Class Reunion

Colin C. Adams, Francis C. Oakley Third Century Professor of Mathematics

Mathematical Intelligencer, Vol. 28, No. 2, 22-23 (2006)

The Cohomology of Proof

Colin C. Adams, Francis C. Oakley Third Century Professor of Mathematics

Mathematical Intelligencer, Vol. 28, No. 3, 29-30 (2006)

Journey to the Center of Mathematics

Colin C. Adams, Francis C. ™s solution.Oakley Third Century Professor of Mathematics

Mathematical Intelligencer, Vol. 28, No. 4, 13-16 (2006)

The Great Pi/e Debate

Colin C. Adams, Francis C. Oakley Third Century Professor of Mathematics

Mathematical Association of America (2006)

Cusp Size Bounds from Surfaces in Hyperbolic 3-Manifolds

Colin C. Adams, Francis C. Oakley Third Century Professor of Mathematics

with A. Colestock*, J. Fowler*, W.D. Gillam*, E. Katerman*

Transactions of the A.M.S.,Vol. 258, No. 2, 727-741 (2005)

Totally Geodesic Surfaces in Hyperbolic Knot Complements I

Colin C. Adams, Francis C. Oakley Third Century Professor of Mathematics

with E. Schoenfeld*

Geometriae Dedicata,Vol. 116, 723-247 (2005)

Hyperbolic Knots

Colin C. Adams, Francis C. Oakley Third Century Professor of Mathematics

Ed. by W. Menasco and M. Thistlethwaite

The Handbook of Knot Theory, Elsevier (August 2005)

The Mathematical Ethicist

Colin C. Adams, Francis C. Oakley Third Century Professor of Mathematics

Mathematical Intelligencer, Vol. 27, No. 2, 24-25 (2005)

Math Talk

Colin C. Adams, Francis C. Oakley Third Century Professor of Mathematics

with Lew Ludwig

Mathematical Intelligencer, Vol. 27, No. 3, 31-32 (2005)

Phone Interview

Colin C. Adams, Francis C. Oakley Third Century Professor of Mathematics

Mathematical Intelligencer, Vol. 27, No. 4, 26-28 (2005)

The Theorem Blaster

Colin C. Adams, Francis C. Oakley Third Century Professor of Mathematics

Mathematical Intelligencer, Vol. 28, No. 1, 17-18 (2006)

Brunnian Clothes: Not for the Bashful

Colin C. Adams, Francis C. Oakley Third Century Professor of Mathematics

T. Fleming and C. Koegel

American Mathematical Monthly, 111, No. 9, 741-748 (2004)

Brunnian links are links such that the removal of any one component trivializes the link. Can you make Brunnian clothes? This paper discusses the consequences.

Integers Causing Bipolar Orientation Reversal of Appendage Motion

Colin C. Adams, Francis C. Oakley Third Century Professor of Mathematics

Math Horizons, Vol. XII, Issue 3, 10-11, 24 (2005)

Try to draw a six in the air as you move your foot in a counterclockwise direction. This paper is an investigation of the positive integers that cause orientation reversal upon air depiction.

A Proof of God

Colin C. Adams, Francis C. Oakley Third Century Professor of Mathematics

Mathematical Intelligencer, Vol. 26, No. 3, 13-15 (2004)

What is someone claimed a proof of God? And what if it was right?

Mangum, P.I.

Colin C. Adams, Francis C. Oakley Third Century Professor of Mathematics

Mathematical Intelligencer, Vol. 26, No. 4, 22-24 (2004)

The adventures of a principal investigator on an NSF grant.

Trial and Error

Colin C. Adams, Francis C. Oakley Third Century Professor of Mathematics

Mathematical Intelligencer, Vol. 27, No. 1, 37-40 (2005)

What happens if the courts get involved in grading.

The Mathematical Ethicist

Colin C. Adams, Francis C. Oakley Third Century Professor of Mathematics

Mathematical Intelligencer, Vol. 27, No. 2, 24-25 (2005).

A sample of the column to which ethically challenged mathematicians turn.

Why Knot

Colin C. Adams,Mark Hopkins Professor of Mathematics

Key Curriculum Press,(2004)

First book in a series of expository comic style books about mathematics with attached toys.

Detecting Incompressible Boundary in 3-Manifolds

Colin C. Adams,Mark Hopkins Professor of Mathematics

Geometriae Dedicata,99, 47-60 (2003)

A construction is presented which can be utilized to prove incompressibility of boundary in a 3-manifold. This can be utilized to show that a candidate arc c is not an unknotting tunnel for a cusped 3-manifold. It can also be used to show that a "tubed surface" is incompressible.

Cleanliness of Geodesics in Hyperbolic 3-Manifolds

Colin C. Adams,Mark Hopkins Professor of Mathematics

A. Colestock, J. Fowler, W.D. Gillam, E. Katerman

Pacific Journal of Mathematics,213, no. 2, 201-211 (2004)

This paper is an investigation of geodesics in cusped hyperbolic 3-manifolds. We derive conditions guaranteeing the existence of geodesics avoiding the cusps and use these geodesics to show that in "almost all" finite volume hyperbolic 3-manifolds, infinitely many horoballs in the universal cover corresponding to a cusp are visible in a fundamental domain of the cusp when viewed from infinity.

The Three Little Pigs

Colin C. Adams,Mark Hopkins Professor of Mathematics

Mathematical Intelligencer,25, No. 3, 27-28 (2003)

What happens when the three little pigs decide to get Ph.D.'s in mathematics.

Don't Touch the Button

Colin C. Adams,Mark Hopkins Professor of Mathematics

Mathematical Intelligencer,25, No. 4, 32-34 (2003)

A new faculty member tries to deal with an unusual chair.

Rumpled Stilskin

Colin C. Adams,Mark Hopkins Professor of Mathematics

Mathematical Intelligencer,26, No. 1, 22-24 (2004)

What happens when a father claims his daughter can turn coffee into theorems.

This Theorem is Big

Colin C. Adams,Mark Hopkins Professor of Mathematics

Mathematical Intelligencer,26, No. 2, 57, (2004)

When Hollywood discovers mathematics, this is the result.

Cusp Densities of Hyperbolic 3-Manifolds

Colin C. Adams, Mark Hopkins Professor of Mathematics

Proceedings of the Edinburgh Mathematics Society, 45, 277-284 (2002)

The cusp density of a hyperbolic 3-manifold is the ratio of the largest possible volume in a set of cusps with disjoint interiors to the volume in the manifold. It is known that all cusp densities fall in the interval [0, .853É]. It is shown that the cusp densities of finite volume orientable hyperbolic 3-manifolds are dense in this interval.

An Introduction to the Supercrossing Index of Knots and the Crossing Map

Colin C. Adams, Mark Hopkins Professor of Mathematics

C. Lefever, J. Othmer, S. Pahk, A. Stier, and J. Tripp

Journal of Knot Theory and Its Ramifications, 11, No. 3, 445-459 (2002)

This paper is an introduction to supercrossing index for knots and links, which is related to crossing index in the same way that N. Kuiper's superbridge index is related to bridge index. A variety of results on supercrossing index and the associated crossing map are given.

La Forma dell' Universo: Dieci Possibilita

Colin C. Adams, Mark Hopkins Professor of Mathematics

J. Shapiro

Le Scienze, 73-82 (2003)

An Italian translation of "The Shape of the Universe: Ten Possibilities", which appeared in American Scientist in 2001.

How to Ace Calculus: The Streetwise Guide

Colin C. Adams, Mark Hopkins Professor of Mathematics

J. Hass and A. Thompson

A Chinese version of this humorous supplement to calculus.

Dr. Yeckel and Mr. Hyde

Colin C. Adams, Mark Hopkins Professor of Mathematics

How mathematics can take over someone's life and turn them into a creature unlike anything seen before.

The Red Badge of Courage

Colin C. Adams, Mark Hopkins Professor of Mathematics

The battle lines are drawn. It is the students against the professors as they confront the final exam and fight for their lives.

A Difficult Delivery

Colin C. Adams, Mark Hopkins Professor of Mathematics

How the birth of a new theorem parallels human birth.

Wiling Away the Hours

Colin C. Adams, Mark Hopkins Professor of Mathematics

How did Andrew Wiles come up with his proof of Fermat's Last Theorem? Just what was he doing in the attic all those years.

Waist Size for Cusps in Hyperbolic 3-Manifolds

Colin C. Adams, Mark Hopkins Professor of Mathematics

Topology, 41, No. 2, 257-270 (2002)

The waist size of a cusp in orientable hyperbolic 3-manifold is the length of the shortest nontrivial curve in the maximal cusp boundary generated by a parabolic isometry. A variety of results on waist size are proved. In particular, it is shown that the smallest possible waist size, which is 1, is realized only by the cusp in the figure-eight knot complement.

Alternating Knots in S x I

Colin C. Adams, Mark Hopkins Professor of Mathematics

T. Fleming, M. Levin, and A. Turner

Pacific Journal of Mathematics, 203, No. 1, 1-22 (2002)

One of the Tait conjectures, which was stated 100 years ago and proved in the 1980s, said that reduced alternating projections of alternating knots have the minimal number of crossings. We prove a generalization of this for knots in S x I, where S is a surface. We use a combination of geometric and polynomial techniques.

The Shape of the Universe: Ten Possibilities

Colin C. Adams, Mark Hopkins Professor of Mathematics

J. Shapiro

American Scientist, 89, 443-453 (2001)

Recent observations suggest the global geometry of the spatial universe may be Euclidean. Mathematicians have proved that a Euclidean universe must be one of 18 possiblities. Eight of these are unlikely for physics reasons. The remaining ten are described in this paper.

A Deprogrammer's Tale

Colin C. Adams,Mark Hopkins Professor of Mathematics

Mathematical Intelligencer, 23, No. 3, 13-14 (2001)

What to do when your child gets hooked on mathematics.

