41. Annals Of Mathematics, II. Series, Vol. 151, No. 2, Pp. 459-515, 2000 Review from Zentralblatt MATH The authors prove the ``double bubble conjecture''in $\bbfR^3$, in the particular case in which the two volumes are equal. http://www.emis.de/journals/Annals/151_2/3.html

Annals of Mathematics, II. Series Vol. 151, No. 2, pp. 459-515 (2000) Previous Article Next Article Contents of this Issue Other Issues ... EMIS Home Double bubbles minimize Joel Hass and Roger Schlafly Review from Zentralblatt MATH Zbl 0970.53009 )], see the review below. Reviewed by Robert Finn Keywords: double bubble conjecture; equal volumes; spherical cap Classification (MSC2000): Full text of the article: PDF file (526 kilobytes) Electronic fulltext finalized on: 27 Apr 2001. This page was last modified: 22 Jan 2002. Johns Hopkins University Press ELibM for the EMIS Electronic Edition Metadata extracted from Zentralblatt MATH with kind permission

42. Research Supervision Generalized the recent proof of the double bubble conjecture from R3 to R 4 and certain higher dimensional cases. Papers, Component http://www.williams.edu/Mathematics/fmorgan/student2.html

Research Supervision Ph.D. Students Julian Lander , MIT, 1984. Gave the first positive results in general codimension on when a minimizing surface inherits the symmetries of the boundary. Thesis, "Area-minimizing integral currents with boundaries invariant under polar actions," Trans. Amer. Math. Soc. 307 (1988), 419-429. Benny Cheng , MIT, 1987. Proved new families of very symmetric cones to be area-minimizing, such as the cone over the unitary matrices Un (n >= 4), by extending the theory of coflat calibrations. Thesis, "Area-minimizing equivariant cones and coflat calibrations," MIT. "Area minimizing cone type surfaces and coflat calibrations," Indiana U. Math. J. 37 (1988), 505-535. Gary Lawlor , Stanford, 1988. Proved the five-year-old angle conjecture on which pairs of m-planes are area-minimizing. Developed a curvature criterion for area minimization and classified all area-minimizing cones over products of spheres. Gave the first example of nonorientable area-minimizing cones. Thesis, "A sufficient criterion for a cone to be area-minimizing," Memoirs of the AMS 91, No. 464 (1991), 1-111. Mohamed Messaoudene , MIT, 1988. Analyzed the mass norm in the first nonclassical case

43. Published SMALL Papers Ben Reichardt, Cory Heilmann, Yvonne Lai, Anita Spielman (1999 Geometry Group), Proofof the double bubble conjecture in R4 and certain higher dimensional cases http://www.williams.edu/Mathematics/published.html

Published SMALL Papers C. Adams and W. Sherman, "Minimum Ideal Triangulations of Hyperbolic 3-Manifolds", Discrete Comput. Geom. 6:135-153 (1991). C. Adams, J. Brock, J. Bugbee, T. Comar, K. Faigin,A. Huston, A. Joseph, D. Pesikoff, "Almost Alternating Links", Topology and its Applications, 46 (1992) 151-165. C. Adams, B. Brennan, D. Greilsheimer, A. Woo, "Stick Numbers and Composition of Knots and Links", Journal of Knot Theory and its Ramifications, Vol. 6, No.2 (1997)149-161. C. Adams, K. Foley, J. Kravis, R. Dorman, S. Payne, "Alternating Graphs", Journal of Combinatorial Theory, Series B, v. 77, n. 1, September 1999, p. 96-120. C. Adams, T. Fleming, M. Levin, A. Turner, "Alternating Knots in S x I",Pacific Journal of Mathematics, Vol. 203, No. 1 (2002) 1-22. C. Adams, E. Furstenberg, J. Li, J. Schneider, Exploring Knots, Mathematics Teacher, Vol. 90, No. 8, Nov. 1997, 640-646, 652. C. Adams, C. Lefever*, J. Othmer*, S. Pahk*, A. Stier* and J. Tripp*, "An Introduction to the Supercrossing Index of Knots and the Crossing Map", Journal of Knot Theory and its Ramifications, Vol. 11, No. 3(2002) 445-459. C. Adams, A. Colestock*, J. Fowler*, W.D. Gillam*, E. Katerman*, "Clean Geodesics in Hyperbolic 3-Manifolds", to appear in Pacific Journal of Mathematics.

