Greater Hudson Valley
Mathematical Speakers Bureau

The Greater Hudson Valley Mathematical Speakers Bureau, formed among liberal arts colleges in eastern New York, Connecticut and western Massachusetts, provides member institutions a source of undergraduate-level talks given by talented mathematicians residing at nearby colleges. In addition to providing interesting, accessible talks for undergraduates, the Bureau provides an opportunity for faculty members of neighboring colleges to interact and discuss common research or pedagogical interests.

Current GHVMSB institutions: Bard College, Mount Holyoke College, Skidmore College, Sarah Lawrence College, Smith College, Trinity College, Union College, Vassar College and Williams College.

If you have any questions or suggestions regarding the GHVMSB, please contact Dan King at Sarah Lawrence College (dking@mail.slc.edu). Please note: Edith Starr, GHVMSB co-organizer, of Vassar College is on sabbatical this year.

List of available speakers for 1999-00 Academic Year

G-RATED TALKS -- for general undergraduate audiences.

"Bus Tours of the Universe and Beyond, by Travel Agent Mel Slugbate"

SPEAKER: Colin Adams

AFFILIATION: Williams College

E-MAIL: Colin.Adams@williams.edu

PHONE: 413-597-3300

PRE-REQUISITES: No special math or travel experience is necessary!

REQUIRED EQUIPMENT: Overhead projector and a Macintosh PowerPC with projection.

TIME/DATE PREFERENCES: Teaching 10-1, MWF in the Fall.

ABSTRACT: Ever wondered if the earth is doughnut-shaped? Ever wanted to ride in a bus, sipping multi-colored drinks, as you tour the universe? Ever felt that your wallet was making your pants sag? Then this is the talk for you. Acclaimed travel expert Mel Slugbate will share his accumulated wisdom on travel, and mention a few reasonably priced tours of possible models of the universe. In this introduction to the idea of manifolds, he will answer such questions as: "How can you tell what surface you are living on?" "When is it safe to eat something that is moving?" "Is the universe infinite in extent, and if so, is the tour round-trip?"

"Why Knot?"

SPEAKER: Colin Adams

AFFILIATION: Williams College

E-MAIL: Colin.Adams@williams.edu

PHONE: 413-597-3300

PRE-REQUISITES: None, suitable for a general audience.

REQUIRED EQUIPMENT: Overhead projector

TIME/DATE PREFERENCES: Teaching 10-1, MWF in the Fall.

ABSTRACT: Knots have appeared everywhere from art, literature and religion to seafaring. We will discuss their past history as well as their future, focussing on the role they play in mathematics and the applications of that role to DNA and synthetic chemistry.

"Curvature of Polyhedra"

SPEAKER: Ethan Bloch

AFFILIATION: Bard College

E-MAIL: bloch@bard.edu

PHONE: 914-758-7266

PRE-REQUISITES:

REQUIRED EQUIPMENT: None

TIME/DATE PREFERENCES: Monday - Thursday

ABSTRACT: In classical differential geometry, the proper way to compute the curvature of smooth surfaces was defined by Gauss. A few centuries earlier, Descartes defined a simple way to compute the curvature at the vertices of simplicial surfaces (that is, surfaces made up of triangles, such as a tetrahedron). In this talk we will explain how to define curvature for simplicial surfaces, demonstrate some properties of such curvature (including a "Gauss-Bonnet" type theorem), and discuss how to extend this definition to more general polyhedra.

"Factoring Polynomials"

SPEAKER: Tom Garrity

AFFILIATION: Williams College

E-MAIL: tgarrity@williams.edu

PHONE: 413-597-2399

PRE-REQUISITES: High school algebra

REQUIRED EQUIPMENT: Chalkboard only

TIME/DATE PREFERENCES:

ABSTRACT: How can you factor polynomials in more than one variable? We will discuss a fairly new method that first translates this algebra problem into a geometric one and then changes the geometric problem into one of graph theory.