Why Knot: Knots, Molecules and Stick Numbers

Colin C. Adams, Mark Hopkins Professor of Mathematics

Plus Magazine, on-line science magazine, 15 (2001)

An exposition of the ideas behind knot theory, stick number of knots and its implications for chemistry.

Hiring Season

Colin C. Adams, Mark Hopkins Professor of Mathematics

Mathematical Intelligencer, 23, No. 3, 21-22 (2001)

The difficulties in hiring in a math department.

Fields Medalist Stripped

Colin C. Adams, Mark Hopkins Professor of Mathematics

Mathematical Intelligencer, 23\4, No. 1, 36 (2002)

What happens when a mathematician takes drugs to enhance his abilities.

Homotopy on the Range

Colin C. Adams, Mark Hopkins Professor of Mathematics

Mathematical Intelligencer, 24, No. 2, 19 (2002)

Algebraic topology meets a cattle drive.

Overcoming Math Anxiety

Colin C. Adams, Mark Hopkins Professor of Mathematics

Mathematical Intelligencer, 23, No. 1, 45-50 (2001)

Math anxiety as a disease.

U-Substitution

Colin C. Adams, Mark Hopkins Professor of Mathematics

Mathematical Intelligencer, 23, No. 2, 29 (2001)

A technique of integration on the basketball court.

How to Ace the Rest of Calculus: the Streetwise Guide

Colin C. Adams, Mark Hopkins Professor of Mathematics
J. Hass and A. Thompson

W.H. Freeman and Company (April, 2001)

The second in the humorous How to Ace Series, covering the rest of calculus.

The Knot Book

Colin C. Adams, Mark Hopkins Professor of Mathematics

W.H. Freeman and Company (January, 2001)

Updated paperback edition.

The Pepsi Putnam Challenge

Colin C. Adams, Mark Hopkins Professor of Mathematics

Mathematical Intelligencer, 22, No. 3, 37-38 (2000)

The Putnam exam with product placement.

The S.S. Riemann

Colin C. Adams, Mark Hopkins Professor of Mathematics

Mathematical Intelligencer, 22, No. 4, 41-42 (2000)

A theorem meets its iceberg.

Systoles for Hyperbolic Knot and Link Complements

Colin C. Adams, Mark Hopkins Professor of Mathematics
Alan Reid

Proceedings of the Cambridge Philosophical Society, 128, 103-110 (2000)

The systole length of a 3-manifold is the length of the shortest geodesic in the manifold. We prove that if L is a knot in the 3-sphere S³ with hyperbolic complement then the systole length of the complement is at most 7.35534.... Improved bounds are obtained for particular types of knots and for manifolds containing particular surfaces.

Into Thin Air

Colin C. Adams, Mark Hopkins Professor of Mathematics

Mathematically Bent Column, Mathematical Intelligencer, 22, 1, 21-22 (2000)

Mathematics as mountain climbing.

Research Announcement

Colin C. Adams, Mark Hopkins Professor of Mathematics

Mathematically Bent Column, Mathematical Intelligencer22, 2, 26-27 (2000)

Mathematics through the personals.

Alternating Graphs

Colin C. Adams, Mark Hopkins Professor of Mathematics
K. Foley, J. Kravis '98, R. Dorman, S. Payne

Journal of Combinatorial Theory, Series B77, 1, 96-120 (September 1999)

In this paper we generalize the concept of alternating knots to alternating graphs and show that every abstract graph has a spatial embedding that is alternating. We also prove that every spatial graph is a subgraph of an alternating graph. We define n-composition for spatial graphs and generalize the results of Menasco on alternating knots to show that an alternating graph is n-composite for n=0,1,2,3 if and only if it is "obviously n-composite'" in any alternating projection. Moreover, no closed incompressible pairwise incompressible surface exists in the complement of an alternating graph. We then generalize results of Kauffman, Murasugi, and Thistlethwaite to prove that the crossing number of an even-valent rigid-vertex alternating spatial graph is realized in every reduced alternating projection with no uncrossed cycles and, if the graph is not 2-composite, the crossing number is not realized in any non-alternating projection. We give examples showing that this result does not hold for graphs with vertices of odd valence or graphs with uncrossed cycles.

Isoperimetric Cusps on Hyperbolic 3-Manifolds

Colin C. Adams, Mark Hopkins Professor of Mathematics

Michigan Mathematics Journal,46, 3, 515-531 (1999)

Given a choice C of one of a set of specified isometry types of maximal cusps, it is shown that three disjoint simple closed curves can be removed from any closed 3-manifold or two disjoint simple closed curves can be removed from a cusped hyperbolic 3-manifold so that the resulting manifold M' is hyperbolic and one of the cusps in M' is isometric as a maximal cusp to C. The isometry types of the possible maximal cusps include ones of volume and If one more curve is removed, maximal cusps of volumes and can also be realized. A similar result is proved for totally geodesic boundaries as well.

Simple Geodesics in Hyperbolic 3-Manifolds

Colin Adams, Chair and Mark Hopkins Professor of Mathematics
J. Hass, University of California, Davis
P. Scott, University of Michigan

Bulletin of the London Mathematical Society31, 81-86, (1999)

Among orientable hyperbolic 2-manifolds, the thrice-punctured sphere is the only example that contains no simple closed geodesics. We are interested in determining which hyperbolic 3-manifolds do and do not contain simple closed geodesics. We prove that the Fushsian group corresponding to the thrice-punctured sphere again generates the only example of a complete non-elementary hyperbolic 3-manifold such that it does not contain a simple geodesic. Moreover, we will prove that a hyperbolic 3-orbifold that is not obtained from a particular set of Fushsian groups will always contain a simple geodesic.

Isoperimetric Curves on Hyperbolic Surfaces

Colin C. Adams, Chair and Mark Hopkins Professor of Mathematics
Frank Morgan, Professor of Mathematics

Proceedings of the American Mathematical Society127, 5, 1347-1356, (1999)

Least-perimeter enclosures of prescribed area on hyperbolic surfaces are characterized.

Maximal Cusps, Collars and Systoles for Hyperbolic Surfaces

Colin C. Adams, Chair and Mark Hopkins Professor of Mathematics

Indiana University Mathematics Journal47, 2, 419-437, (1998)

It is proved that a maximal cusp in any orientable hyperbolic surface has area at least 4, with the lower bound of four realized only for a cusp in the three-punctured sphere. For a maximal cusp in any other hyperbolic surface with p punctures, it is shown that it has area at most 6|c (S) | - (P-1), and that there is a metric that realize this upper bound. Moreover, over all possible hyperbolic metrics, the area a of the maximal cusp takes on all values such that 4 < A 6|c (S) | - (P-1). If S is a punctured orientable hyperbolic surface other than the thrice-punctured sphere, and if S is endowed with any complete hyperbolic metric, then it is proved that there exists a choice of a maximal set of cusps in the hyperbolic surface with total area greater than 5p/2 if p is even and 5p/2 + 3/2 if p is odd. These universal lower bounds lower bounds are best possible. Applications of these results to collars and systoles are included.

The Newest Inductee in the Number Hall of Fame

Colin C. Adams, Chair and Mark Hopkins Professor of Mathematics

Mathematics Magazine71, 5, 341-349, (1998)

The wards ceremony to induct the Gieseking Constant (the volume of an ideal regular hyperbolic tetrahedron) in the Number Hall of Fame, including basic background on hyperbolic geometry.

This article explains how teachers can use the mathematical theory of knots to involve their students with mathematics in the classroom.

The Knot Book(Japanese Translation)

Colin C. Adams, Mark Hopkins Professor of Mathematics

W.H. Freeman and Co., New York, NY, Japanese Translation, (January,1998)

Real Estate in Hyperbolic Space: Investment Opportunities for the 90's

Colin C. Adams, Mark Hopkins Professor of Mathematics

Mathematical Sciences Research Institute, CD-ROM, 1, (Spring, 1998)

Knotted Tilings

Colin C. Adams, Chair and Mark Hopkins Professor of Mathematics

The Mathematics of Long Range Aperiodic Order, ed. by R. Moody, Kluwer Academic Pub. 1-8, (1997)

This paper is a discussion of tilings of 3-space by knotted tiles. A proof of the fact that for any topological shape in Euclidean 3-space with one boundary component, there exists a tiling by tiles, all of which are congruent and all of which have that topological shape is reviewed. The work of S. Oh is then extended to show that for any topological shape in the 3-sphere with one boundary component, there exists of tiling of the 3-sphere by eight congruent tiles, all of that topological shape. Finally, it is shown that given a topological shape in 3-space with one bound ary component, there exists a tiling of Euclidean 3-space by tiles that fall into two congruence classes, such that all of the tiles have that same topological shape and such that the tiling is aperiodic in the sense that it has no translational symmetry.

Stick Numbers and Composition of Knots and Links

Colin C. Adams, Chair and Mark Hopkins Professor of Mathematics
Bevin Brennan '97, Deborah Greilsheimer '97, and Alexander Woo '97

Journal of Knot Theory and its Ramifications, 6, 32, 149-161, (1997)

The concept of stick number for knots and links is addressed under various restrictions concerning the length of the sticks, the angles between sticks, and placements of the vertices. In particular, we focus on the effect of composition on the various stick numbers. Ultimately, we determine the traditional stick number for an infinite class of knots, which are the (n,n-1)-torus knots together with all of the possible compositions of such knots. The exact stick number was previously known for only seven knots.

Exploring Knots

Colin C. Adams, Chair and Mark Hopkins Professor of Mathematics
Eric Furstenberg '97, Jie Li, and Jodi Schneider

Mathematics Teacher, 90, 8, 640-646, 652, (November, 1997)

A CD-ROM version of a presentation given in the Conversations Series at the Mathematical Sciences Research Institute, Berkeley, CA, Dec. 11, 1996.

