Programs of Past Congresses

First Congress
Weaverville, CA
May 13-14, 1972

Inaugural Session: Comparison of curricula, texts, departmental organizations and instructional methods; plans for the presentation of formal papers at the second congress.

In attendance: from Humboldt State College: Charles Biles, Ronald Levine, Nick Mousouris, Marshall Ruchte, Roy Ryden, Charles Snygg; from Southern Oregon College: Art Clemmons, Richard Montgomery, Sheldon Rio, Bob McCoy, Ron Steffani, Dana Sudbourough.

Organizers: Richard Montgomery, Roy Ryden

Second Congress
Whiskeytown National Recreation Area
May 18-20, 1973

A Measure Approach to Calculus. Roy Ryden, CSU Humboldt. A currently nontypical, but extremely illuminating, approach to beginning calculus.

An Interesting Compactification of P-Spaces, and How It Arose. Richard Montgomery, Southern Oregon College.

Open Forum; Mini-Topics:

  1. Honors Courses in Your Curriculum: Yes, No and Where?
  2. Keeping Mathematically Alive (or Maintaining Mathematical Momentum) in the State of Jefferson
  3. Mathematics for ‘Others’
  4. Is There a Computer in Your Program?

Campfire Activities.

Organizers: Richard Montgomery, Roy Ryden

Third Congress
Whiskeytown National Recreation Area
May 10-12, 1974

Some New Applications of Mathematics to Biology. Ron Levine, Humboldt State University.

Discreteness and Periodicity in Topological Abelian Groups. Robert Hooper, University of Nevada, Reno.

Informal Session; Suggested topics:

  1. Academic-Administration Subversion.
  2. Maintaining Mathematical Momentum in the State of Jefferson.
  3. The State of Jefferson was lost on the Playing Fields of Carson City, Sacramento, and Salem.
  4. What can we do about High School Mathematics at the College Level?
  5. You name it!

Annual Business Meeting and Campfire Activities.

Organizers: Richard Montgomery, Roy Ryden

Fourth Congress
Whiskeytown National Recreation Area
May 16-18, 1975

Character Theory of Finite Groups. William Fisher, CSU Chico.

Curve Fitting for the Biologist, Exponential and Restricted Exponential Growth—Some Elementary Observations by a Non-Statistician. Art Clemmons, Southern Oregon College.

Informal Session—a discussion of curriculum, honors courses, capstone courses, textbooks, finances, etc.—anything which comes to mind.

Annual Business Meeting. Campfire Activities.

Organizers: Richard Montgomery, Roy Ryden

Fifth Congress
Whiskeytown National Recreation Area
May 21-23, 1976

Convergence Space: A Generalization of Topological Space. Edwin Wagner, University of Nevada, Reno.

Some Fascination Applications of the Calculus of Variations. Bob Hunt, Humboldt State University

Informal Session—a discussion of curriculum, honors courses, capstone courses, textbooks, finances, etc.—anything which comes to mind.

Annual Business Meeting. Campfire Activities.

Organizers: Richard Montgomery, Roy Ryden

Sixth Congress
Whiskeytown National Recreation Area
May 21-22, 1977

Computers Are Everywhere. Ken Larson, Southern Oregon State College.

Toward Symmetry with Steiner. Rick Luttman, Sonoma State College. A brief account of the theory of Steiner symmetrization and an investigation into convergence questions relating to sequences of Steiner symmetrizations.

K.F. Roth, Fields Medalist. Roy Ryden, Humboldt State University. A short explanation of why Roth was awarded the Fields Medal in 1958, and a brief discussion of the impact of Roth's work on mathematics.

Informal Session—a discussion of curriculum, honors courses, capstone courses, textbooks, finances, etc.—anything which comes to mind.

Annual Business Meeting. Campfire Activities.

Organizers: Richard Montgomery, Roy Ryden

Seventh Congress
Whiskeytown National Recreation Area
May 20-21, 1978

Some Mathematical Applications in Forestry—Spatial Distributions. Howard B. Stauffer, California State University, Hayward, and, formerly, the Pacific Forest Research Center, Victoria, B.C.

On Things Related to Schröder Series, Normal Forms for Diffeomorphisms & Small Divisor Theory. Buck Ware, California State University, Chico.

Newly Emerging Materials on Applications of Undergraduate Mathematics: Current States & Instructional Implications. Richard G. Montgomery, Southern Oregon State College.

Informal Roundtable Session: Open discussion of common concerns.

Annual Business Meeting & Worthwhile Campfire Activities

Organizer: Richard Montgomery

Eighth Congress
Whiskeytown National Recreation Area
May 19-20, 1979

Looking Back at Arithmetic Densities. Marshall Ruchte, Humboldt State University. A discussion of the major results of the “old days” and an attempt to look at what has happened since.

Random Permutations. Gerald Kimble, University of Nevada, Reno. A discussion of randomly generated permutations and an indication why this subject is of interest in computer science.

The usual “informal roundtable session” which is an open discussion covering common concerns.

Organizer: Roy Ryden

Ninth Congress
Whiskeytown National Recreation Area
May 16-18, 1980

How to Ask Sensitive Questions: Randomized Response Designs. Reider Peterson, Southern Oregon State College. Information on sensitive issues can be obtained by randomly asking either the sensitive question or an alternate innocuous question.

A Potpourri of Problems From My Diary. David Klarner, SUNY at Binghamton and Humboldt State University. A presentation of a small collection of partially solved tractable problems.

A Short History of the State of Jefferson. Roy Ryden, Humboldt State University. (Based on information from Stan Mottaz, Humboldt State University.)

Open “under the oaks” discussion on mathematics and mathematics education at universities and colleges in the State of Jefferson.

Bonfire, etc.