44. Mathematical Recreations A notorious case is the double bubble conjecture, which states that the shapeformed when two bubbles coalesce consists of three spherical surfaces. http://www.fortunecity.com/emachines/e11/86/bubble.html

web hosting domain names email addresses related sites ... Mathematical Recreations by Ian Stewart Double Bubble,Toil and Trouble The dodecahedron has 20 vertices, 30 edges and 12 faces- each with five sides. But what solid has 22.9 vertices, 34.14 edges and 13.39 faces -each with 5.103 sides? Some kind of elaborate fractal , perhaps? No, this solid is an ordinary, familiar shape, one that you can probably find in your own home. Look out for it when you drink a glass of cola or beer, take a shower or wash the dishes. I've cheated, of course. My bizarre solid can be found in the typical home in much the same manner that, say, 2.3 children can be found in the typical family. It exists only as an average. And it's not a solid; it's a bubble. Foam contains thousands of bubbles, crowded together like tiny, irregular polyhedra-and the average number of vertices, edges and faces in these polyhedra is 22.9, 34.14 and 13.39, respectively. If the average bubble did exist, it would be like a dodecahedron, only slightly more so. Coalescing Bubbles that enclose unequal volumes have shapes that remain a mathematical challenge.

45. Trailing After Double Bubbles: Science News Online, Jan. 29, 2000 A proof of the doublebubble conjecture for the case in which the two bubbles'volumes are unequal appears within reach. The double bubble conjecture. http://www.sciencenews.org/20000129/note17ref.asp

Math Trek The Math Game Food for Thought Sickening Food Science Safari Let's Go to Lascaux TimeLine 70 Years Ago in Science News Week of Jan. 29, 2000; Vol. 157, No. 5 Trailing after double bubbles A proof of the double-bubble conjecture for the case in which the two bubbles' volumes are unequal appears within reach. References: Morgan, F. 2000. The double bubble conjecture. Abstracts of Papers Presented to the American Mathematical Society 21(January):129. Further Readings: Morgan, F. 2000. The Math Chat Book . Washington, D.C.: Mathematical Association of America. Peterson, I. 1995. Toil and trouble over double bubbles. Science News 148(Aug. 12):101. Sources: Michael Hutchings Department of Mathematics Stanford University Stanford, CA 94305 Frank Morgan Department of Mathematics Williams College Williamstown, MA 01267 Web site: http://www.williams.edu:803/Mathematics/fmorgan/ Manuel Ritoré Departmento de Geometria y Topologia Universidad de Granada E-18071 Granada Spain Web site: http://www.ugr.es/~ritore/ From Science News Vol. 157, No. 5

46. Is A Double Soap Bubble Stable? Is a double soap bubble stable? LA Slobozhanin J. Iwan D. Alexander. lion@mae.cwru.edu.The double bubble conjecture has been proved recently. http://www.eng.abdn.ac.uk/~apm2002/abstracts/Slobozhanin_Alexander-1/Slobozhanin

Next: About this document ... Is a double soap bubble stable? L.A. Slobozhanin - J. Iwan D. Alexander lion@mae.cwru.edu The double bubble conjecture has been proved recently. This conjecture refers to the following minimal surface problem: for two regions with prescribed volumes, what is the surface that encloses and separates these regions and has the least surface area? According to the conjecture, this surface is an equilibrium standard double bubble consisting of three spherical segments that meet at equal angles along a common circle. From the viewpoint of stability theory, this means that an equilibrium soap double bubble is stable to perturbations that preserve the volumes of each bubble. In this paper, the stability to perturbations that do not conserve the volumes of bubbles comprising the standard double bubble has been examined. The analysis is performed on the principle of minimum free energy. It is shown that the double bubble shape remains stable also to these perturbations, and, thus, it is stable to arbitrary perturbations. L. A. Slobozhanin, Department of Mechanical and Aerospace Engineering and National Center for Microgravity Research on Fluids and Combustion, Case Western Reserve University, Cleveland, Ohio 44106, USA