"Classical Mathematics. Renaissance, Baroque and Romantic Mathematics. Also Pop, Punk, and New Age Mathematics"

SPEAKER: Jim Henle

AFFILIATION: Smith College

E-MAIL: jhenle@smith.edu

PHONE: (413) 585-3867

PRE-REQUISITES:

REQUIRED EQUIPMENT: Tape player, overhead projector

TIME/DATE PREFERENCES: On Sabbatical Spring Semester

ABSTRACT: A discussion of artistic periods in mathematics leading to the conclusion that mathematics corresponds closest to music of all the arts.

"Mathematical Aspects of the Music of Bach"

SPEAKER: Victor E. Hill IV

AFFILIATION: Williams College

E-MAIL: vhill@williams.edu

PHONE: 413-597-2428

PRE-REQUISITES:

REQUIRED EQUIPMENT:

TIME/DATE PREFERENCES:

ABSTRACT:

"Rigor and Non-Rigor in the Work of Regiomontanus"

SPEAKER: Victor E. Hill IV

AFFILIATION: Williams College

E-MAIL: vhill@williams.edu

PHONE: 413-597-2428

PRE-REQUISITES:

REQUIRED EQUIPMENT:

TIME/DATE PREFERENCES:

ABSTRACT: Johannes Mueller of Koenigsberg (1436-1476), who adopted the pen name of Regiomontanus, was probably the most significant and influential mathematician of the 15th century. The printing press and observatory that he set up at Nuremberg were intended to advance the interest of both science and literature. His major work, "De triangulis omnimodis" (1464, pub. 1533), may be regarded as the first mathematical treatise in Western Europe to rise above the elementary and imprecise writings of the preceding centuries. Still, this book curiously combines Greek deductive reasoning with arguments that are often at best not rigorous and at worst simply incorrect. This talk compares those elements in selected proofs from the work of Regiomontanus.

"Zero and the Null Set: A Mathematical Talk on Nothing"

SPEAKER: Victor E. Hill IV

AFFILIATION: Williams College

E-MAIL: vhill@williams.edu

PHONE: 413-597-2428

PRE-REQUISITES:

REQUIRED EQUIPMENT:

TIME/DATE PREFERENCES:

ABSTRACT: The concept of nothing, zero, came curiously late into the history of mathematics, and it gained intellectual acceptance against much resistance. Still, this idea turns out to be an intriguing thread through the long development of mathematical thought. The relationship between zero and the empty set leads to the remarkable construction in the 20th century of a full set theory out of the mathematical concept of nothing. In this talk, Dr. Hill traces the history of "nothing" in mathematics from prehistory to the present, with literary references from Homer to Hemingway.

"Story Problems: their History, Purposes, and Solutions"

SPEAKER: Victor E. Hill IV

AFFILIATION: Williams College

E-MAIL: vhill@williams.edu

PHONE: 413-597-2428

PRE-REQUISITES:

REQUIRED EQUIPMENT:

TIME/DATE PREFERENCES:

ABSTRACT: "Story problems" (or "word problems") have often been feared or hated by generations of students. Yet these problems have a fascinating history, sometimes progressing over centuries from the practical to the absurd, and often presenting additional problems of interpretation, assumptions, or cultural context. In this talk (which assumes only a bit of high school algebra as background), Dr. Hill categorizes these problems from PRACTICE to PREPOSTEROUS and surveys, in particular, the possibility that a given problem may, upon inspection, turn out to have many legitimate solutions."

"Fixed Points and Fermat's Little Theorem"

SPEAKER:Brenda Johnson

AFFILIATION: Union College

EMAIL: johnsonb@union.edu

PHONE: 518-388-6162 (or 388-6246)

PRE-REQUISITES:

REQUIRED EQUIPMENT:

TIME/DATE PREFERENCES:

ABSTRACT: The French mathematician Pierre de Fermat, famous for his "Last Theorem" recently proved by Andrew Wiles, produced many important results in number theory. After his "Last Theorem", the best known is probably his "Little Theorem": if p is a prime number and a is an integer, then a^p - a will always be divisible by p. Many proofs of this result have been written since Fermat first reported it in 1640. I will discuss a proof that draws on elementary ideas from dynamical systems, and go on to show how these ideas can be used to prove other results in number theory.