Organizer: Roy Ryden

Tenth Congress
Whiskeytown National Recreation Area
May 16, 1981

Measuring Tuna Population. Roland Lamberson, Humboldt State University. A presentation of a mathematical model which is designed to catch an accurate measure of tuna population.

Spline Functions. Clem Falbo, Sonoma State University. A description of a new method for solving “variable coefficient” linear differential equations. A Spline solution yields a computational algorithm.

An Interdisciplinary Course Based on Gödel, Escher, Bach: An Eternal Golden Braid. Harry Coonce, Mankato State University. A report concerning a course involving faculty and students from various disciplines centered around the recent best-seller by Douglas Hofstadter.

Informal Discussion: Topics include teaching of undergraduate mathematics, textbooks, curricula, subterfuge, etc.

Campfire Activities—Annual Business Meeting.

Organizer: Roy Ryden

Eleventh Congress
Whiskeytown National Recreation Area
May 14-16, 1982

Rational Languages and Syntactic Monoids. Simon Goberstein, California State University, Chico. A theorem due to S. Eilenberg (1976) establishing a one to one correspondence between pseudovarieties of finite monoids and certain families of rational languages.

Mathematical Modeling of Subsurface Water Flow. David Ellis, San Francisco State University. Subsurface water is a major source of water supply in most areas of the United States. In many places, these supplies are in jeopardy due to a combination of increased demand and deterioration of water quality. A two-dimensional model of unconfined flow in saturated regions in the form of a nonlinear time dependent boundary value problem is presented.

Mathematics Teaching and Multiembodiment. John Englehardt, Southern Oregon State College.

“Under-the-oaks”: Informal, spontaneous sharing of the concerns and status of mathematics and mathematics education in the State of Jefferson.

Campfire: Annual “business” meeting of the Congress.

Organizer: Richard Montgomery

Twelfth Congress
Whiskeytown National Recreation Area
May 20-22, 1983

Efficient Computation in GF(p) & in Z/pnZ. Jay J. Thomas, Electrical Engineering, Naval Postgraduate School. When p is a Mersenne prime, the basic operations in GF(p) can be easily and naturally implemented in hardware. These operations can be extended to GF(pn) and to the ring of integers modulo pn. One application is the exact computation of the inverse and the characteristic polynomial of an ill-conditioned matrix.

Fifty-Two Things to Remember When Playing Blackjack. Peter Griffin, Sacramento State University. A “spectrum of opportunity” arises when “twenty-one” is played without shuffling the deck. Here's how to apply multivariate statistical methods and lose your shirt!

A Number-Theorist's adventures in Computerland. Kent Wooldridge, Computer Science, Chico State University. A recent convert to the other side discusses the kinds of mathematics that computer science people do.

Under-the-Oaks: Open discussion of the concerns & status of mathematics & mathematics education in the State of Jefferson.

Campfire: Annual “business meeting.”

Organizer: Buck Ware.

Thirteenth Congress
Whiskeytown National Recreation Area
May 18-20, 1984

Differentiation among the Square-Integrable Functions. Jack Ladwig, Chico State University. Enlarging the set of allowable solutions to differential equations—via the Fourier transform—provides the framework for current numerical solution techniques.

Finding Stability in Matrix Games. Martin Flashman, Humboldt State University. Matrix games provide a simple model for non-cooperative decision making. A solution concept that involves stability, the Nash equilibrium, yields a convenient method for analyzing matrix games. An algorithm for finding equilibria for two-person matrix games is discussed.

Intuitive/Counterintuitive Statistics. Jo Service, CSU Long Beach. Some ideas from cognitive science on why statistics and probability are so tricky for many (most?) people to master.

Under-the-Oaks: Open discussion of the concerns & status of mathematics & mathematics education in the State of Jefferson.

Organizer: Buck Ware

Fourteenth Congress
Whiskeytown National Recreation Area
May 17-19, 1985

Mathematical Stories. Les Lange, San Jose State University. A potpourri of examples of nice mathematics selected from approximation theory, number theory, linear algebra—and even a couple of unusual examples from financial mathematics.

Reflections and the Foundation of Geometry. James Smith, San Francisco State University. In one approach to the foundations of geometry, axioms are developed to describe groups of motions, and then the geometries are reconstructed from those groups.

The Tenets of Structured Programming. Howard Stauffer, Humboldt State University. Illustrated by examples (& “counterexamples”) taken from Pascal.

Under-the-Oaks. Open discussion of the concerns & status of mathematics & mathematics education in the State of Jefferson.

Campfire: Annual “business meeting.”

Organizer: Buck Ware.

Fifteenth Congress
Whiskeytown National Recreation Area
May 16-18, 1986

Binomial Coeffients. Neville Robbins, San Francisco State University. An exploration of some properties of the binomial coefficients, ranging from the familiar to the arcane; a look at a new number-theoretic function involving the binomial coefficients; and some questions regarding triangular numbers.

A Simplified Model of Poker. Tory Parsons, Chico State University. A (vastly oversimplified) model of poker is used to illustrate the mathematical analysis of games of strategy involving chance.

Publishing: Tradition and Innovation. Menton Sveen, Houghton-Mifflin.

Under-the-Oaks: Open discussion of the concerns & status of mathematics & mathematics education in the State of Jefferson. Theme this year: the standard calculus sequence.

Campfire: Annual “business meeting.”

Organizer: Buck Ware.

Sixteenth Congress
Whiskeytown National Recreation Area
May 15-17, 1987

Error Detecting and Correcting. George Converse, Southern Oregon State College. From simple parity and Hamming codes to self-checking checkers and unidirectional error codes.

Convexity and Nearest Point Functions. Marc Marsh, Sacramento State University. Does a subset M of a Hilbert space always have a unique point nearest to a given point outside? It does if M is closed and convex. In an infinite dimensional space the converse is an open problem.