Splitting Versus Unlinking

Colin C. Adams, Professor of Mathematics

Journal of Knot Theory and its Ramifications, 5, 295-299, (1996)

We prove there exists a link consisting of two components, each of which is individually unknotted, such that the link can be split with a single crossing change, however any such crossing change must knot one of the components.

Unknotting Tunnels for Two-Bridge Knots and Links

Colin C. Adams, Professor of Mathematics
Alan Reid

Commentarii Mathematici Helvetici, 71, 617-627, (1996)

We give a complete classification of the unknotting tunnels in 2-bridge link complements, proving that only the upper and lower tunnels are unknotting tunnels. Moreover, we show that the only strongly parabolic tunnels in 2-cusped hyperbolic 3-manifolds are exactly the upper and lower tunnels in 2-bridge knot and link complements. From this, it follows that the upper and lower tunnels in 2-bridge knot and link complements must be isotopic to geodesics of length at most ln(4), where length is measured relative to maximal cusps. Moreover, the four dual unknotting tunnels in a 2-bridge knot complement, which together with the upper and lower tunnels form the set of all known unknotting tunnels for these knots, must each be homotopic to a geodesic of length at most 6ln(2).

Unknotting Tunnels in Hyperbolic 3-Manifolds

Colin C. Adams, Professor of Mathematics

Math Ann., 302, 177-195 (1995)

In this paper, we examine the relationship between the unknotting tunnels of a hyperbolic 3-manifold and the hyperbolic structure on that manifold. In particular, we begin by proving that the maximal cusp volume of a 1-cusped hyperbolic 3-manifold with tunnel number greater than one is at least 3v3/4. This implies that the volume of the entire manifold is at least 3v0/2, where v0 = 1.01494... . Several interesting corollaries of a purely topological nature follow. These results point toward a new invariant for any compact 3-manifold, which is the least volume of any hyperbolic knot complement within it. We investigate this "parent volume", determining the manifold of least parent volume and the manifold with boundary of least parent volume. We also give necessary conditions for a vertical geodesic to be an unknotting tunnel and use this to classify all of the isotopy classes of unknotting tunnels for the Whitehead link and a few other links. Moreover, in addition to other applications, we obtain a universal upper bound of ln(4) on the length of the shortest representative in the isotopy class of an unknotting tunnel for a 2-cusped hyperbolic 3-manifold. Finally, we begin to delineate some restrictions on the hyperbolic structure of a tunnel number one hyperbolic 3-manifold.

Tilings of Space by Knotted Tiles

Colin C. Adams, Professor of Mathematics

Mathematical Intelligencer, Vol. 17, No. 2, 41-51 (1995)

It is demonstrated that Euclidean 3-space can be tiled with tiles, all congruent to a solid torus, knotted into the shape of any given knot. In fact, given any compact polyhedral submanifold of Euclidean 3-space with one boundary complement, it forms the topological type of a tile that can be used to tile Euclidean 3-space. These results generalize to Euclidean n-space, also.

Dehn Filling Hyperbolic 3-Manifolds

Colin C. Adams, Professor of Mathematics

Pacific Journal of Mathematics, Vol. 65, No. 2, 217-238 (1994)

Define a complete family of parent (ancestor) manifolds to be a set of compact 3-manifolds such that every closed orientable 3-manifold can be obtained by one (or more) Dehn fillings of the manifolds in the family. In 1983, R. Myers proved that the set of 1-cusped hyperbolic 3-manifolds is a complete family of parent manifolds. We prove this result in a new way and then go on to prove:

Theorem 1.1. a) Let Vo be any positive real number. Then the set of 1-cusped hyperbolic 3-manifolds of volume greater than Vo is a complete family of parent manifolds. b) Let V1 be any positive real number. Then the set of 1-cusped hyperbolic 3-manifolds of cusp volume greater than V1 is a complete family of parent manifolds. c) The set of 2-cusped hyperbolic 3-manifolds containing embedded totally geodesic surfaces is a complete family of ancestor (actually grandparent) manifolds. d) For any positive integer N, the set of hyperbolic 3-manifolds, each of which shares its volume with N or more other hyperbolic 3-manifolds, is a complete family of ancestor manifolds.

As a corollary to Theorem 1b, we prove that there exists a complete family of parent manifolds such that at most one Dehn filling on each manifold in the family yields a manifold of finite fundamental group.

A Two-Dimensional Minkowski ?(x) Function

Olga R. Beaver, Professor of Mathematics

with T. Garrity, Professor of Mathematics

Journal of Number Theory,107, 105-134 (2004)

A function from a triangle to itself is defined that has both interesting number theoretic and analytic properties. This function is shown to be a natural generalization of the classical Minkowski ?(x) function. It is shown there exists a natural class of pairs of cubic irrational numbers in the same cubic number field that are mapped to pairs of rational numbers, in analog to ?(x) mapping quadratic irrationals on the unit interval to rational numbers on the unit intervals. It is shown that this new function satisfies an analog to the fact that ?(x), while increasing and continuous, has derivative zero almost everywhere.

Small Area Estimation in a Watershed Erosion Assessment Survey

Carsten Botts, Assistant Professor of Statistics

with J.D. Opsomer and J.Y. Kim

Journal of Agricultural, Biological and Environmental Statistics, Vol. 3, 139-152 (2003)

A Methodological Study for Accuracy Assessment of GAP Land over Maps

Carsten Botts, Assistant Professof of Statistics

with Sarah M. Nusser, Erwin E. Klaas, and Robin McNeely

Gap Analysis Bulletin, No 10 (2001)

A Shrinkage Estimator for Spectral Densities

Carsten Botts, Assistant Professor of Statistics

with Michael J. Daniels

Biometrika, Vol. 93, 179-195 (2006)

On Diophantine Approximation Along Algebraic Curves

Edward B. Burger, Professor of Mathematics

with Ashok Pillai '05

The Proceedings of the American Mathematical Society, 136, 11-19 (2008)

Shrinking the Period Lengths of Continued Fractions While Still Capturing Convergents

Edward B. Burger, Professor of Mathematics

with Jesse Gell-Redman, Ross Kravitz '06, Daniel Walton, and Nicholas Yates '07

The Journal of Number Theory, 128, 144-153 (2008)

Here we prove that every real quadratic irrational a can be expressed as a periodic non-simple continued fraction having period length one. Moreover, we show that the sequence of rational numbers generated by successive truncations of this expansion is a sequence of convergents of a. We close with an application relating the structure of a quadratic a to its conjugate.

On Newton's Method and Rational Approximations to Quadratic Irrationals

Edward B. Burger, Professor of Mathematics

Bulletin of the Canadian Mathematical Society, 47, 12 - 16 (2004)

In 1988 Rieger exhibited a differentiable function having a zero at the golden ratio (Ð1+Ã5)/2 for which when Newton's method for approximating roots is applied with an initial value x0=0, all approximates are so-called "best rational approximates"Ñin this case, of the form F2n/F2n+1, where Fn denotes the nth Fibonacci number. Recently this observation was extended by Komatsu to the class of all quadratic irrationals whose continued fraction expansions have period length 2. Here we generalize these observations by producing an analogous result for all quadratic irrationals and thus provide an explanation for this phenomena.

Diophantine Inequalities and Irrationality Measures for Certain Transcendental Numbers

Edward B. Burger, Professor of Mathematics

The Indian Journal of Pure and Applied Mathematics, 32 1591-1599 (2001)

Here we construct U-numbers having pre-subscribed diophantine structure for which effective measures of irrationality are computed.

Math Forum Ñ I Couldn't Keep my Distance: A Mathematical Seduction

Edward B. Burger (as Drew Aderburg), Professor of Mathematics

The Mathematical Association of America Math Horizons, 12-15 (February 2002)

Here in a humor style, we give an introduction to p-adic numbers and construct infinite series that possess incredible convergence properties.

On a Quantitative Refinement of the Lagrange Spectrum

Edward B. Burger, Professor of Mathematics

Amanda Folsom, Alexander Pekker, Rungporn Roengpitya '01, Julia Snyder '02

Acta Arithmetica, 102, 55-82 (2002)

Here we answer a question posed by Davenport in 1947 regarding diophantine inequalities exhibiting only finitely many solutions. As a consequence of the results introduced here, a quantitative version of the classical Lagrange spectrum is found.

Diophantine Inqualities and Irrationality Measures of Certain Transcendental Numbers

Edward B. Burger,Professor of Mathematics

The Indian Journal of Pure and Applied Mathematics, 32, 1591-1599 (2001)

Here we construct U-numbers having pre-subscribed diophantine structure for which effective measures of irrationality are computed.

"Math Forum" - I Couldn't Keep my Distance: A Mathematical Seduction

Edward B. Burger (as Drew Aderburg), Professor of Mathematics

The Mathematical Association of America Math Horizons, 12-15 1591-1599 (February 2002)

Here in a humor style, we give an introduction to p-adic numbers and construct infinite series that possess incredible convergence properties.

On a Quantitative Refinement of the Lagrange Spectrum

Edward B. Burger, Professor of Mathematics

Amanda Folsom, Alexander Pekker, Rungporn Roengpitya '01, Julia Snyder '02

Acta Arithmetica, 102, 55-82 (2002)

Here we answer a question posed by Davenport in 1947 regarding diophantine inequalities exhibiting only finitely many solutions. As a consequence of the results introduced here, a quantitative version of the classical Lagrange spectrum is found.

Exploring the Number Jungle: A Journey into Diophantine Analysis

Edward B. Burger, Professor of Mathematics

The American Mathematical Society, 155 (2000)

In this text the reader is led through a journey of discovery to the high points of diophantine analysis and related areas of number theory.