"Problems in Democracy: The Mathematics of Social Choice and Arrow's Impossibility Theorem."

SPEAKER: Dan King

AFFILIATION: Sarah Lawrence College

E-MAIL: dking@mail.slc.edu

PHONE: (914) 395-2424

PRE-REQUISITES: None

REQUIRED EQUIPMENT: Overhead projector and blackboard

TIME/DATE PREFERENCES:

ABSTRACT: Voting is the vehicle by which decisions are made in a democratic society. When there are only two alternatives, the most democratic method for determining the winner is easy: majority rules. When there are three or more alternatives the fairest method is not so evident. This talk examines several methods by which groups of individuals with divergent opinions may collectively agree upon a single social choice. Featured will be Arrow's Impossiblity Theorem which states that, in a sense, there is no ideal social choice mechanism.

"Paradoxes of Prediction and Cooperation: Newcomb's Problem and the Prisoner's Dilemma"

SPEAKER: Dan King

AFFILIATION: Sarah Lawrence College

E-MAIL: dking@mail.slc.edu

PHONE: (914) 395-2424

PRE-REQUISITES: None

REQUIRED EQUIPMENT: Overhead projector and blackboard

TIME/DATE PREFERENCES:

ABSTRACT: Fascinating and perplexing, paradoxes challenge our most cherished beliefs. Left unresolved, these paradoxes pose a serious threat to the very foundations of our social, moral and political philosophies. Two intriguing paradoxes in game theory will be discussed, Newcomb's Problem and the Prisoner's Dilemma. Together we will attempt to resolve these paradoxes and bring peace to the world...and to ourselves.

"The Soap Bubble Geometry Contest"

SPEAKER: Frank Morgan

AFFILIATION: Williams College

E-MAIL: Frank.Morgan@williams.edu

PHONE:

PRE-REQUISITES: None

REQUIRED EQUIPMENT: One or ideally two overhead transparency projectors and screens. Bucket of cold water (for dipping wire frames). 3'x6' table. Roll of paper towels. Cloth handtowel.

TIME/DATE PREFERENCES:

ABSTRACT: Mathematicians (with the help of undergraduates) are just beginning to understand the geometry of soap bubbles on their way to understanding the universe. After reporting on the latest news and rumors, we'll present a guessing contest, complete with demonstrations and prizes. You don't need to know anything. Friends and families welcome.

"The Double Soap Bubble Conjecture"

SPEAKER: Frank Morgan

AFFILIATION: Williams College

E-MAIL: Frank.Morgan@williams.edu

PHONE:

PRE-REQUISITES: None

REQUIRED EQUIPMENT: One or ideally two overhead transparency projectors and screens.

TIME/DATE PREFERENCES:

ABSTRACT: It is well known that a round soap bubble provides the least-perimeter way to enclose a fixed volume of air. The Double Bubble Conjecture says that the familiar double soap bubble provides the least-area way to enclose a separate two given volumes of air. The planar case was proved by undergraduates. The case of equal volumes in R3 was proved by computer by Hass, Hutchings, and Schlafly. There are rumors of a proof of the general case in R3 (and perhaps in R4 in further work by undergraduates).

"Statistics and Causality"

SPEAKER: Jerry Reiter

AFFILIATION: Williams College

E-MAIL: Jerome.P.Reiter@williams.edu

PHONE: 413-597-3519

PRE-REQUISITES: Introductory Statistics course including regression

REQUIRED EQUIPMENT: transparency projecto

TIME/DATE PREFERENCES:

ABSTRACT: Increasingly, causal questions are being answered with statistics. For both scientists and consumers, it has become important to understand how valid causal studies can be designed and how suspicious studies can be identified. This talk aims to further these understandings by explaining the statistical principles and techniques that underlie valid studies of causal relationships. It explains how randomized experiments yield reliable estimates of causal effects and describes techniques used to estimate causal effects in non-randomized studies.