Some Applied Number Theory. Howard Swann, San Jose State University. On the knapsack problem and public key crypto systems; cracking a code with partial information.

Under-the-Oaks: Open discussion of the concerns & status of mathematics & mathematics education in the State of Jefferson. Theme this year: calculators and computers in the classroom.

Campfire: Annual “business meeting.”

Organizer: Jack Ladwig.

Seventeenth Congress
Whiskeytown National Recreation Area
May 20-22, 1988

Can You See the Fish After It Jumped? Charlie Hamaker, Saint Mary's College. This intriguing question is answered by examining the Radon transform of the outgoing wave, which contains all the information “you can hope to get.”

Projective Geometry and the Symbolic Method. Joel Stein, Chico State University. A symbolic calculus suitable for proving theorems in projective geometry is illustrated with several examples from the plane and 3-space.

On Teaching Mathematics. Roy Ryden, Humboldt State University.

Under-the-Oaks: Open discussion of the concerns & status of mathematics & mathematics education in the State of Jefferson. Postponed from last year: Calculators and Computers in the Classroom.

Campfire: Annual business meeting.

Organizer: Jack Ladwig.

Eighteenth Congress
Whiskeytown National Recreation Area
May 19-21, 1989

So You Think You Know Trig? Rick Luttman, Sonoma State University. In the vein of Period Three Implies Chaos, the iteration of a trigonometric function is examined for periodic, and eventually or asymptotically periodic, points. The other points form an uncountable set. Chaos.

Cellular Automata. Bob Hooper, University of Nevada, Reno. At each point of a lattice in the plane is a cell. At any time each cell is in a particular state. The states change with time. An example of this is the game Life of J. H. Conway.

Classrooms in Four Cultures. Gregory Karpilovsky, Chico State University. Reflections on teaching mathematics in the Soviet Union, Australia, South Africa, and the United States.

Under-the-Oaks: Open discussion of the concerns & status of mathematics & mathematics education in the State of Jefferson. This year: The availability, quality, and utility of mathematical software.

Campfire: Annual business meeting.

Organizer: Jack Ladwig.

Nineteenth Congress
Whiskeytown National Recreation Area
May 18-20, 1990

Thomsen's Equation For Line Reflections. Jim Smith, San Francisco State University. The shortest nontrivial equation holding for reflections in the sides of an arbitrary triangle has 22 terms. The methods involved include transformational geometry and combinatorics of paths in a hexagonal plane lattice.

Romancing the Triangle. Ken Yanosko, Humboldt State University. Shades of Pythagoras, one can generate all triangles with integer side lengths in which one angle is a rational multiple of another.

Challenges of the Nineties. Marcia Sward, Mathematical Association of America. How mathematical organizations and individuals can participate creatively and fully as our community is challenged to re-examine basic assumptions including who can do mathematics, how it should be taught, and how it relates to the environment, economy, politics and health.

Under-the-Oaks: Open discussion of the concerns & status of mathematics & mathematics education in the State of Jefferson. This year: Follow up to “Challenges of the Nineties.”

Campfire: Annual business meeting.

Organizer: Jack Ladwig.

Twentieth Congress
Whiskeytown National Recreation Area
May 17-19, 1991

Is There a Shift in the Wind? Eric Hayashi, San Francisco State University. Shift operators are among the simplest and most concrete of linear operators, yet they play a major role in analyzing a wide class of operators. Jordan works for finite dimensions; what about the infinite cases?

A Star is Born. Kemble Yates, Southern Oregon State College. In 1902 James Jeans theorized that stars form in an interstellar gas cloud as a result of gravitational instabilities. The historical evolution of this model and a version incorporating suspended dust particles is presented.

Under-the-Oaks. Topic: The State of Jefferson. Rummaging through political fact and 19 years of Congress memorabilia.

Campfire—annual business meeting.

Organizer: Richard Montgomery.

Twenty-First Congress
Whiskeytown National Recreation Area
May 15-17, 1992

Accelerating the Convergence of Chebyshev Series. Lisa Yates, Southern Oregon State College. This talk analyzes the role of non-linear changes of variable in the acceleration of the convergence of Chebyshev series for entire functions. Attention is focused on those functions which satisfy the differential equation u'' - xnu = 0, n = 0,1,2,... . It is shown how the “optimal” change of scale may be determined.

Biodegradable Matrices: Generic Decompositions Expose Roots Quickly and Safely. Jeffrey Haag, Humboldt State University. While matrices are generally not biodegradable, they do admit various forms of decomposition. Prolonged decomposition exposes their latent roots (eigenvalues). Current favored forms of decomposition include QR (slow, safe) and LU (fast, risky). Hybrid decompositions that are typically fast and stable are discussed.

Gödel, Escher & Bach by Hofstadter. Diane Johnson, Humboldt State University. A brief introduction to the interplay between the ideas of three great men. Music and pictures are provided.

Under-the-Oaks.

Campfire (annual pseudo-business meeting).

Organizer: Richard Montgomery.

Twenty-Second Congress
Whiskeytown National Recreation Area
May 14-16, 1993

The Maximum Valency of Regular Graphs with Given Odd Girth. Gou-Hui Zhang, Sonoma State University. A graph G is a finite nonempty set V(G) of objects called vertices and a set E(G) of 2-element subsets of V(G) called edges. The odd girth of a graph G gives the length of a shortest odd cycle in G, if it has one. A regular graph of order n, degree k, and odd girth g is called an (n,k,g)-graph. Let f(k,g) denote the smallest n for which there exists an (n,k,g)-graph. This talk considers the problem of determining the number f(k,g), and discusses some different variations of the problem.