On Periods of Elements from Real Quadratic Number Fields

Edward B. Burger, Professor of Mathematics
A.J. van der Poorten

Canadian Math. Soc. Proceedings Series, 27, 35-43 (2000)

Here we provide another proof of a theorem of E. Burger regarding a necessary and sufficient condition on the partial quotients of two real quadratic irrational numbers to insure that they are elements of the same quadratic number field over Q.

Diophantine Olympics and World Champions: Polynomials and Primes Down Under

Edward B. Burger, Professor of Mathematics

The American Mathematical Monthly, 107, 822-829 (2000)

Using a lighthearted Olympic theme, we prove that there exist transcendental numbers whose best approximates have denominators that are values of certain non-constant polynomials evaluated at integers and transcendental numbers whose approximates have denominators who are all primes.

Abandoning Dead Ends: Embracing Lively Beginnings

Edward B. Burger Associate Professor of Mathematics
Michael Starbird

Mathematical Association of America Focus, 9, 10-11 (2000)

Here we examine what mathematics can add to the life of the mind far beyond the discipline itself.

On a Question of Mordell and a Spectrum of Linear Forms

Edward B. Burger Associate Professor of Mathematics

Journal of the London Mathematical Society, 62, 701-715 (2000)

Here we consider a question raised by Mordell regarding lattice point-free parallelepipeds and find a spectrum of optimal regions in dimension 2.

The Heart of Mathematics: An Invitation to Effective Thinking

Edward B. Burger Associate Professor of Mathematics
Michael Starbird

Key College Press in cooperation with Springer-Verlag, (2000)

In this new innovative textbook, the authors put special emphasis on the deep ideas of mathematics and present the subject in a lively and entertaining fashion. Throughout the text, the authors present the subject as a way of thinking that can be used to solve problems, analyze situations, and model the world around us. The book is accompanied by a kit including 3D glasses, CD-ROM, and other hands-on manipulatives.

On Real Quadratic Number Fields and Simultaneous Diophantine Approximation

Edward B. Burger Associate Professor of Mathematics

Monatshefte fur Mathematik, 128, 201-209 (2000)

Jere we provide a necessary and sufficient condition on the partial quotients of two real quadratic irrational numbers to insure that they are elements of the same quadratic number field over Q. Such a condition has implications to simultaneous diophantine approximation. In particular, we deduce an improvement to Dirichlet's Theorem in this context which, as an immediate consequence, shows the Littlewood Conjecture to hold for all numbers from the same quadratic number field.

On Simultaneous Diophantine Approximation in the Vector Space Q + Qa

Edward B. Burger Associate Professor of Mathematics

The Journal of Number Theory, 82, 12-24 (2000)

Beginning with an improvement to Dirichlet's Theorem on simultaneous approximation, in this paper we introduce an algorithm, in sympathy with the classical continued fraction algorithm, to generate the sequence of best approximates to certain simultaneous systems. We then produce best possible upper bounds for the associated diophantine inequalities. In addition, we consider implications within the geometry of numbers and investigate the converse of the sharpened version of Dirichlet's result. Finally, we close with some remarks on the Littlewood Conjecture in this context.

On Diophantine Approximation below the Lagrange Constant

Edward B. Burger Associate Professor of Mathematics
Jonathan Todd '96

Fibonnaci Quarterly, 38, 136-144 (2000)

Let a be a real quadratic irrational and m(a) the Lagrange constant for a. For any c, 0 < c < m(a), we explicitly compute the finite list of positive integers q satisfying q||aq|| < c, and determine the optimal lower bound for c to ensure solvability of the inequality. This extends and generalizes results of Tognetti, Van Ravenstein and Winley.

Pleasures vs. Problems

Edward B. Burger Associate Professor of Mathematics
Michael Starbird

The Notices of the American Mathematical Society, 47, (2000)

In this opinion piece, the authors examine how the mathematics community could inspire and entice others to explore mathematics.

Mathematics Beyond the Ivy-Colored Hall

Edward B. Burger Associate Professor of Mathematics

Mathematical Intelligencer, 20 (1998)

Fermat's Last Theorem, the Four Color Conjecture, and Bill Clinton for April Fools' Day

Edward B. Burger Assistant Professor of Mathematics
Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

Amer. Math. Monthly, 104, 246-255, (1997)

In the spirit of April Fools' Day, we celebrate important mistakes made in 19th century mathematics and their important implications. Specifically, we give "proofs" of Fermat's Last Theorem, the Four Color Conjecture, and the "fact" that one of the authors is Bill Clinton.

Small Solutions to Systems of Linear Congruences Over Number Fields

Edward B. Burger Assistant Professor of Mathematics

Rocky Mountain J. of Math, 26, 875-888, (1996)

Here we give new, sharp upper bounds for the size of solutions to systems of linear congruences over number fields. Such theorems generalize and improve results of Aubry, Thue, Brauer, Reynolds, and Cochrane. Fundamental to our method is an analysis of ideals through the geometry of numbers of the adele space.

Uniformly Approximable Numbers and the Uniform Approximation Spectrum

Edward B. Burger Assistant Professor of Mathematics

J. of Number Theory, 61, 194-208, (1996)

We say a real number a is uniformly approximable if the upper bound in Dirichlet's theorem, from diophantine approximation, of 1/(Q+1)q may be sharpened to c(a)/(Q+1)2 for all sufficiently large Q. We begin by showing that the set of uniformly approximable numbers is precisely the set of badly approximable numbers. In addition, the optimal lower bound of c(a), referred to as the uniform approximation constant, is explicitly given. This allows us to introduce the notion of a uniform approximation spectrum. We conclude with a determination of the smallest values of this new spectrum and a comparison of this spectrum with other spectra.

On the Structure of Quadratic Irrationals Associated with Generalized Fibonacci and Lucas Numbers

Edward B. Burger Assistant Professor of Mathematics
Christopher Kollett '95

Fibonacci Quarterly, 34, 200-212, (1996)

Here we determine the explicit structure for irrational numbers of the form (Fn(a)/Lm(a))fa, where Fn and Lm are generalized Fibonacci and Lucas numbers, respectively, and fa is the generalized golden ratio. This work extends and generalizes previous results of Long and Jordan, and also resolves two open questions regarding the structure of the periods for such a class of numbers.

Does (sum of)1/n! Really Converge? Infinite Series and p-adic

Edward B. Burger Assistant Professor of Mathematics
Thomas Struppeck

Amer. Math. Monthly, 103, 565-577, (1996)

Here we begin with an entertaining introduction to p-adic analysis and then consider infinite series which converge simultaneously at all spaces of Q. This result answers a question posed by Koblitz. We also prove a transcendence result.

On Liouville Decompositions in Local Fields

Edward B. Burger Assistant Professor of Mathematics

Proc. of the Amer. Math. Soc., 124, 3305-3310, (1996)

In 1962 Erdos proved that every real number may be decomposed into a sum of Liouville numbers. Here we consider more general functions which decompose elements from an arbitrary local field into Liouville numbers. Several examples and applications are given. As an illustration, we prove that for any real numbers a₁, a₂, ... , a_N, not equal to 0 or 1, there exist uncountably many Liouville numbers L such that a^L₁ , a^L₂ , ..., a^L_N are all Liouville numbers.

Inhomogeneous Inequalities Over Number Fields

Edward B. Burger Assistant Professor of Mathematics

Illinois Journal of Mathematics, 38: 452-470 (1994)

Here we produce a number field analogue of a theorem due to Khintchine regarding transference between homogeneous and inhomogeneous diophantine approximation. We also produce, in this general setting, a quantitative version of a theorem of Kronecker on inhomogeneous approximation. A qualitative number field analogue of Kronecker's result was first given bCantor. Our results also provide inhomogeneous analogues of results of Mahler. We examine these issues in the context of the ring of S-integers of a number field. Basic to our method are some new inequalities from the geometry of numbers over the associated adéle ring. In particular, new bounds are given for the inhomogeneous minimum.

Transcendence and Irrationality Measures

Edward B. Burger Assistant Professor of Mathematics

Proceedings of the Seminar on the Theory of Numbers, Caen, France (1994)

Let S be a nonempty set of nonnegative integers and e > 0. Here we prove the existence of U2-numbers a such that for each n Œ S there exist infinitely many distinct rationals p/q so that |(a+n)² - p/q| << q^{-q(1+e)log q} and for each nonnegative integer m oe S and rational p/q with q sufficiently large, q^{-q(1+e)log q} << |(a+m)² - p/q|, where the implied constants only depend upon e. Thus for n Œ S we have that (a+n)² is Liouville and for m oe S we have effective measures of irrationality for (a+m)².

Sur Les Quotients Partiels de U-Numbres Dans un Corps de So(',e)ries Formelles

Edward B. Burger Assistant Professor of Mathematics

Comptes Rendus de L'Aco(',a)do(',e)mie des Sciences Paris (So(',e)rie I), 319: 421-426 (1994)

Nous généralisons des résultats du premier auteur et T. Struppeck à un corps de séries de Laurent formelles. Soient A et B, deux sous ensembles de K[z]. Nous démontrons l'existence d'une infinité non dénombrable de U2-nombres (selon la classification de Mahler dans ce contexte), a dans K((1/z)), tels que les quotients partiels du développement en fraction continue sont tous dans A, tandis que (a + b)² si car(K) != 2, ou (a +b) - 1/(a +b) si car(K) = 2 sont des nombres de Liouville pour chaque b dans B. De plus, nous contrôlons la distribution asymptotique des quotients partiels de ces nombres a.

Minimal Edge Piecewise Linear Knots

Jorge Calvo, Visiting Assistant Professor of Mathematics
Kenneth C. Millett, University of California, Santa Barbara

Ideal Knots, (Andrzej Stasiak, Vaevolod Katrich, and Louis Kauffman, eds.)
Series on Knots and Everything, 19, World Scientific, Singapore, 107-128,(1999)

Continuous Foldability of Polygonal Paper

Satyan Devadoss, Assistant Professor of Mathematics

Erik Demaine, Joe Mitchell and Joe O'Rourke

Proceedings of the 16th CCCG Conference, 64-67 (2004)

We prove that any given well-behaved folded state of a piece of paper can be reached via a continuous folding process starting from the unfolded paper and ending with the folded state. The argument is an extension of that originally presented in Demaine-Mitchell.