"Symmetry, Rigid Motions, & Polygons"

SPEAKER: David Vella

AFFILIATION: Skidmore College

E-MAIL: dvella@skidmore.edu

PHONE: 518-580-5291

PRE-REQUISITES: A casual acquaintance with basic high-school geometry and addition of vectors in the plane. It would also help if the audience members have heard of some basic terminology such as "group", "subgroup", "subgroup generated by...", although no actual group theory is used beyond these basic terms.

REQUIRED EQUIPMENT: overhead projector and/or blackboard

TIME/DATE PREFERENCES: Flexible

ABSTRACT: Did you know that if the midpoints of the sides of an arbitrary quadrilateral are connected, the resulting quadrilateral must be a parallelogram? There are elementary proofs (both synthetic and analytic) of this fact of Euclidean geometry, but these proofs give no insight as to what happens if the quadrilateral is replaced by an arbitrary polygon. In this talk it is shown that by paying heed to the underlying symmetry of the problem and by taking a transformational approach, one is led to a more satisfactory understanding of both the original problem and its generalizations.

"A Mountain out of a Molehill With An Application to Computer Handwriting Recognition"

SPEAKER: Steve C. Wang

AFFILIATION: Harvard University (formerly Williams College)

E-MAIL: scwang@stat.harvard.edu

PHONE: (617) 495-1600

PRE-REQUISITES: None

REQUIRED EQUIPMENT: Overhead Transparency Projector, Chalkboard

TIME/DATE PREFERENCES: None

ABSTRACT: How has a seemingly innocuous theorem led to a rift in the way people think about and practice statistics? I will derive this theorem -- Bayes theorem -- from first principles and describe the philosophical and practical controversies it has provoked. I will also demonstrate some of its applications to card games, disease testing, and computer handwriting recognition.

"Real Estate in Hyperbolic Space: Investment Opportunities for the Next Millennium, by Mel Slugbate"

SPEAKER: Colin Adams

AFFILIATION: Williams College

E-MAIL: Colin.Adams@williams.edu

PHONE: 413-597-3300

PRE-REQUISITES: No previous math or real estate background assumed! Recommended for students and faculty alike! Siskel and Ebert say, "Two fingers up!"

REQUIRED EQUIPMENT: Overhead projector

TIME/DATE PREFERENCES: Teaching 10-1 MWF in the Fall.

ABSTRACT: The sky-high stock market got you nervous? What goes up must come down? Antsy about stocks, bonds and mutual funds? Afraid of risky investments in Euclidean space? Then real estate in hyperbolic space is for you. We will discuss the enormous potential of this new investment opportunity and describe the many fascinating properties of hyperbolic space that make it such an attractive place to live. This is the financial equivalent of the 1980's junk bond. Don't miss it. Bring your checkbook and credit references!

"How to Always Win at Limbo: You can sum some of the series some of the time and some of the series none of the time but can you sum some of the series all of the time?"

SPEAKER: Edward Burger

AFFILIATION: Williams College

E-MAIL: eburger@williams.edu

PHONE: 413-597-2454

PRE-REQUISITES: Have heard of infinite series.

REQUIRED EQUIPMENT: One overhead and one blackboard.

TIME/DATE PREFERENCES:

ABSTRACT: The talk will revolve around the basic question: What does it mean to be small and for two things to be close to one another? We'll take a strange look at infinite series and ask what it means for them to converge. In fact, we'll even attempt to build some very exotic infinite series: we'll either success or fail... you'll have to come to the talk to find out what happens. Will you be at the edge of your seats? Perhaps, but if not, then you'll probably fall asleep and either way, after the talk, youi'll feel refreshed and great. No matter what, you'll learn a sneaky way to always win at Limbo.