Solving Equations and Polynomial Equations. David Meredith, San Francisco State University. Numerical equation solving can be difficult, and a proof exists in the literature that it is in general impossible. Various difficulties in one variable equation solving are discussed; and Graeffe's method, a general algorithm for solving polynomial equations, is presented.

Are Two Better Than One? Brian Jersky, Sonoma State University. When students are first introduced to the concept of confidence intervals, they often ask whether a smaller interval could be obtained by randomly dividing the sample into two groups, calculating two intervals based on these smaller samples, and declaring the parameter to be somewhere in the overlap of the two intervals, with the same specified confidence level as for the original single sample. This talk examines under what conditions this idea may be helpful.

Under-the-Oaks.

Campfire (annual pseudo-business meeting).

Organizer: David Ellis.

Twenty-Third Congress
Whiskeytown National Recreation Area
May 20-22, 1994

Various Applications of Stochastic Processes. Allison Kimber, Humboldt State University. Examples are given of how stochastic processes are useful in a variety of fields where the effects of randomness are important.

"Ever Heard of Dual Billiards?" and Other Things to Say When the Cue Ball Hits the Floor. Wendy Brunzie, University of California at Davis. A hands-on look at the dynamical system called the dual billiards map.

Student Portfolios: A Path to Deeper Understanding. Sherry Ettlich, Southern Oregon State College. Writing assignments that have been used to encourage student reflection and synthesis of concepts.

Discussion Under-the-Oaks.

Campfire (annual pseudo-business meeting).

Organizer: Richard Montgomery.

Twenty-Fourth Congress
Whiskeytown National Recreation Area
May 19-21, 1995

Wave Propagation in Random Media. Ram Vedantham, University of California at Davis. A model which may be used for sound waves in a turbulent medium is derived. Attaching stochastic properties to the medium permits a theoretical investigation of the moments of the solution. The flavor is given via specific cases using numerical and analytical methods.

Did Plutarch Get Archimedes' Wishes Right? Les Lange, San Jose State University. A brief, corrective detective tale about a famous passage from ancient mathematical literature. In answering the title question several beautiful geometric ratios of historical interest are used and an old moral is repeated.

Using Calculus to Explore How Our Bodies Feel the World. Luke Simcik, University of California at Davis. The derivative is intimately woven into the design of our senses and leads to a consistent explanation for such paradoxes as water that feels both hot and cold at the same time.

Under the Oaks with Art Clemons moderating a panel discussion of mathematics as a general education requirement.

Family Campfire and annual pseudo-business meeting.

Organizer: Richard Montgomery.

Twenty-Fifth Congress
Whiskeytown National Recreation Area
May 17-19, 1996

Mixing Circles. Kathy Hahn, Hayward State University. Why does the eagle circle his hunting grounds? Why do spiders spin round webs? Do round ponds support more life than elliptical ones? These questions relate to the isoperimetric property. Santalo's proof using properties of mixed areas is given. Mixed areas in the plane and mixed volumes in space possess many beautiful and useful qualities, some of which are revealed.

Maximum Determinants of Binary Matrices. Gordon Latta, Naval Postgraduate School. The largest value attainable for the determinant of a matrix whose entries are all zeros and ones remains an open question. To obtain a lower bound, ideas from n-dimensional geometry, optimal control theory, and quadratic programming are used.

The Changing High School Curriculum. Diane Resek, San Francisco State University. The high school curriculum is changing. One changed curriculum is described and how the changes might affect college teachers is discussed.

Under the Oaks: Discussion of issues and events affecting the teaching and practice of mathematics in the State of Jefferson.

Family Campfire and annual pseudo-business meeting.

Organizers: Jeff Haag, Kemble Yates.

Twenty-Sixth Congress
Whiskeytown National Recreation Area
May 16-18, 1997

Student Talks: Undergraduate Projects. Their hard work has paid off and, now, here are the results. Students from near and far present short talks on projects they have done.

From Murphy's Camp to the First Morning of Creation. David Nash, Southern Oregon State College. Tall—but true!—tales for the State of Jefferson. Or, how Murphy's Camp led to Einstein, the Atomic Age, CAD/CAM at General Motors, and the Big Bang—with consequences for geometry and operator theory.

Under the Oaks—Discussion of issues and events affecting the teaching and practice of mathematics in the State of Jefferson. This year: Is calculus being reformed or deformed?

Family Campfire and annual five minute business meeting.

Organizers: Jeff Haag, Kemble Yates.

Twenty-Seventh Congress
Whiskeytown National Recreation Area
October 3-5, 1997

Wavelets, Frames, and Applications. Li Shidong, San Francisco State University. An expository talk on the basic theories of and some new developments in wavelets and frames. Highlights of applications in numerical analysis, data compression, irregular sampling, Gabor time-frequency representations in signal processing, etc., are discussed.

Convexly Arranged Sets and Mutually Transverse Matchings. Dusty Sabo, Southern Oregon University. Given a set S with n points in general position in Rn, a matching is a collection of line segments determined by points in S such that each point is an endpoint of no more than one segment in the matching. Many possible matchings on a k-subset of these points are possible. One might try to find a convex k-gon, a matching on a k-subset that has maximal segment length, or the largest possible matching on S so that each pair of its segment intersect, and so on. Some known results and some open conjectures are presented.

Discussion Under the Oaks led by Andrew Phelps, Humboldt State University. Personal reminiscences of the new math, and implications for today.

Family Campfire and annual five minute business meeting.

Organizers: Jeff Haag, Kemble Yates.

Twenty-Eighth Congress
Whiskeytown National Recreation Area
October 9-11, 1998

Real Data—Real Connections: Enhance your Classes with the CBL/CBR and a TI-83. Stuart Moskowitz, Humboldt State University. Real data is more meaningful and more interesting for our students. With the CBL/CBR data collector, graphs of moving objects are plotted, and then piecewise functions to model the movement are found.