Combinatorial Equivalence of Real Moduli Spaces

Satyan Devadoss,Assistant Professor of Mathematics

Notices of the American Mathematical Society, 51, 620 - 628 (2004)

A well-known construction of associahedra comes from truncations of simplices. Motivated by compactifications of point configurations, we show associahedra as truncations of certain products of simplices. This is then used to provide a combinatorial construction of the real moduli space of spheres relating it to blow-ups of the braid hyperplane arrangement.

A Space of Cyclohedra

Satyan Devadoss, Assistant Professor of Mathematics

Intro Stats

Richard D. De Veaux, Professor of Statistics

with Paul Velleman and David Bock

Stats: Data and Models

Richard D. De Veaux, Professor of Statistics

with Paul Velleman and David Bock

Stats: Modeling the World

Richard D. Deveaux, Professor of Statistics

with David Bock and Paul Velleman

How to Lie with Bad Data

Richard D. Deveaux, Professor of Statistics

with D.J. Hand

Intro Stats

Richard D. Deveaux, Associate Professor of Statistics

Paul Velleman and David Bock

Bagging and Boosting

Richard D. Deveaux, Associate Professor of Statistics

Resizing Triathlons for Fairness

Richard D. Deveaux Associate Professor of Statistics

with H. Wainer

J.H. Albert, J. Bennett, J.J. Cochran, Editors

Reducing Junk Mail Using Data Mining Techniques

Richard D. Deveaux, Associate Professor of Statistics

with Herb Edelstein

R. Peck, Editor

Comparison of Tree Based Methods on Mammography Data

Richard D. Deveaux, Associate Professor of Statistics

with Thu Hoang Tu Boa Ho, David Cheung, Huan Liu, Editors

Stats: Data and Models

Richard D. De Veaux, Professor of Mathematics

Dave Bock and Paul Velleman

Do Purported Over-The-Counter Memory Enhancing Agents Improve Memory: A Double-Blind, Placebo-Controlled Trial of Ginkgo Biloba

Richard D. De Veaux, Professor of Mathematics

Paul R. Solomon, Professor of Psychology

Felicity Adams '93, Amanda Silver '96, Jill Zimmer '96

Data Mining: A View from Down in the Pit

Richard D. De Veaux, Professor of Mathematics

A personal perspective on data mining.

Comments on Data Mining et Statistiques by Besse, LeGall, Rimbaud et Sarpy

Richard D. De Veaux, Professor of Mathematics

Discussion of the scope of Data Mining and how it differs from Statistics.

Curriculum Guidelines for Bachelor of Arts Degrees in Statistical Science

Richard D. De Veaux, Professor of Mathematics

C. Acuna, T. Tharpey, G. Cobb

Results from the NSF workshop on curriculum for a major in Statistics.

Ginkgo for Memory Enhancement A Randomized Controlled Trial

Richard D. De Veaux, Professor of Mathematics

Paul R. Solomon, Felicity Adams, Amanda Silver, Jill Zimmer

A paper showing no memory improvement for healthy participants taking recommended doses of Ginkgo Biloba.

Data Mining et Statistique: Discussion de l'article de Besse et al

Richard D. De Veaux, Professor of Mathematics

Journal de la Societe Francaise de Statistique, 142, no. 1 (2001)

Data Mining: A View from Down in the Pit

Richard D. De Veaux, Professor of Mathematics

Stats Magazine, 3-9 (2002)

A Statistical Basis for Using Radial Basis Functions for Process Control

Richard D. De Veaux, Assistant Professor of Mathematics
Lyle H. Ungar

American Control Conference Proceedings, (1995)

Radial Basis Function (RBF) neural networks offer an attractive equation form for use in model-based control because they can approximate highly nonlinear plants and yet are well suited for linear adaptive control. We show how interpreting RBFs as mixtures of Gaussians allows the application of many statistical tools including the EM algorithm for parameter estimation. The resulting EMRBF models give uncertainty estimates and warn when they are extrapolating beyond the region where training data was available.

Multicollinearity: A Tale of Two Non-parametric Regressions

Richard D. De Veaux, Assistant Professor of Mathematics
Lyle H. Ungar

Selecting Models from Data: AI and Statistics IV, 293-302 (1994)

The most popular form of artificial neural network, feedforward networks with sigmoidal activation functions, and a new statistical technique, multivariate adaptive regression splines (MARS) can both be classified as nonlinear, nonparametric function estimation techniques, and both show great promise for fitting general nonlinear multivariate functions. In comparing the two methods on a variety of test problems, we find that MARS is in many cases both more accurate and much faster than neural networks. In addition, MARS is interpretable due to the choice of basic functions which make up the final predictive equation. This suggests that MARS could be used on many of the applications where neural networks are currently being used.

However, MARS exhibits problems in choosing among predictor variables when multicollinearity is present. Due to their redundant architecture, neural networks, however, do not share this problem, and are better able to predict in this situation. To improve the ability of MARS to deal with multicollinearity, we first use principal components to reduce the dimensionality of the input variables before invoking MARS. Using data from a polymer production run, we find that the resulting model retains the interpretability and improves the accuracy of MARS in the multicollinear setting.

Radial Basis Function Neural Networks for Process Control

Richard D. De Veaux, Assistant Professor of Mathematics
Lyle H. Ungar
Tom Johnson

CIMPRO Proceeding, 357-364 (1994)

Radial basis function (RBFs) neural networks provide an attractive method for high dimensional nonparametric estimation for use in nonlinear control. They are faster to train than conventional feedforward networks with sigmoidal activation networks ("backpropagation nets"), and provide a model structure better suited for adaptive control. This article gives a brief survey of the use of RBFs and then introduces a new statistical interpretation of radical basis functions and a new method of estimating the parameters, using the EM algorithm. This new statistical interpretation allows us to provide confidence limits on predictions made using the networks.

Statistical Approaches to Fault Analysis in Multivariate Process Control

Richard D. De Veaux, Assistant Professor of Mathematics
Lyle H. Ungar
Johnathon M. Vinson

American Control Conference Proceedings, (1994)

After a brief review of some statistical approaches to multivariate process control, we present a technique for determining root causes when information is available on likely out of control scenarios or fault types. We utilize linear dimension reduction techniques such as principal component analysis or partial least squares to limit the number of latent variables to study. While using historical in control data is important in establishing control means and limits, these data often have less structure for dimension reduction than do data which come from known fault types. If these latter dast are available, the expanded data set can be analyzed for dimension reduction, using the in control data to set limits in the reduced set. When a sequence of points is then seen to be beyond the control limits, the distance to the nearest known fault type is measured. If the dimensions can be reduced to two, these can be plotted as well. The new problem is classified into one of the existing fault types when its distance to it becomes smaller than a pre-specified criterion. If it remains out of control, but fails to approach an existing fault type, a new fault paradigm is created. Our approach is demonstrated on a simulated chemical process.

A Dual Approach to Defining the Triangle Sequence (a Type of Multidimensional Continued Fraction Algorithm, Initially Developed in A Dual Approach to Triangle Sequences: A Multidimensional Continued Fraction Algorithm

Thomas Garrity, Professor of Mathematics

with S. Assaf, L. Chen, T. Cheslack-Postava '00, B. Cooper '01, A. Diesl, M. Lepinski and A. Schuyler

Recommended Resources in Algebraic and Differential Geometry

Thomas Garrity, Professor of Mathematics

Kristine Fowler, Editor

Review of John Adams Mathematics in Nature

Thomas Garrity, Professor of Mathematics

On Relations of Invariants for Vector-Valued Forms

Thomas Garrity, Professor of Mathematics

Zachary Grossman

Electronic Journal of Linear Algebra, 11, 22 - 40 (2004)

An algorithm is given for computing explicit formulas for the generators of relations among the invariant rational functions for vector-valued bilinear forms. These formulas have applications in the geometry of Riemannian submanifolds and in CR geometry.

A Two-Dimensional Minkowski ?(x) Function

Thomas Garrity, Professor of Mathematics

Olga R. Beaver, Professor of Mathematics

Journal of Number Theory,107, 105 - 134 (2004)

A function from a triangle to itself is defined that has both interesting number theoretic and analytic properties. This function is shown to be a natural generalization of the classical Minkowski ?(x) function. It is shown there exists a natural class of pairs of cubic irrational numbers in the same cubic number field that are mapped to pairs of rational numbers, in analog to ?(x) mapping quadratic irrationals on the unit interval to rational numbers on the unit interval. It is also shown that this new function satisfies an analog to the fact that ?(x), while increasing and continuous, has derivative zero almost everywhere.

All the Mathematics you Missed [But Need to Know for Graduate School]

Thomas Garrity, Professor of Mathematics

Cambridge University Press (2001)

Here we consider a question raised by Mordell regarding lattice point-free parallelepipeds and find spectrum of optimal regions.

On Periodic Sequences for Algebraic Numbers

Thomas Garrity, Professor of Mathematics

J. of Number Theory, 88, no. 1, 86-103 (2001)

For each positive integer n > 1, a new approach to expressing real numbers as sequences of nonnegative integers is given. The N = 2 case is equivalent to the standard continued fraction algorithm. For n = 3, it reduces to a new iteration of the trian gle. Cubic irrationals that are roots of x3 + k x2 + x - 1 are shown to be precisely those numbers with purely periodic expansions of period length one. For general positive integers n, it reduces to a new iteration of an n dimensional simplex.