"Plants and Symmetry"

SPEAKER: Chris Gole

AFFILIATION: Smith College

E-MAIL: cgole@math.smith.edu

PHONE: (413) 585 3875

PRE-REQUISITES:

REQUIRED EQUIPMENT: Computer projector

TIME/DATE PREFERENCES: Monday, Friday (fall), Tuesday (spring)

ABSTRACT: Even to the casual observers, plant forms across many different species display a certain developmental similarity. Pine cones, cacti, broad leaf perennials and many others - all share a common growth pattern of their leaves, flowerets or other botanical elements. When viewed from above, individual elements emerge from the central shoot in double families of spirals and frequently the number of spirals in these families are two successors in the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13,...)! This phenomenon, central to the multi-disciplinary field of Phyllotaxis, has received relatively scarce attention from Mathematicians. I will indicate in this talk how the theory of dynamical systems can be brought to good use in this context.

"When Mathematics Resembles Astrology"

SPEAKER: Jim Henle

AFFILIATION: Smith College

E-MAIL: jhenle@smith.edu

PHONE: (413) 585-3867

PRE-REQUISITES:

REQUIRED EQUIPMENT:

TIME/DATE PREFERENCES: On Sabbatical Spring Semester

ABSTRACT: Examples of elementary mathematical statements that can't be proved or disproved without assuming the existence of incomprehensibly large infinite sets.

"Are there odd perfect numbers?"

SPEAKER: John McCleary

AFFILIATION: Vassar College

E-MAIL: mccleary@vassar.edu

PHONE: (914) 437-5526

PRE-REQUISITES:

REQUIRED EQUIPMENT: Overhead projector

TIME/DATE PREFERENCES: Tuesdays

ABSTRACT: The search for odd perfect numbers may be one of the oldest mathematical journeys known. In this talk I will describe the class of perfect numbers, what we know about the even ones, and then describe what is known about the odd perfect numbers. The proofs are elementary and fall into certain classes of ideas.

"On Inscribed and Circumscribed Circles: Jacobi, Poncelet, Steiner, and Abel"

SPEAKER: John McCleary

AFFILIATION: Vassar College

E-MAIL: mccleary@vassar.edu

PHONE: (914) 437-5526

PRE-REQUISITES:

REQUIRED EQUIPMENT: Overhead projector

TIME/DATE PREFERENCES: Tuesdays

ABSTRACT: A triangle naturally has an inscribed and a circumscribed circle. If we ignore the generating triangle, can we take the pair of circles and recover the triangle? Given radii of a pair of circles, one inside another, and the distance between their centers, can we decide if a pair of circles comes from a triangle? The relations between these quantities for triangles and for other polygons was studied by Euler, Fuss, and er and deeply generalized by Poncelet in his great Traite des Propri\'et\'es Projectives. The famous Porisme de Poncelet treats the problem of inscribed and circumscribed polygons for pairs of conic sections. Jacobi studied this theorem and gave a remarkable proof of Poncelet's result based on his investigations of elliptic functions. In this talk, we present the problem of inscribed and circumscribed circles for triangles using elementary geometry and trigonometry to arrive at the primary algebraic relation, and introduce Jacobi's ideas---the nineteenth century view surrounding this problem turns out very surprising.

"Polynomial (In)equalities and Beyond"

SPEAKER: Charles Steinhorn

AFFILIATION: Vassar College

E-MAIL: steinhorn@vassar.edu

PHONE: 914-437-5524 or 5525

PRE-REQUISITES: None

REQUIRED EQUIPMENT: blackboard and chalk, and an overhead projector.

TIME/DATE PREFERENCES: Tuesday or Thursday only.