47. Stcon00 The double bubble conjecture says that the familiar double soap bubble providesthe leastarea way to enclose and separate two given volumes of air. http://home.moravian.edu/public/math/ClubsEvents/Conference/Archives/stcon00.htm

MORAVIAN COLLEGE STUDENT MATHEMATICS CONFERENCE Saturday, February 26, 2000 The conference is sponsored by the Moravian College chapter of Pi Mu Epsilon, with support from USG and Lehigh Valley Association of Independent Colleges (LVAIC): Allentown College Lehigh University Cedar Crest College Moravian College Lafayette College Muhlenberg College Additional Information Fees, tentative schedule, directions and deadlines. Call For Papers Presentation details. Link to last year's conference Link to 2001's conference Collier Hall of Science Moravian College Bethlehem, PA Invited Address: "Bursting the Double Bubble" By Frank Morgan Professor of Mathematics Williams College Computer graphics image by John M. Sullivan , University of Illinois, Urbana Champaign. For more information about Double Bubbles, please click here Abstract: A single round soap bubble provides the least-area way to enclose a given volume of air. The Double Bubble Conjecture says that the familiar double soap bubble provides the least-area way to enclose and separate two given volumes of air. Much media attention focussed on the recent proof using computers for the case of equal volumes, which in turn can be traced to undergraduate research. Now there are rumors from Spain of a proof for arbitrary volumes in R3, and an extension to R4 by undergraduates. Call For Papers The Moravian College chapter of Pi Mu Epsilon invites you to the fourteenth annual MORAVIAN COLLEGE STUDENT MATHEMATICS CONFERENCE on February 26, 2000, a unique opportunity for undergraduate students in the Tri-State area to meet and discuss mathematics. The day begins with a lively

48. Michigan Undergraduate Mathematics Conference 2002 Double Bubble No More Trouble in November 2000 Math Horizons; Proof of thedouble bubble conjecture in March 2001 American Mathematical Monthly. http://www.calvin.edu/academic/math/mumc2002/

Fourth Annual Michigan Undergraduate Mathematics Conference A conference for Undergraduate Mathematics Students and Faculty February 16, 2002 Calvin College Grand Rapids, Michigan [Featured Speaker] [Keynote Address] [Student Talks] [Poster] ... Who Wants to Be a Mathematician? See You Next Year in Dearborn! The 2003 MUMC will be held at the University of MichiganDearborn on Saturday, February 15, 2003. Visit the conference website for additional information. There are a number of other Undergraduate Mathematics Conferences across the country. Here are links to a few of them: Rose-Hulman Conference on Undergraduate Mathematics Hudson River Undergraduate Mathematics Conference (April 27, 2002) Featured Speaker Frank Morgan from Williams College will be our feature speaker. Frank Morgan is currently the Second Vice-President of the Mathematical Association of America and has long been involved in undergraduate research projects and has advised numerous students and groups of students at both graduate and undergraduate levels. At Williams College, where he currently teaches in the Mathematics and Statistics Department, he was the founding director of the very successful SMALL undergraduate research project Professor Morgan works in minimal surfaces and studies the behavior and structure of minimizers in various dimensions and settings. (If you don't know what that means, think of soap bubbles as 2-dimensional surfaces in a 3-dimensional space.) He has written four books:

49. University Of Pittsburgh: Department Of Mathematics The problem of two bubbles, known as the double bubble conjecture, was solvedonly recently by J. Hass, M. Hutchings, and R. Schlafy.(The double bubble http://www.math.pitt.edu/articles/kelvin.html

Table of Contents Fall 2001 Cannonballs and Honeycomb: Kelvin If we turn to the next page after the Kepler conjecture in Kepler's Six-Cornered Snowflake , we find a discussion of the structure of the bee's honeycomb. The rhombic dodecahedron was discovered by Kepler through close observation of the honeycomb. The honeycomb is a six-sided prism sealed at one end by three rhombi. By sealing the other end with three additional rhombi, the honeycomb cell is transformed into the rhombic dodecahedron. Figure 8 The cannonball packing of balls leads to honeycomb cells. It is also related to more general foam problems. If we tile space with hollow rhombic dodecahedra, and imagine that each has walls made of a flexible soap film, we have an example of a foam. Kelvin The problem of foams, first raised by Lord Kelvin, is easy to state and hard to solve. How can space be divided into cavities of equal volume so as to minimize the surface area of the boundary? The rhombic dodecahedral example is far from optimal. Lord Kelvin proposed the following solution. Truncated octahedra fill space (see Figure 9).