Global Structures on CR Manifolds via Nash Blow-ups

Thomas Garrity, Professor of Mathematics

Michigan Math. J., 48, 281-294 (2000)

A generic compact real codimension two submanifold X of complex (n+2) space will have a CR structure at all but a finite number of points (failing at the complex jump points J). The main theorem of this paper gives a method of extending the CR structure on the non-jump points X-J to the jump points. We examine a Gauss map from X-J to an appropriate flag manifold F and take the closure of the graph of this map in X x F. This is a version of a Nash blow-up. We give a clean criterion for when this closure is a smooth manifold and see that the local differential properties at the points X-J can now be naturally extended to this new smooth manifold, allowing global techniques from differential geometry to be applied to compact CR manifolds. As an example, we find topological obstructions for the manifold to be Levi nondegenerate.

The Equivalence Problem for Higher-Codimensional CR Structures

Thomas Garrity, Associate Professor of Mathematics
Robert Mizner

Pacific Journal of Mathematics, 177, 211-235, (1997)

The equivalence problem for CR structures can be viewed as a special case of the equivalence problem for G-structure. This paper uses Cartan's methods (in modernized form) to show that a CR manifold of codimension 3 or greater with suitably generic Levi form admits a canonical affine connection, and consequently that the automorphisms of the CR manifold constitute a Lie group.

The most difficult technical step is to construct a smooth moduli space for generic vector-valued hermitian forms, which is tied to the CR manifold via the Levi map. The techniques used to construct this space are drawn from the classical invariant theory of complex projective hypersurfaces.

Vector-Valued Forms and CR Geometry

Thomas Garrity, Associate Professor of Mathematics
R. Mizner

CR-Geometry and Determined Systems, Advanced Studies in Pure Mathematics, 25, 110-121, (1997)

Vector-valued forms arise in the study of various higher codimensional geometries. This note gives an overview of how the invariant theory of the Levi form (a vector-valued form) can be used to understand higher codimensional CR-structures.

Invariants of Vector-Valued Bilinear and Sesquilinear Forms

Thomas Garrity, Associate Professor of Mathematics
Robert Mizner, Assistant Professor of Mathematics

Linear Algebra and Its Applications, 218: 225-237 (1995)

First steps in the algebraic invariant theory of vector-valued bilinear and sesquilinear forms are made. In particular, explicit formulas for generators of all invariant rational functions for such forms are derived. These formulas, and certain analogues, have applications to the geometry of Riemannian submanifolds, distributions, and CR structures.

President Garfield and the Pythagorean Theorem

Victor E. Hill IV, Thomas T. Read Professor of Mathematics

Math Horizons,, 9-11, 15 (February 2002)

When James A. Garfield was in the House of Representatives in 1876, he produced a new proof of the Pythagorean Theorem on right triangles. This article is concerned with his background and interests in mathematics as well as his experiences as a student at Williams. 1854-56.

Groups and Characters

Victor E. Hill IV, Thomas T. Read Professor of Mathematics

Chapman & Hall Publishers, (November 1999)

Group representation and character theory is both elegant and practical, with important applications to quantum mechanics, spectroscopy, crystallography, and other fields in the physical sciences. This text offers an easy-to-follow introduction to the theory of groups and of group characters. Group theory is covered through the Sylow Theorems and the full subgroup structure of A5. Numerous character tables are worked out in full, along with a presentation of real and induced characters that lead to the table for S5.

Efficacy and Safety of Galantamine in Patients with Dementia with Lewy Bodies: A 24-Week Open-Label Study

Stewart Johnson, B Professor of Mathematics

with K. Edwards, D. Royall, L. Hershey, D. Lichter, A. Hake, M. Farlow and F. Pasquier

Optimality of Stasis and Small Switching Cycles in Planar Systems with Two-Valued Controls

Stewart Johnson, Professor of Mathematics

Efficacy and Safety of Galantamine in Patients with Dementia with Lewy Bodies: A 12-Week Interim Analysis

Stewart Johnson,Professor of Mathematics

K. Edwards, L. Hershey, L. Wray, EM. Bednarczyk, D. Lichter, M. Farlow

Dementia and Geriatric Cognitive Disorders, 17 (2004)

Observations on the neurochemistry of dementia with Lewy bodies (DLB) have suggested that cholinesterase inhibitors (ChEIs) might be beneficial in treating some clinical symptoms of DLB. A 24-week, multicenter open-label study was designed to assess the safety and efficacy of the ChEI galantamine in patients with DLB, and an interim analysis of results was performed at 12 weeks. Efficacy analyses were performed on data from 25 patients. Scores on the Neuropsychiatric Inventory (NPI-12) improved (decreased) by 7.52 points over the 12 weeks (marginally significant, p = 0.061). NPI-12 scores decreased by half in 12 of the 25 patients. Highly significant improvement was observed in scores on the NPI-4 subscale (delusions, hallucinations, apathy, and depression: p = 0.003). Scores on the Clinician's Global Impression of Change (CGIC) improved by 0.95 points (significant, p = 0.02). Improvements also were found in secondary efficacy variables, including cognitive, functional, activities of daily living, sleep and confusion assessments. Motor scores, as measured by the UPDRS motor subscale, showed mild improvement, which demonstrates that galantamine has no adverse effect on parkinsonian symptoms. Adverse events generally were transient and of mild-to-moderate intensity. Two of the 25 patients discontinued galantamine because of nausea and anorexia. One serious adverse event was recorded, but it was judged to be unrelated to the study medication.

The Safety and Efficacy of Galantamine in Patients with Dementia with Lewy Bodies: an Interim Analysis

Stewart Johnson, Professor of Mathematics

K. Edwards, L. Hershey, D. Lichter, E. Bednarczyk

Annual Meeting of the American College of Neuropsychiatrists (2003) Poster session at the 2003 annual meeting of the American College of Neuropsychiatrists.

Simple Hybrid Systems

Stewart Johnson, Assistant Professor of Mathematics

International Journal of Bifurcations and Chaos, Vol. 4, No. 6, 1655-1665 (1994)

A hybrid system is one which can instantaneously switch between a number of phase portraits. Switching occurs when trajectories hit prescribed switching curves. This type of system is highly applicable to digital controls such as robotic controls. A working definition of such a system is given, a method for stabilizing periodic orbits is given, and several examples are explored.

Testing Marginal Homogeneity for Multivariate Ordinal Data

Bernhard Klingenberg, Assistant Professor of Statistics

with B. Solari, A. Salmaso, and F. Pesarin

Many assessment instruments used in the evaluation of toxicity, safety, pain or disease progression consider multiple clinical endpoints to fully capture the presence and severity of treatment effects of medical products. This papers develops methods for a valid statistical analysis for these endpoints.

Regression Models for Long Binary Time Series

Bernhard Klingenberg, Assistant Professor of Statistics

Time series of binary variables present unique statistical challenges due to serial correlation and uneven sampling intervals, which we tackle with autocorrelated random effects in generalized linear mixed models. We illustrate our approach and model assessment tools with an analysis of the series of winners in the traditional boat race between the universities of Oxford and Cambridge, re-evaluating a long-held belief about the effect of the weight of the crew on the odds of winning.

Multivariate Extensions of NcNemar's Test

Bernhard Klingenberg, Assistant Professor of Statistics

with Agresti, A.

Several generalizations of the univariate McNemar statistic to test marginal homogeneity of success rates in dependent samples are explored.

Multivariate Tests Comparing Two Vectors of Binomial Probabilities, with Application to Safety Studies for Drugs

Bernhard Klingenberg, Assistant Professor of Statistics

Testing marginal homogeneity of success rates in two independent samples across several endpoints simultaneously is explored in this article.

Controlling Formal Fibers of Principal Prime Ideals

Susan Loepp, Professor of Mathematics

with A. Dundon, D. Jensen, J. Provine, and J. Rodu

In this paper, we show that given a complete local ring T and a finite set of prime ideals C satisfying certain conditions, there exists a local domain A whose completion is T and such that A contains a principal prime ideal whose maximal elements in its formal fiber are exactly C. We also show that in some special cases, A is an excellent ring.

Chains of Rings with Local Formal Fibers

Susan Loepp, Professor of Mathematics

with K. Chen and J. Provine

Susan Loepp, Associate Professor of Mathematics

Semilocal Generic Formal Fibers

Susan Loepp, Associate Professor of Mathematics

Philippa Charters '03

Characterization of Completions of Excellent Domains of Characteristic Zero

Susan Loepp, Associate Professor of Mathematics

In this paper, we show that a complete local ring T containing the integers is the completion of a local excellent integral domain if and only if it is reduced, equidimensional, and no integer of T is a zero divisor.

Generic Formal Fibers of Polynomial Ring Extensions

Susan Loepp,Associate Professor of Mathematics

C. Rotthaus

Let R be a local integral domain with maximal ideal M and S the domain R[X] localized at the ideal (M,X) where X is an indeterminate. In this paper, we explore the relationship between the dimension of the generic formal fiber of R and the dimension of the generic formal fiber of the domain S. Specifically, we show that if R is a universally catenary local domain such that the dimension of the generic formal fiber of S is dimR, then the dimension of the generic formal fiber of R is dimR Ð 1. We also provide counter-examples showing that the converse does not hold.

On the Completeness of Factor Rings

Susan Loepp, Associate Professor of Mathematics

It was previously unknown whether or not there existed an integral domain A such that A/I is complete for all nonzero ideals I, but the dimension of the generic formal fiber of A is not zero. In this paper, we construct such integral domains that are not only excellent, but we also show that the generic formal fiber of A can be forced to be local and conclude that the dimension of the generic formal fiber can be controlled.

Some Results on Tight Closure and Completion

Susan Loepp, Associate Professor of Mathematics

C. Rotthaus

In this paper, we construct examples of nonexcellent local domains for which tight closure and completion do not commute. In addition, we construct an example of a complete local normal Gorenstein domain which is not F-regular but is the completion of an F-regular local ring.