ABSTRACT: Most calculus students know that a (non-zero) polynomial in one variable of degree n has at most n real number zeroes. Descartes knew that there are significantly better bounds for polynomials of high degree which consist of a small number of monomials: for example, a polynomial in one variable of arbitrary degree which is the sum of 3 monomials has no more than 7 zeroes. This talk begins with this theme of finding bounds for the number of zeroes of polynomials. It then moves on to similar questions for polynomial inequalities in one or more variables, and concludes with some discussion

"Taylor Series of Composite Functions and Combinatorial Identities"

SPEAKER: David Vella

AFFILIATION: Skidmore College

E-MAIL: dvella@skidmore.edu

PHONE: 518-580-5291

PRE-REQUISITES: The audience member should be familiar with the Taylor series of a function, and be willing to wade through some elaborate looking notation. It would also help if the audience members were familiar with the notion of a partition of an integer n.

REQUIRED EQUIPMENT: overhead projector and blackboard

TIME/DATE PREFERENCES: Flexible

ABSTRACT: The familiar chain rule from calculus can be extended to second and higher derivatives. One possible applicationof this endeavor is the computation of the Taylor coefficients of a composite function f(g(x)) in terms of the Taylor coefficients of f and of g. The resulting theorem serves as a machine for generating dozens of combinatorial identities involving binomial coefficients, Stirling numbers, Bernoulli numbers, Euler numbers, and Bell numbers. We obtain new proofs of old results as well as (we hope!) some new identities.

"On Writing Numbers"

SPEAKER: Tom Garrity

AFFILIATION: Williams College

E-MAIL: tgarrity@williams.edu

PHONE: 413-597-2399

PRE-REQUISITES: Linear Algebra

REQUIRED EQUIPMENT: Chalkboard only

TIME/DATE PREFERENCES:

ABSTRACT: How should we express a real number? Decimal expansions are the best if we want to be able to add and multiply two numbers. Further, decimal expansions are also (eventually) periodic precisely when the number is rational. Are there other ways to expand real numbers so that their algebraic properties are captured by some type of periodicity? We will discuss a recent generalization of continued fractions for writing real numbers. In this new method for expressing real numbers, periodicity will mean that the number is a cubic irrational. Many basic questions remain open.

"The Burnside Counting Theorem and Group Characters"

SPEAKER: Victor E. Hill IV

AFFILIATION: Williams College

E-MAIL: vhill@williams.edu

PHONE: 413-597-2428

PRE-REQUISITES: This talk assumes a basic background in linear algebra, but does not require a background in group theory.

REQUIRED EQUIPMENT:

TIME/DATE PREFERENCES:

ABSTRACT: In 1911, W. F. Burnside published (in a form that is only scarcely recognizable when compared to modern notation) a remarkable theorem relating two abstract algebraic concepts that were then still in their youth: what we now refer to as orbits and group characters. Polya in 1957 and Liu in 1968, among others, showed how these concepts can be applied to more recent problems in mathematics. In this talk, Dr. Hill shows how the Burnside Counting Theorem can begin with a simple problem relating to the planning of a set of children's blocks and can extend to basic questions in group character theory, with applications to spectroscopy in chemistry.

"Lights Out: Solving an Electronic Game as a Paradigm for doing Mathematics"

SPEAKER: Benjamin Lotto

AFFILIATION: Vassar College

E-MAIL: belotto@vassar.edu

PHONE: 914 437 7180

PRE-REQUISITES: Basic linear algebra---solving linear systems using vectors and matrices

REQUIRED EQUIPMENT: Computer projection system

TIME/DATE PREFERENCES:

ABSTRACT: Lights Out is an addictive puzzle manufactured by Tiger Electronics. We will work out a mathematical solution to Lights Out and use the solution process to illustrate how mathematics is done.

"Buffon's Needle and Buffon's Noodle"

SPEAKER: David Robbins

AFFILIATION: Trinity College

E-MAIL: David.Robbins@mail.trincoll.edu

PHONE: 860-297-2293

PRE-REQUISITES: Calculus, maybe the notion of a sample space

REQUIRED EQUIPMENT: Overhead (two if possible, but one ok), blackboard

TIME/DATE PREFERENCES:

ABSTRACT: Exercise: Throw a needle of length 1 unit onto a plane ruled with parallel lines 2 units apart. What is the probability that the needle will cross a line? (Clearly, it can't cross more than one!) This is the classical Buffon needle problem. What can we say if the needle is replaced by a piece of cooked spaghetti? We will discuss these and some related questions in geometrical probability.