50. MAA: Math Horizons--Subscribe On March 18, 2000 an international team of mathematicians announced a proof of thedouble bubble conjecture, which says that the familiar double soap bubble http://www.mathcs.carleton.edu/math_horizons/teasers11-00.html

Double Bubble No More Trouble On March 18, 2000 an international team of mathematicians announced a proof of the Double Bubble Conjecture, which says that the familiar double soap bubble provides the least-area way to enclose and separate two given volumes of air. The two spherical caps are separated by a third spherical cap, all meeting at 120-degree angles. (If the volumes are equal, the separating surface is a flat disc.) This result is the culmination of ten years of remarkable progress by a number of mathematicians including several undergraduate students. The first step was the realization that the problem is actually quite difficult. Suppose You Want to Vote Strategically Be honest. There have been times when you voted strategically to try to force a personally better election result; I have. The role of manipulative behavior received brief attention during the 2000 US Presidential Primary Season when the Governor of Michigan failed on his promise to deliver his state's Republican primary vote for George Bush. His excuse was that the winner, John McCain, strategically attracted cross-over votes of independents and Democrats. TopSpin on the Symmetric Group When I was about 10 I remember getting a puzzle in my stocking which consisted of a 4 x 4 grid with 15 square pieces in it. Of course, there was one space in the grid that held no piece, and you could slide the pieces around so that a piece next to the "hole" could be slid into that space. This particular puzzle had the pictures of four comic book figures when solved. However, you could move the pieces around to give some of the figures different heads, which added a great deal of fun for me. The box the puzzle came in gave some "impossible" positions, and I recall that at the time I wondered how they knew this. Today I still look for puzzles like these whenever I visit a toy store. Now, though, I find that the mathematics behind the puzzles intrigues me as much as the challenge of solving them.

51. Thomas C. Hales - The Kepler Conjecture The problem of two bubbles, known as the double bubble conjecture, was solved onlyrecently by J. Hass, M. Hutchings, and R. Schlafy.\footnote * % {The double http://pear.math.pitt.edu/PittMathZine/2001/fall/articles/kelvin.html

Cannonballs and Honeycombs Thomas C. Hales If we turn to the next page after the Kepler conjecture in Kepler's Six-Cornered Snowflake , we find a discussion of the structure of the bee's honeycomb. The rhombic dodecahedron was discovered by Kepler through close observation of the honeycomb. The honeycomb is a six-sided prism sealed at one end by three rhombi. By sealing the other end with three additional rhombi, the honeycomb cell is transformed into the rhombic dodecahedron. The cannonball packing of balls leads to honeycomb cells. It is also related to more general foam problems. If we tile space with hollow rhombic dodecahedra, and imagine that each has walls made of a flexible soap film, we have an example of a foam. Kelvin Lord Kelvin found that by warping the faces of the truncated octahedra ever so slightly, he could obtain a foam with smaller surface area than the cells of the truncated octahedra. This was Lord Kelvin's proposed solution. It satisfies the conditions Plateau discovered %more than a century ago for minimal soap bubbles. Everyone seemed satisfied with Kelvin's solution; only a proof of optimality was missing. The Kepler conjecture and the Kelvin problem are both special cases of a more general foam problem. Phelan and Weaire ask us to imagine that the soapy film walls have a measurable thickness. We interpolate between the Kepler and Kelvin problems with a parameter $w$ (measuring the wetness of the film) that gives the fraction of space filled by the thick film walls, and $1-w$ is the fraction filled by the cavities. If the foam is perfectly dry, then $w=0$, and the film walls are surfaces. The Kelvin problem asks for the most efficient design. When the foam becomes sufficiently wet, $w$ is close to $1$, and the cavities of the foam can be independently molded. The isoperimetric inequality dictates that they minimize surfaces area by forming into perfect spheres. The Kepler problem asks for the smallest value of $w$ for which every cavity is a perfect sphere.