Generic Formal Fibers of Polynomial Rings

Susan Loepp, Associate Professor of Mathematics

Journal of Pure and Applied Algebra, 163, 93-106 (2001)

In this paper, we construct an excellent regular local ring A whose generic formal fiber is local and we show that the generic formal fiber of the polynomial ring over A in finitely many variables can also be controlled.

Some Results on Tight Closure and Completion

Susan Loepp, Associate Professor of Mathematics

C. Rotthaus

Journal of Algebra, 246, 859-880 (2001)

In this paper, we construct examples of nonexcellent local domains for which tight closure and completion do not commmute. In addition, we construct an example of a complete local normal Gorestein domain which is not F-regular but is the completion of an F-regular local ring.

Formal Fibers at Height One Prime Ideals

Susan Loepp, Assistant Professor of Mathematics

Journal of Pure and Applied Algebra, 148, 191-207 (2000)

In this paper, we construct examples of local unique factorization domains so that not only can the dimension of the generic formal fibers of these domains be controlled, but also the dimension of the formal fibers for infinitely many height one prime ideals.

Excellent Rings with Local Generic Formal Fibers

Susan Loepp, Assistant Professor of Mathematics

Journal of Algebra, 201, 573-583, (1998)

Until recently it was thought that excellent rings cannot have local generic formal fibers. In this paper, however, we construct a class of excellent rings that in fact do possess this property. In addition, we show that the dimension of the generic formal fiber ring can be controlled.

Constructing Local Generic Formal Fibers

Susan Loepp, Assistant Professor of Mathematics

Journal of Algebra, 187, 16-38, (1997)

Until recently, it was not known whether rings with local generic formal fiber rings exist. In this paper, we answer this question by constructing rings possessing local generic formal fiber rings. In fact, our construction shows that all possible dimensions of generic formal fiber rings are realized. Moreover, we show that for a complete local unique factorization domain, T, and a chain of prime ideals p₀ Ì p₁ Ì ... pn of T (satisfying minor conditions), it is possible to construct a corresponding chain of unique factorization domains A_n Ì A_n-1 Ì ... Ì A₀ all with completion T such that the generic formal fiber ring of A_i is local with maximal ideal p_i.

Regularized Semiclassical Radial Propagator for the Coulomb Potential

Robert S. Manning, Visiting Assistant Professor of Mathematics
Gregory S. Ezra

Phys. Rev. A, 50: 954 (1994)

We derive a regularized semiclassical radial propagator for the Coulomb potential, a case for which standard approaches run into well-known difficulties associated with a non-Cartesian radial coordinate and a potential singularity. Following Kleinert [Path Integrals in Quantum Mechanics, Statistics and Polymer Physics (World Scientific, Singapore, 1990)], we first perform a quantum-mechanical regularization of the propagator. The semiclassical limit is then obtained by stationary phase approximation of the resulting integrals. The semiclassical propagator so derived has the standard Van Vleck-Gutzwiller form for the radial Coulomb problem with a potential correction (Langer modification) term included. The regularized semiclassical propagator is applied to compute the autocorrelation function for a Gaussian Rydberg wave packet.

Invariants of Vector-Valued Bilinear and Sesquilinear Forms

Robert Mizner, Assistant Professor of Mathematics
Thomas Garrity, Associate Professor of Mathematics

Linear Algebra and Its Applications, 218: 225-237 (1995)

First steps in the algebraic invariant theory of vector-valued bilinear and sesquilinear forms are made. In particular, explicit formulas for generators of all invariant rational functions for such forms are derived. These formulas, and certain analogues, have applications to the geometry of Riemannian submanifolds, distributions, and CR structures.

Rotation in a Normed Plane

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

with Jack Cook and Jonathan Lovett

Given a norm on a plane, we show that if you can isometrically rotate a generic "irrational" unit rhombus along with its diagonals, then the norm is Euclidean (up to linear equivalence).

Soap Bubble Cluster Pressures

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

with Manuel A. Fortes and M. Fatima Vaz

We provide theoretical estimates and Surface Evolver experiments on the pressures of bubbles in planar clusters.

Soap Bubble Clusters

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

Although soap bubble clusters and froths provide simple models of diverse physical phenomena, the underlying mathematics is deep and still not understood.

Review of Riemannian Geometry: A Modern Introduction by Isaac Chavel

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

with Cesar Silva. The SMALL Program at Williams College. Proc. Conf. Promoting Und. Res. Math. (Joseph A. Gallian, ed.), Amer. Math. Soc., 2007.

When Soap Bubbles Collide

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

with Colin Adams and John M. Sullivan

In Polytopes, Small Balls About Some Vertex Minimize Perimeter

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

On the Isoperimetric Problem in Euclidean Space with Density

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

with Cesarr Rosales, Vincent Bayle, and Antonio Canete–et

Problem 852 on "Eigenvalues of a Sum"

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

http://front.math.ucdavis.edu/math.SG/edu/0504537.

Geometric Measure Theory: a Beginner's Guide

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

Academic Press, Russian Edition (2006)

A Russian translation and update, marking the fifth edition of the advanced graduate text.

Isoperimetric Estimates in Products

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

Ann. Global Anal. Geom., 110, 73-79 (2006)

In a product M₁xM₂ of Riemannian manifolds, least perimeter required to enclose given volume among general regions is at least 1/2 times that among regions of product form, assuming that the isoperimetric profiles of M₁ and M₂ are concave.

Myers' Theorem with Density

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

Kodai Math. J., 29, 454-460 (2006)

We provide generalizations of theorems of Myers and others to Riemannian manifolds with density.

Review of Singular Sets of Mininizers for the Mumford-Shah Functional by Guy David

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

SIAM Review, 48, 187-189 (2006)

Geometry Lessons, Interview with Jeffrey Hildner

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

The Christian Sci. J., 124, 52-55 (2006)

Mathematical principles governing the shape of soap bubbles provide an analogy for God as divine Principle governing the universe.

Real Analysis

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

American Mathematical Society, (2005)

Based on a one-semester core real analysis course at Williams.

Real Analysis and Applications (Including Fourier Series and the Calculus of Variations

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

American Mathematical Society, (2005)

Streamlined, complete theory, plus applications in Fourier series and the calulus of variations, including physics (least action and Lagrange's equations), economics (optimal production and maximal utility), Riemannian geometry, and general relativity.

Clusters with Multiplicities in R2

Frank Morgan, Webster Atwell Class of 1921 Professor of Matheamtics

Pacific J. of Math, 221, 123-146 (2005)

Perimeter-minimizing planar double soap bubbles in which regions are allowed to overlap with multiplicities meet in fours, fives, and sixes as well as threes. We further provide certain generalizations to immiscible fluids and higher dimensions, and an associated theory of calibrations. We work in the category of flat chains with coefficients in a normed group.

Hexagonal Surfaces of Kapouleas

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

Pacific J. of Math, 220, 379-387 (2005)

For the "hexagonal norm on R3, for which the isoperimetric shape is a hexagonal prism rather than a round ball, we give analogs of the compact immersed constant-mean-curvature surfaces of Kapouleas.

Regularity of Area-Minimizing Surfaces in 3D Polytopes and of Invariant Surfaces in Rn

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

J. Geom. Anal., preprint at arvix.org, 321-341 (2005)

In (the surface of) a convex polytope P3 in R4, an area-minimizing surface avoids the vertices of P and crosses the edges orthogonally.

In a smooth Riemannian manifold M with a group of isometries G, an area-minimizing G-invariant oriented hypersurface is smooth (except for a very small singular set in high dimensions). Already in 3D, area-minimizing G-invariant unoriented surfaces can have certain singularities, such as three orthogonal sheets meeting at a point. We also discuss flat chains modulo and soap films.

Manifolds with Density

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

Notices Amer. Math Soc., 52 853-858 (2005)

We discuss the category of Riemannian manifolds with density and present easy generalizations of the volume estimate of Heinze and Karcher and thence of the isoperimetric inequality of Levy and Gromov.

Geometric Measure Theory and the Proof of the Double Bubble Conjecture

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

Amer. Math Soc., (2005)

Global Theory of Minimal Surfaces (Proc. Clay Inst., 2001 Summer School, MSRI)

Notes by Ritore based on Morgan's course at MSRI.

Teaching Tips

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

Amer. Math Soc., (2005)

Easy ways to be a better teacher.

A Round Ball Uniquely Minimizes Gravitational Potential Energy

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

Amer. Math. Soc., 133, 2733-2735 (2005)

We prove that among measurable bodies in R3 of mass m0 and density at most 1, a round ball of unit density uniquely minimizes gravitational potential energy.

Streams of Cylindrical Water

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

Math. Intelligencer, 26, 70-72 (2004)

Just as isotropic surface energy produces round water droplets and unstable undulating streams, crystalline energy produces cylindrical droplets and crystalline unduloids.Cylindrical Surfaces of Delaunay

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

Rend. Circ. Mat. Palermo, 53, 469-477 (2004)

For the cylindrical norm on R3, for which the isoperimetric shape is a cylinder rather than a round ball, there are analogs of the classical Delaunay surfaces of revolution of constant mean curvature.

Planar Wulff Shape is Unique Equilibrium

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

Proc. Amer. Math. Soc., 133, 809-813 (2005)

In R2, for any norm, an immersed closed rectifiable curve in equilibrium for fixed area must be the Wulff shape (possibly with multiplicity).

Planar Clusters and Perimeter Bounds

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

Aladar Heppes

Phil. Mag., 85, 1333-1345 (2005)

We provide upper and lower bounds on the least-perimeter way to enclose and separate n regions of equal area in the plane. Along the way, inside the hexagonal honeycomb, we provide minimizers for each n.

A Note on Cross-Profile Morphology for Glacial Valleys

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

Short Communications, Earth Surface Processes and Landforms, 30, 513-514 (2005)

We provide an improvement to the Hirano-Aniya catenary model for the cross-profile morphology of a glacial valley.