The Cosmic Ruler: How Astronomers Measure Distances in the Universe. Kemble Yates, Southern Oregon University. The time-honored technique of “bootstrapping” has allowed astronomers to parlay knowledge of the size of the Earth to compute the size of the Universe. A nice variety of relatively simple mathematics is presented which allows each step up the cosmological distance ladder.

Discussion Under the Oaks led by Eric York, Southern Oregon University. An Introduction to the History of the Cayley-Hamilton Theorem. The Cayley-Hamilton Theorem states that any (square) matrix is a root of its characteristic equation. Some of the development of this landmark result is explored, with excursions into the history of matrix theory in general. Some interesting twists occur along the way. Elementary matrix arithmetic is the only prerequisite.

Family Campfire and annual five minute business meeting.

Organizers: Jeff Haag, Kemble Yates.

Twenty-Ninth Congress
Whiskeytown National Recreation Area
October 8-10, 1999

Synchronization of Biological Oscillators. Sunil Tiwari, Sonoma State University. A robust model for a population of integrate-and-fire oscillators will be presented. The model evolves to a final state in which all of the oscillators are firing synchronously for a wide variety of initial conditions.

Singular Value Decompositions Can Grade Your Papers. David Meredith, San Francisco State University. Singular values for any nonzero matrix are defined, and a nice proof of their existence is presented. A surprising application: computer grading of essay exams!

Discussion under the Oaks led by Jack Ladwig, Chico State University. Topic: What math do we expect our entering freshman to know, and how will it change in the future? Some work from the joint UC/CSU partnership Mathematics Diagnostic Testing Project is presented, and a discussion follows.

Family Campfire and annual five minute business meeting.

Organizers: Jeff Haag, Kemble Yates.

Thirtieth Congress
Whiskeytown National Recreation Area
September 29 - October 1, 2000

Hunting for a Uniform Distribution at Whiskeytown. Daniel Kim, Southern Oregon University. What is the expected area of a randomly-formed triangle within a square of side length r? How can we represent the quantity in terms of r? These types of questions are extended to other geometric objects in other domains. Various thoughts associated with these processes are also explored.

Genetic Algorithms and an Interesting Application to BioMathematics. David Ellis, San Francisco State University. Many scientific and engineering tasks can be formulated as optimization problems in which the minimum or maximum of an objective function with real parameters is sought. When the objective “cost” function is nonlinear and contains many real variables, conventional deterministic search methods such as the simplex method, the method of Nelder and Mead, and the Levenberg-Marquardt algorithms can be ineffective. The alternative of genetic algorithms (GA) is described, and it is shown how GA's work in fitting certain mathematical models to experimental data produced by polymerase chain reactions (PCRs).

Discussion Under the Oaks led by Jeff Haag. Topic: How has technology changed what we do in the classroom?

Family Campfire and annual five minute business meeting.

Organizers: Jeff Haag, Kemble Yates.

Thirty-First Congress
Whiskeytown National Recreation Area
October 5-7, 2001

Making Connections: Minimal Surfaces and Complex Analysis. Jim Fischer, Oregon Institute of Technology. Minimal Surfaces are surfaces in R3 whose mean curvature is everywhere equal to zero. Nature provides many examples via soap films spanning wire frames. These soap film surfaces have the property that of all surfaces with a prescribed boundary (the wire frame), the soap film is the one with least area. After an introduction to surfaces and mean curvature, the connection is made between minimal surfaces and complex analysis using the Weierstrass-Enneper Representation Theorem. This is a surprising and useful result that allows the establishment of several properties of minimal surfaces as well as the solving of some important related problems. Along the way connections are made with variational calculus and non-linear partial differential equations.

Trace Fields of Knots. Thomas Mattman, California State University, Chico. In knot theory, knots are classified through the use of invariants. A knot invariant is a mathematical structure (e.g., number, polynomial, group) associated to the knot such that equivalent knots have equal values of the invariant (but not necessarily conversely). For example, equivalent knots have the same Jones polynomial but it's still unknown whether or not there is any knot which has the same Jones polynomial as the unknot. A hyperbolic knot is one whose complement in S3 is a hyperbolic manifold. This hyperbolic structure gives rise to a knot invariant called the trace field which is a simple extension of the rational numbers. Recent progress in calculating this invariant for some classes of knots is discussed.

Discussion Under the Oaks led by Will Bagnall, Arcata High School. Topic: “A Day in the Life of a High School Math Teacher—i.e., I Should've Got My PhD.” Dave Barry once said, “How can we expect today's young people to understand mathematics when so many of them can't even point their baseball caps in the right direction?” Have you ever wondered what's going on in the high school math classes? What are some of the challenges that teachers of mathematics at the K-12 level face? What would middle/high school math teachers like to say to their college math professors after teaching in the public schools for a while? Here's your chance to discuss subject competence, curriculum, teacher training, public school politics, students, ... with a REAL high school math teacher!

Family Campfire and annual five minute business meeting.

Organizers: Jeff Haag, Kemble Yates.

Thirty-Second Congress
Whiskeytown National Recreation Area
October 4-6, 2002

Probing for Primes. Ken Yanosko, Humboldt State University. In recent years, a renewed interest in large prime integers has arisen, both among amateur mathematicians and in the world of commercial cryptography. Some tests for primality are examined, and an attempt is made to set a record for the largest “Jefferson” prime.

Space Oddities: The Curious Realm of Manifolds. Curtis Feist, Southern Oregon University. Manifolds are often held up as “natural” objects for study. But for such supposedly natural objects, they certainly have an impressive array of special cases and pathological examples. Here's a look at a few of my favorites

The Paths of Moons around Suns. Sam Brannen, Sonoma State University. As a moon revolves around its planet and its planet revolves about its sun, what path does the moon make around the sun? We will show that the path depends on two factors: the relative distances from the moon to the planet and from the planet to the sun, and the number of times the moon orbits the planet during one planetary “year.” In some cases, the path has loops, in other cases the path has cusps, and surprisingly, in some cases the path can actually be convex.