''Signature in Linear Algebra and Topology''

SPEAKER: Ranja Roy

AFFILIATION: Union College

E-MAIL: royr@union.edu

PHONE: 518-388 6395

PRE-REQUISITE: Basis Linear Algebra and Calculus

REQUIRED EQUIPMENT: Overhead projector and blackboard

TIME/DATE PREFERENCES: Thursday

ABSTRACT: The definition of signature in the topological set up is simply a generalization of the concept of signature of a symmetric matrix. In this talk we will explain the transition using examples from geometry.

"What We Don't Know about Cutting a Cake Fairly"

SPEAKER: William Zwicker

AFFILIATION: Union College

EMAIL: zwickerw@union.edu

PHONE: 518-388-6160 (or 388-6246)

PRE-REQUISITE:

REQUIRED EQUIPMENT:

TIME/DATE PREFERENCES:

ABSTRACT: Suppose we divide up a cake -- one for which people differ on the value of a piece. The result is "envy free" if no one would prefer another's share. Recently, there has been notable progress in developing two cake cutting methods. Discrete schemes allow only steps such as, "Portia cuts piece A (so as to halve it, in her eyes)," while moving-knife schemes allow, "Colin passes two knives slowly over the cake (so that the region between them is always exactly half the cake, in his eyes)." We'll discuss the new results, and some conjectures inspired by them. One of these attempts to pin down the exact nature of the advantage that continuous schemes have over discrete ones.

"Hypergame . . . or, I Stubbed my Toe on the Foundations of Mathematics"

SPEAKER: William Zwicker

AFFILIATION: Union College

EMAIL: zwickerw@union.edu

PHONE: 518-388-6160 (or 388-6246)

PRE-REQUISITE:

REQUIRED EQUIPMENT:

TIME/DATE PREFERENCES:

ABSTRACT: While making a bonus question for a course in Game Theory, I discovered a very peculiar game, which seems to lead to conflicting results. Does Hypergame truly create a contradiction in mathematics? Does Hypergame even exist?

"The Mathematics of Political Power"

SPEAKER: William Zwicker

AFFILIATION: Union College

EMAIL: zwickerw@union.edu

PHONE: 518-388-6160 (or 388-6246)

PRE-REQUISITE:

REQUIRED EQUIPMENT:

TIME/DATE PREFERENCES:

ABSTRACT: When elected representatives vote "yes" or "no" on proposed legislation, constitutional change, etc., the voting systems range from simple majority rule, to weighted versions in which legislators from more populous districts cast more votes, to complex bicameral systems with presidential vetoes and veto overrides, such as the US federal system. A key design question is whether the actual difference in influence among the legislators came out as intended. The traditional approach is to use a mathematical "voting power index," but the known indices differ sharply from each other. Can the issue be resolved? Some recent results indicate two promising lines of research: axioms for power indices, and the use of indices that assign intervals to measure power, rather than single numbers.

"Randomness and Modular Arithmetic"

SPEAKER: Charles Steinhorn

AFFILIATION: Vassar College

E-MAIL: steinhorn@vassar.edu

PHONE: 914-437-5524 or 5525

PRE-REQUISITES: None, just mathematical sophistication

REQUIRED EQUIPMENT: blackboard and chalk

TIME/DATE PREFERENCES: Tuesday or Thursday only.

ABSTRACT: The random graph can be understood as the undirected graph whose vertices are the natural numbers such that edges between vertices are determined by flipping a coin. The first part of the talk focuses on the random graph from a naive probabilistic point of view. The second part of the talk deals with a particular class of finite graphs that ``approximate'' the random graph. The surprise here is that these finite graphs are constructed using modular arithmetic.