Proof of the Double Bubble Conjecture

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

Robert Hardt, Editor

Six Themes on Variation, Amer. Math. Soc., 59-77 (2004)

Reprint of a American Mathematical Monthly article on our 2002 proof of the Double Bubble Conjecture.

Fractals and Geometric Measure Theory: Friends and Foes

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

Michael L. Lapidus and Michiel van Frankenhuijsen, Editors

Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Proc. Symp. Pure Mathematics, 72, 93-96 (2004)

Mandelbrot's fractals, like good friends, inspire more general and realistic geometries. But later, like foes, they thwart efforts to prove that solutions to geometric problems are well behaved.

Kepler's Conjecture and Hales's Proof

Frank Morgan, Webster Atwell Class of 1921 Professor of Mathematics

Notices Amer. Math. Soc., 52, 44-47 (January, 2005)

A review of G. Szpiro's book on Kepler's Conjecture and a discussion of Hales's recent proof.

Regularity of Isoperimetric Hypersurfaces in Riemannian Manifolds

Frank Morgan,Dennis Meenan ('54) Centennial Professor of Mathematics

Trans. AMS, 355, 5041 - 5052 (2003)

We add to the literature the well-known fact that an isoperimetric hypersurface S of dimension at most six in a smooth Riemannian manifold M is a smooth submanifold. If M is merely C1,1, then S is still C1,1/2.

Periodicity of Pressures in Periodic Foams

Frank Morgan,Dennis Meenan ('54) Centennial Professor of Mathematics

D. Weaire, N. Kern, S.J. Cox, J.M. Sullivan

Proc. Roy. Soc. London A, 460, 569 - 579 (2004)

We show that the periodic foams in equilibrium have periodic pressures. Also we show that a planar equilibrium foam with congruent bubbles must be a fully periodic arrangement of hexagons.

Soap on a Hope

Frank Morgan,Dennis Meenan ('54) Centennial Professor of Mathematics

The Last Word, New Scientist, 57 (January 17-23, 2004)

Illustrated response to a question about the existence of torus bubbles.

Area-Minimizing Surfaces in Cones

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

Comm. Anal. Geom., 10, 971-983 (2002)

We show that a k-dimensional area-minimizing surface can pass through an acute conical singularity if and only if k ³ 3. The larger k, the more acute the conical singularity can be.

Instabilities of Cylindrical Bubble Clusters

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

Kenneth A. Brakke

Eur. Phys. J.E, 9, 453-460 (2002)

We use the second variation formula to compute instabilities for certain cylindrical bubble clusters and compare to earlier simulations, experiments, and computations of Cox, Weaire, and Fortes.

Proof of the Double Bubble Conjecture

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

Michael Hutchings, Manuel Ritore, and Antonio Ros

Ann. Math., 155, 459-489 (2002)

We prove that the standard double bubbles provides the least-area way to enclose and separate two regions of prescribed volume in R3.

Isoperimetric Regions in Cones

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

Manuel Ritore

Trans. AMS, 354, 2327-2339 (2002)

We consider cones C over Mn and prove that if the Ricci curvature of M is at least n-1, then geodesic balls about the vertex minimize perimeter for given volume.

Math Chat

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

MAA Web Page, Semi Monthly (1998)

Column with questions, answers, and prizes. Available at the Mathematical Association of America web page at www.maa.org.

An Isoperimetric Comparison Theorem for Schwarzschild Space and Other Manifolds

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

Hubert Bray

Proc. AMS, 130, 1467-1472 (2002)

We give a very general isoperimetric comparison theorem, which implies for example that geodesic spheres in the Schwarzschild space minimize area for given volume, which in turn has applications to the Penrose Inequality in general relativity.

Hexagonal Economic Regions Solve the Location Problem

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

Amer. Math. Monthly, 109, 165-172 (2002)

We show in a certain mathematical sense that congruent regular hexagons solve the local problem, i.e. provide optimal market regions about centers of production.

Small Perimeter-Minimizing Double Bubboles in Compact Surfaces are Standard

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

Electronic Proceedings of the 78th Annual Meeting of the Louisiana/Mississippi Section of the MAA, Univ. of Miss. (2001)

We prove that in a smooth, compact, two-dimensional submanifold of Rn, the least-perimeter way to enclose and separate two regions of small prescribed areas is a standard double bubble, consisting of three constant-curvature curves meeting in threes at 120 degrees. This paper is largely superseded by the next one, which proves that small stable double bubbles are standard.

The Standard Double Bubble is the Unique Stable Double Bubble in R3

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

Wacharin Wichiramala

Proc. AMS, 130, 2745-2751 (2002)

We prove that the only equilibrium double bubble in R2 which is stable for fixed areas is the standard double bubble. This uniqueness result holds for small stable double bubbles in surfaces, where it is new even for perimeter-minimizing double bubbles.

The Perfect Shape for a Rotating Rigid Body

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

Mathematics Magazine, 75, 30-32 (2002)

The energy-minimizing shape for a rotating rigid body is not an oblate spheroid but a stationary ball with a small, distant planet.

Whad'Ya Know

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

Joseph Corneli, Paul Holt, Nicholas Leger, and Eric Schoenfeld

MAA Focus (2001)

A humorous account of the appearance of Morgan, his Geometry Group, and other mathematicians on the popular program on Public Radio International, in Madison, Wisconsin during the MathFest.

Proof of the Double Bubble Conjecture

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

The Amer. Math. Monthly, 108, 193-205 (March 2001)

An expository account of our proof that the standard double bubble provides the least-area way to enclose a separate two regions of prescribed volume in R3.

Riemannian Geometry: a Beginner's Guide

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

A.K. Peters, revised printing (2001)

Starting with an extrinsic approach to curvature, this book provides a short, intuitive, direct introduction to Riemannian geometry, including topics from general relativity, global geometry, and current research on norms more general than area. The second edition includes many new problems and new sections on the isoperimetric problem and on double Wulff crystals.

Calculus Lite

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

A.K. Peters, Third Edition, (2001)

This lean text covers single-variable calculus in under 300 pages by (1) getting right to the point, and stopping there, and (2) introducing some standard preliminary topics, such as trigonometry and limits, by using them in the calculus.

100-Year-Old Kelvin Conjecture Disproved by Weaire and Phelan

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

Math Horizons,(September 1999)

Popular account of surprising new way to partition space into unit volumes, upsetting the 100-year-old Kelvin conjecture.

Wulff Clusters in R2

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics
Christopher French '95, Scott Greenleaf

J. Geom. Anal. 7, 593-611, (1999)

The first existence and regularity results on the cheapest way to enclose and separate planar regions of prescribed areas, where cost is given by a general norm F, thus generalizing the Wulff shape for enclosing a single region.

The Hexagonal Honeycomb Conjecture

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

Trans. AMS 351, 1753-1763, (1999)

Contrary to popular belief, it remains conjectural that the planar hexogonal honeycomb provides the least-perimeter way to enclose and separate infinitely many regions of unit area. We prove existence for two formulations of the problem. Many questions remain.

The Isoperimetric Problem on Surfaces

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics
Hugh Howards '92, Michael Hutchings

Amer. Math. Monthly, 106, 430-439, (1999)

A survey of old and new results, including a proof that horizontal circles provide the least-perimeter way to enclose given area in a paraboloid of revolution.

Isoperimetric Curves on Hyperbolic Surfaces

Frank Morgan, Professor of Mathematics
Colin Adams, Chair and Mark Hopkins Professor of Mathematics

Proceedings of the American Mathematical Society, 127, 5, 1347-1356,(1999)

Least-perimeter enclosures of prescribed area on hyperbolic surfaces are characterized.

Instability of the Wet X Soap Film

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics
Kenneth Brakke

J. of Geom. Anal., 8, 749-767 (1998)

We show that adding slight thickness to a soap film shaped like an X leaves it unstable, although adding much thickness makes it stable. Analogous questions about other singularities remain controversial.

Recollections of Fred Almgren

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

J. Geom. Anal.,8, (1998)

Recollections of student and colleagues.

Math Chat

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

MAA Web Page,(1998-present)

Biweekly column with questions, answers, and prizes. Available at the MAA web page at www.maa.org.

Immiscible Fluid Clusters in R² and R³

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

Mich. Math. J., 45, 441-450, (1998)

We prove that an energy-minimizing planar cluster of immiscible fluids consists of finetely many circular arcs meeting at finitely many points, as long as the interfacial energies satisfy strict triangle inequalities. For R3, we generalize soap bubble cluster regularity.

Math Chat

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

MAA Web Page (www.maa.org) (first and third Thursday of each month,1998)

Biweekly column with questions, answers, and prizes.

Math Chat

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

MAA Web Page, Semi-Monthly, (1998)

Column with questions, answers, and prizes. Available at the Mathematical Association of America web page at www.maa.org.

Riemannian Geometry: a Beginner's Guide

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

A.K. Peters, (1998)

Starting with an extrinsic approach to curvature, this book provides a short, intuitive, direct introduction to Riemannian geometry, including topics from general relativity, global geometry, and current research on norms more general than area.

Lower semicontinuity of Energy of Clusters

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

Proc. Royal Soc., 127A, 819-822, (1997)

We discuss existence and lowersemicontinuity for clusters of materials minimizing an energy given by a collection of norms jij on the interfaces between regions Ri and Rj. Following Ambrosio and Braides, we exhibit a problem for which the triangle inequality holds but existence fails, and we state a new sufficient condition for lowersemicontinuity, which may be necessary.

Calculus Lite

Frank Morgan, Dennis Meenan ('54) Centennial Professor of Mathematics

AK Peters, Wellesley, second edition, (1997)

This lean text covers single-variable calculus in under 300 pages by (1) getting right to the point, and stopping there, and (2) introducing some standard preliminary topics, such as trigonometry and limits, by