Discussion Under the Oaks led by Kemble Yates, Southern Oregon University. Topic: “Senior Capstones: Meaningful Closure to the Undergraduate Experience.”

Family Campfire and annual five minute business meeting.

Organizers: Jeff Haag, Kemble Yates.

Thirty-Third Congress
Whiskeytown National Recreation Area
October 3-5, 2003

Mathematics and Protein Folding. Ben Ford, Sonoma State University. The understanding of protein structure is the next big frontier in molecular biology. Proteins are the tools by which DNA does its work; a protein's function is determined largely by its physical shape (and many diseases are caused by mis-folding). The folded (least energy) shape of a protein is determined by its coding DNA sequence—but the resulting calculus minimization problem has thousands of variables and is not tractable. Many mathematical techniques can be brought to bear on this “protein folding” problem, including statistical, algorithmic, and dynamical systems approaches.

Bifurcation and Stability of Simple Fluid Flows. Elizabeth Burroughs, Humboldt State University. For an applied mathematician, nothing is as exciting as a physically simple system that exhibits complicated dynamical behavior. Starting with basic notions of bifurcation and stability, we look at two physically simple fluid systems, the closed-loop thermosyphon and the differentially heated cavity, and discuss the dynamics that lead each to exhibit periodic behavior.

Discussion Under the Oaks led by Rapti de Silva, Chico State University. Topic: Notions of Proof. What would you accept as a mathematical proof? What kinds of experiences are needed to foster an understanding of and motivate a necessity for mathematical proof? How can we incorporate such experiences into teaching mathematics to college students? This discussion is based on recent research and on participants' experience as learners and teachers of mathematics.

Family Campfire and annual five minute business meeting.

Organizers: Jeff Haag, Kemble Yates.

Thirty-Fourth Congress
Whiskeytown National Recreation Area
October 1-3, 2004

The Importance of Being Complete. Walden Freedman, Humboldt State University. Zabreiko's Lemma, published in 1969, says that if X is a Banach space, that is, a complete normed vector space, then every countably subadditive seminorm on X is continuous. The lemma, which follows from a weak form of the Baire Category Theorem, is not that well known, and after defining our terms carefully, we explore it. We consider examples showing how the lemma can fail to hold for noncomplete spaces. We also explore the lemma's applications in Banach space theory. For example, it can be used to give simple proofs of the Uniform Boundedness Principle, and the Closed Graph Theorem.

Golden Nuggets from Geometry. Rick Luttmann, Sonoma State University. In three decades of service as the classical geometry Associate Editor for the Problems section of the American Mathematical Monthly, the speaker has encountered a large number of interesting new results in geometry. He shares a selection of his favorites.

Discussion Under the Oaks led by Jorgen Berglund, Chico State University. Topic: Blended Programs: An Opportunity for Change. In California, the traditional path to a secondary mathematics teaching credential takes a minimum of five years, four years in an undergraduate degree program and a fifth year in a credential program. This traditional path has long failed to provide California with the number of mathematics teachers needed. As a result, there are a growing number of ways to credential or authorize non-majors to teach secondary mathematics. One response to this reality is to create a four year blended program that combines pedagogy and content. In order to create such a program, we must restructure mathematics programs and mathematics courses, and this will entail facing hard choices. Such a blended program is proposed as a starting point for discussion.

Family Campfire and annual five minute business meeting.

Organizers: Jeff Haag, Kemble Yates.

Thirty-Fifth Congress
Whiskeytown National Recreation Area
September 30, October 1-2, 2005

A Convolution Sieve and the Twin Primes Conjecture. Jack Buttcane, Southern Oregon University. The Twin Primes Conjecture states that there are an infinite number of primes separated by 2 and is one of the oldest unsolved problems in mathematics. A sieve relying on the structure of periodic waveforms under a Twin Primes Fourier transform is described and applied to the twin primes problem. The final result is a necessary and sufficient condition for the Strong Twin Primes Conjecture.

Never Underestimate a Theorem That Counts Something. Tyler Evans, Humboldt State University. In a recent note (March 2005) in the America Mathematical Monthly, the authors show how one may derive classical divisibility theorems such as Fermat's (little) theorem, Wilson's theorem and Lucas' theorem all from a single combinatorial lemma. In this talk, we show that this lemma is a consequence of Burnside's counting theorem from elementary group theory. Using this point of view, we easily derive three new divisibility theorems for which the aforementioned classical results are, respectively, the cases of a prime divisor.

Discussion Under the Oaks led by Brigitte Lahme, Sonoma State University. Topic: Getting Students Ready for College Mathematics. What skills would we like our students in college math classes to have? What skills do our students have, and which skills do college entrance exams test? Looking at some test questions and student responses on the CSU Entry Level Math (ELM) test may give us some insight into what students know. How can we improve college readiness in mathematics? California just introduced the Early Assessment Program (EAP) in High Schools to get students ready for mathematics requirements in the California State University system, and thereby decrease the need for remediation in college. As college faculty and future teachers, what can we do to help high school students get ready for college math?

Family Campfire and annual five minute business meeting.

Organizers: Jeff Haag, Kemble Yates.

Thirty-Sixth Congress
Whiskeytown National Recreation Area
October 6-8, 2006

What Epidemic Models Tell Us About Vaccination Policies. Chris Dugaw, Humboldt State University. We briefly develop and analyze simple differential equation models of epidemics. Then we focus on how these models have been used in practice to eradicate diseases and to determine vaccination policies. We also show that allowing individuals to forgo vaccination puts the entire population (including vaccinated individuals) at risk.

