical 2 year old will proudly count to 10 for the coos and applause of adoring parents. Although many adults readily claim ineptitude in mathematics -- no one ever owns up to an inability to count. Counting is one of our first tools, and it is time to appreciate its full mathematical power. I have selected some of my favorite identities involving binomial coefficients, Fibonacci numbers, Stirling numbers, harmonic numbers (and more) that have elegant counting proofs. They require no special training to be appreciated. Hopefully when you encounter identities in the future, the first question to pop into your mind will not be "Why is this true?" but "What does this count?" - To Have or Have Knot
Blake Mellor, Mathematics Department, Loyola Marymount University
Abstract
A knot is a closed loop embedded in three-dimensional space. The study of knots is a fundamental part of low-dimensional topology. We will look at knots from a combinatorial point of view, and introduce some of the important invariants used to study them, such as the Conway and Jones polynomials. We will introduce the idea of finite type invariants, of which these polynomials are examples. If time permits, we will also look at the new area of virtual knot theory, partially motivated by trying to analyze knots with computers, where there are many very accessible questions which are still open. - Apportionment and the 2000 Election
Michael Neubauer and Joel Zeitlin, Department of Mathematics, CSU Northridge
Abstract
We'll discuss the basics of apportionment theory and show apportionments based on different (hypothetical) sizes of the House of Representatives would have influenced the result of the 2000 presidential election. - Reconstructing Topological Spaces from their Groups of
Homeomorphisms
Victor Brunsden, Mathematics Department, Penn State Altoona
Abstract
We show how to reconstruct a topological space X from algebraic information about its group of homeomorphisms H(X). That is, we consider the question, "Does H(X) = H(Y) as groups imply that X is homeomorphic to Y?" - Sprouts
David Bachman, Mathematics Department, Cal Poly
Abstract
Sprouts is a fun pencil-and-paper game created by John Conway and Mike Patterson in 1971. A complete analysis of the game brings together ideas from game theory, graph theory, topology and computer science. This talk will be accessible to all levels.
CAUTION: Playing sprouts can lead to severe addiction.
2001-2002 Colloquia
2000-2001 Colloquia
1999-2000 Colloquia
1998-1999 Colloquia