Linear Algebra: When are we ever going to use this stuff?. Christopher M. Pavone, California State University, Chico. In this talk we introduce a few useful linear algebra concepts, including a powerful algorithm for computing “dominant” eigenvectors. We then show how these concepts are used by the search engine. It is hoped that you walk away from this talk with a basic understanding of how Google works (at least enough to impress your friends), and a new appreciation for the power of linear algebra.

Discussion Under the Oaks led by Elizabeth Burroughs, Humboldt State University. Topic: How Effective Is a Week-long Professional Development Workshop for K-12 Mathematics Teachers?. We describe the design of an evaluation of a five-day mathematics professional development workshop that was offered to teachers in the Fort Bragg School District during the summer of 2006. This discussion considers some of the ways we measured teachers' growth in the areas of mathematical content and pedagogy and highlights some results of our assessment. This is an undergraduate research project that was conducted as part of HSU's Science of Design REU program.

Family Campfire and annual five minute business meeting.

Organizers: Jeff Haag, Kemble Yates.

Thirty-Seventh Congress
Whiskeytown National Recreation Area
October 5-7, 2007

Number Theory and Trees. Benjamin Levitt, California State University, Chico. A graph is an object that consists of a (non-empty) set of vertices and some edges between them. Graphs satisfying certain criteria are called trees. In this talk, we see how to construct a graph from a positive integer and ask “What kinds of integers give rise to (grow?) trees?” Along the path to answering this question we encounter interesting results from classical number theory.

Definitions and Nondefinability in Euclidean Geometry. James T. Smith, San Francisco State University. This talk is about the axiomatic method, the choice of primitive notions for axiomatic Euclidean geometry, a logical framework for geometry, definitions in that framework, and some delicate undefinability results. It emphasizes advances by Mario Pieri and Alfred Tarski in the era 1900-1935, but provides background starting with Aristotle around 350 B.C.E. and a taste of advanced results as recent as 1991.

Discussion Under the Oaks led by Marianna Bogomolny, Southern Oregon University. Topic: What Can We Learn by Generating Examples? “... One of the students was absolutely super: He answered everything nifty! The examiners asked him what diamagnetism was, and he answered it perfectly ... After the exam I went up to this bright young man ... The first question I asked is, ‘Can you give me some example of a diamagnetic substance?’ ‘No.’ ... Everything was entirely memorized, yet nothing had been translated into meaningful words.” (Richard P. Feynman, Surely You're Joking, Mr. Feynman!, 1985). This phenomenon is not unique to physics. Learners are rarely asked to construct examples for mathematical concepts explicitly, especially in the postsecondary level mathematics courses. Yet, it was shown that constructing examples of objects promotes and contributes to learning. In this discussion, we engage in example generation and explore what can be learned from this activity.

Family Campfire and annual five minute business meeting.

Organizers: Jeff Haag, Kemble Yates.

Thirty-Eighth Congress
Whiskeytown National Recreation Area
October 3-5, 2008*

*This meeting was cancelled because of inclement weather. The entire program was moved to the following year.

Thirty-Eighth Congress
Whiskeytown National Recreation Area
October 2-4, 2009

Two Coloring Problems, Each Related to Schur Numbers. Dusty Sabo, Southern Oregon University. In this talk, each problem involves trying to avoid monochromatic solutions to either a linear equation or a system of linear equations. In the problem involving the system, we work to avoid these solutions and in the other we color an integer interval and try to keep the monochromatic solutions to the equation at a minimum.

A Friendly Introduction to Combinatorial Designs. Izabela Kanaana, Sonoma State University. Combinatorial design theory concerns questions about whether it is possible to arrange elements of a finite set into subsets so that certain “balance” properties are satisfied. Design theory has its roots in eighteenth and nineteenth century recreational mathematics, but today the field has many practical applications in experimental design, tournament scheduling, coding theory, cryptography, optical communications, wireless communications, and group testing, to mention just a few areas. This talk introduces some basic terms and definitions, gives some examples, and presents some general constructions of combinatorial designs using complete graphs and Latin squares.

Discussion Under the Oaks led by Dale Oliver, Humboldt State University. Topic: Algebra for All. In March of 2008 the National Math Advisor Panel recommended in its final report that, “All school districts should ensure that all prepared students have access to an authentic algebra course—and should prepare more students than at present to enroll in such a course by Grade 8.” In July of 2008, the California Board of Education voted to require an eighth grade algebra test for all students. What does all of this mean for the mathematical community? We discuss pros and cons of officially moving the algebra requirement to 8th grade, and how we can best support teachers, future teachers, and students in an “Algebra-for-all-at-8th-grade” climate.

Family Campfire and annual five minute business meeting.

Organizers: Jeff Haag, Kemble Yates.

Thirty-Ninth Congress
Whiskeytown National Recreation Area
October 1-3, 2010

Investigations into the Analysis of Ordinal Data. Kathy Gray California State University, Chico. The aim of our research was to investigate the optimal methods of analysis of ordinal data, including Likert scales. Furthermore, properties of ordinal data were explored. A number of parametric and nonparametric tests were considered and assessed by determining the type I error rate and power across the tests. We developed a randomization test to improve the power of one of the tests. By simulating data we were able to judge the validity of the tests under a variety of sample sizes and distributions. We discovered that while some tests tended to be inadequate, a majority of them produced satisfactory results under most scenarios.

My Favorite Mathematical Paradoxes. Cora Neal, Wells Fargo Bank. The speaker explains several of her favorite mathematical paradoxes, their history, and resolutions, if they exist. The paradoxes cover many topics including algebra, geometry, calculus, set theory, and probability. Her goal is to share at least one paradox with you that you haven't heard before.

Discussion Under the Oaks led by Brad Ballinger, Humboldt State University. Topic: Mathematical Preparation of High School Students. What should math teachers prepare high school students to do? Two of the many possible answers to that question are: (1) succeed in college math classes; (2) succeed outside of college. At this Congress in 2005, Brigitte Lahme led a discussion about college-bound students, citing California's (then brand new) Early Assessment Program (EAP). We discuss some work, related to EAP, that has been done to smooth the high school-college transition. We also consider the mathematical needs of high school students who are not college-bound. Must teachers neglect one of these populations in order to serve the other? What methods can teachers use to serve both?

Family Campfire and annual five minute business meeting.

Organizers: Jeff Haag, Kemble Yates.

Fortieth Congress
Whiskeytown National Recreation Area
September 30 – October 2, 2011

Mathematical Models as Testable Scientific Hypotheses: A Modern Approach to Fitting Mathematical Models to Data. Rob Van Kirk, Humboldt State University. Traditional mathematical modeling has used analytical and computational representations of physical and biological processes to quantify relationships among key variables. Meanwhile, traditional statistical analysis has focused on empirical description of data rather than on process. Over the past 15 years, increased computational power, availability of comprehensive data sets, and advances in theory have led to a modern approach that fits process-based mathematical models to data in a framework that allows robust evaluation of scientific hypotheses. This expository talk introduces this modern approach to mathematical modeling and provides some examples of its application.

The Apportionment Problem. Rick Luttmann, Sonoma State University. The U S Constitution requires that the seats in the House of Representatives be apportioned to the States according to their populations. But Representatives come in whole units while population proportions don't. “So, what's the problem? Just round off! It's third-grade mathematics.” Well, no, it isn't. Though the Founding Fathers apparently didn't realize it, there are intractable difficulties in coming up with a “fair” apportionment scheme. This talk explores various methods that have been used or proposed, along with what's wrong with them. The contribution of mathematics is to establish that there is no “perfect” method.

Discussion Under the Oaks led by Francie Bostwick, Southern Oregon University. Topic: Number Talks and Mathematical Thinking. What should math teachers prepare high school students to do? Two of the many possible answers to that question are: (1) succeed in college math classes; (2) succeed outside of college. At this Congress in 2005, Brigitte Lahme led a discussion about college-bound students, citing California's (then brand new) Early Assessment Program (EAP). We will discuss some work, related to EAP, that has been done to smooth the high school-college transition. We will also consider the mathematical needs of high school students who are not college-bound. Must teachers neglect one of these populations in order to serve the other? What methods can teachers use to serve both?

Family Campfire and annual five minute business meeting.

Organizers: Jeff Haag, Kemble Yates.

Forty-First Congress
Whiskeytown National Recreation Area
October 5-7, 2012

An Alternate View of Finding Eigenvalues. Gregg Waterman, Oregon Institute of Technology. The standard approach to eigenvalues and eigenvectors in texts is to demonstrate algebraic techniques for finding the eigenvalues and eigenvectors of a matrix A, which then allows A to be “diagonalized” as A = PDP-1. We instead use what we know about the geometry of a simple transformation in two dimensions to determine the eigenvalues and eigenvectors of the transformation, and from them we construct the matrix of the transformation. After doing this, we analyze the product PDP-1 for our transformation. This analysis reveals the richness of this simple example; we see a variety of fundamental ideas from a first course in linear algebra.

Bifurcations in the Dynamics of Student Attitudes toward Undergraduate Mathematics Class. Zaur Birkaliev, Chico State University. This study incorporates the ideas of complexity and chaos and is based on a series of daily surveys of 254 undergraduate engineering majors enrolled in 10 intensive mathematics classes over 73 days during the entire semester. The findings suggest a hypothesis that patterns of chaos in the dynamics of student attitudes toward mathematics class might represent not only random noise or measurement error but may develop through series of bifurcations associated with such educational parameters as SAT/ACT math score, GPA, course grade, and gender.

Discussion Under the Oaks led by Irving Lubliner, Southern Oregon University. Topic: N Great Problems, At Least N+1 Great Solutions. Do you enjoy problems that seem impossible and turn out to be simple? How about the ones that seem easy but are devilishly challenging? The speaker shares some of his favorites, including one that took him over 25 years to solve!

Family Campfire and annual five minute business meeting.

Organizers: Jeff Haag, Kemble Yates.

Forty-Second Congress
Prairie Creek Redwoods State Park
October 4-6, 2013

Mathematical Creativity: If you want to get there, don't start from here. David Scott, University of Puget Sound. Many significant developments in mathematics occur when an idea or formula that may not make sense in its original context has a valid interpretation in a related context. The talk contains several results as well as a problem that illustrate this type of creativity.

Functions, Duality, and Mapping Diagrams. Martin Flashman, Humboldt State University. Mapping diagrams provide a valuable alternative to graphs for visualizing functions. Core function concepts can be more easily understood using these diagrams. Concepts of duality have been used to illuminate many mathematical categories such as geometry, algebra, and analysis. The speaker explores the power of mapping diagrams as visualizations for functions and discusses how duality can be used to make connections between these diagrams and traditional graphs.

Discussion Under the Redwoods led by Nick Franceschine, Sonoma State University. Topic: 008: A License to Approximate. Actuaries are business professionals who attempt to forecast the financial consequences of future events. The speaker opens the “black box” to show how actuarial mathematics works, discusses opportunities in the actuarial profession, and talks about how interested students might prepare themselves to enter this interesting field. The format is interactive, with the speaker responding to questions about the profession—and why it’s so hard to become an actuary.

Family Campfire and annual five minute business meeting. Also a memorial tribute for David Meredith.

Organizers: Jeff Haag, Kemble Yates.