52. WPI Mathematical Sciences - Colloquia 2000-2001 Frank Morgan, Mathematics Department, Williams College, March 24 2000 Title Thedouble bubble conjecture 1100 am, Stratton Hall, Room 203; refreshments at 10 http://www.wpi.edu/Academics/Depts/Math/News-Events/colloqdetail99-00.html

1999-2000 Colloquia Peter Smereka, Department of Mathematics, University of Michigan, April 28 2000 Title: Relaxation in Hamiltonian Systems 11:00 a.m., Stratton Hall, Room 203; refreshments at 10:30 a.m. in the department lounge, SH107. David Cai, Courant Institute of Mathematical Sciences, April 21 2000 Title: Dispersive Wave Turbulence and Energy Transfer Cycles 11:00 a.m., Stratton Hall, Room 203; refreshments at 10:30 a.m. in the department lounge, SH107. Barbara Keyfitz, John and Rebecca Moores University Scholar Department of Mathematics, University of Houston (currently visiting the Mathematics Department at Brown University), April 14 2000 Title: Solving Hyperbolic Problems with Elliptic Methods 11:00 a.m., Stratton Hall, Room 203; refreshments at 10:30 a.m. in the department lounge, SH107. Frank Morgan, Mathematics Department, Williams College, March 24 2000 Title: The Double Bubble Conjecture 11:00 a.m., Stratton Hall, Room 203; refreshments at 10:30 a.m. in the department lounge, SH107. Student Talk: News on the Double Bubble Conjecture, including work by undergraduates

53. WPI Mathematical Sciences - Events 1999-2000 In both talks, he described recent results on the double bubble conjecture, whichsays that the familiar double soap bubble is the leastarea way to enclose http://www.wpi.edu/Academics/Depts/Math/News-Events/events,99-00.html

1999-2000 Events Commencement Click here for more photos. Mathematical Sciences Awards Ceremony Marcus Sarkis , and a team composed of Jovanna Baptista, Larissa Gilbreath, and Robert Jaeger received the CIMS MQP Award for their project "Pricing a Waiver of Premium Upon Disability," advised by Ann Wiedie and sponsored by John Hancock Insurance Company . Other participating teams and their projects were Andre Freeman and Matthew Lavoie, "Comparing Heuristics for the Traveling Salesman Problem," advised by Brigitte Servatius , and Elizabeth Hogan and Nicholas Allgaier, "Credibility Analysis for Automobile Cession Strategies," advised by Arthur Heinricher and sponsored by Premier Insurance of Massaschusetts. Jonathan Moussa and Matt Shaw were recognized for their strong performance on the Putnam exam, and two teams composed of Brian Ball , Jonathan Moussa, and James Stickney and Jon Kennedy, Will Kennerly, and Casey Richardson, respectively, were recognized for honors received in the COMAP 2000 Mathematical Contest in Modeling. See news item Undergraduates score in Math Modeling, Putnam competitions

54. Colloquium_abstracts The double bubble conjecture says that the familiar double soap bubble isthe leastarea way to enclose and separate two given volumes of air. http://www.wam.umd.edu/~jda/colloquium2_old.html

Abstracts of Colloquium talks Math Department UMD - College Park Other Seminars (February 2) Brian Marcus Coding Theory and Symbolic Dynamics In this talk we will describe several results and open problems regarding coding aspects of symbolic dynamics. We will begin with two sources of motivation: classification of classical dynamical systems and constraints on sequences recorded in data storage devices. These lead to similar coding problems that have been solved by symbolic dynamics. Then we will introduce the basic concepts of symbolic dynamics and survey some of the fundamental coding problems in the subject. (February 9) Vaughan Jones Planar Algebras The simplest planar algebra is the Temperley Lieb algebra which will be carefully defined as an algebra whose basis is a set of planar diagrams. Many algebras based on planar graphs are occurring and seem to play a vital role in the theory of subfactors. We will present some of these algebras. (February 16) Frank Morgan The Double Soap Bubble Conjecture The ancient Greeks suspected and Schwartz proved in 1884 that a round soap bubble provides the least-area way to enclose a given volume of air. The Double Bubble Conjecture says that the familiar double soap bubble is the least-area way to enclose and separate two given volumes of air. A proof for the case of two equal volumes was announced in August by Hass and Schlafly. The story has two remarkable features:

55. You Can’t Hear The Shape Of A Drum (Carolyn Gordon Et Al http//www.pbs.org/wgbh/nova/proof/wiles.html. double bubble conjectureProved (Michael Hutchings et al. 2000), The double soap bubble http://www.mathsci.appstate.edu/mathclub/math.html

Some Recent Mathematical Results and Open Problems Worth a Million Dollars$$$$$$$!! In 1966 the mathematician Mark Kac asked the question, Can you hear the shape of a drum? That may seem like a strange question at first, but it's no stranger than asking if one can ``see'' the chemistry of a star. In the 1991 solution, mathematicians came up with examples of drums that have different shapes but have exactly the same characteristic vibration frequencies. You wouldn't hear any difference if you listened to these drums with your eyes shut tight. http://www.ams.org/index/new-in-math/hap-drum/hap-drum.html In the margin of a book, next to the statement that x n + y n = z n I have discovered a truly remarkable proof which this margin is too small to contain http://www.pbs.org/wgbh/nova/proof/wiles.html Double Bubble Conjecture Proved (Michael Hutchings et al. 2000) The double soap bubble on the left is the optimal shape for enclosing and separating two chambers of air (a given volume) using the least amount of material (surface area). In 1995 the special case of two equal bubbles was heralded as a major breakthrough on this problem when proved with the help of a computer. The new general case involves more possibilities than computers can now handle. The new proof uses only ideas, pencil, and paper. http://www.maa.org/features/mathchat/mathchat_3_18_00.html

56. Bubbles Science World, Bubble movie. Bubble Mania, Circumference/diameterwith bubbles. double bubble conjecture, Pictures of double bubbles. http://www.math.youngzones.org/bubbles.html

Bubbles, Bubbles Everywhere Everyone has fun blowing bubbles. But did you know that bubbles are mathematical? Bubble geometry Geometrical bubbles on wire frames Bubble prints Preserve your bubbles as art Antibubbles The opposite of a bubble Soap bubbles Create geometric art with soap films Zometool bubbles All about bubbles Exploratorium All about bubbles Activites for exploring bubbles Bubblesmith's gallery Bubble pictures WOW! Science World Bubble movie Bubble Mania Circumference/diameter with bubbles Double bubble conjecture Pictures of double bubbles

57. CRC Concise Encyclopedia Of Mathematics On CD-ROM: D Double Bubble; double bubble conjecture; Double Contraction Relation;Double Cusp; Double Exponential Distribution; Double Exponential http://www.math.pku.edu.cn/stu/eresource/wsxy/sxrjjc/wk/Encyclopedia/math/d/d.ht

d'Alembert's Equation d'Alembert Ratio Test d'Alembert's Solution d'Alembert's Theorem ... Dynkin Diagram Eric W. Weisstein

58. BEAUTY For n = 2, the problems are called the double bubble conjecture and the solutionboth to the planar problem and the area problem is known to be the Double http://members.fortunecity.com/jonhays/beauty.htm

web hosting domain names email addresses related sites BEAUTY AND WOMEN'S BELLY-BUTTONS RIDDLE: What do a SQUARE, a CIRCLE, and THE GOLDEN MEAN (a.k.a GOLDEN SECTION, a.k.a. GOLDEN RATIO, a.k.a. DIVINE PROPORTION) have in common? (You may ask a Graduate Math Student for help on this.) ANSWER: They use IRRATIONAL NUMBERS for OPTIMAL or MAXIMIN ("mostest for leestest") REAL IZATION of BEAUTY. Dig? OK, I'll redig that one. (And belly-buttons will pop out.) A SQUARE IS THAT POLYGON WHICH ACHIEVES MAXIMUM AREA WITH MINIMUM BOUNDARY (CALLED "PERIMETER", COMPARABLE TO THE "CIRCUMFERENCE OF A CIRCLE") ACHIEVING MAXIMIN BEAUTY A CIRCLE IS THAT PLANE FIGURE WHICH ACHIEVES MAXIMUM AREA WITH MINIMUM BOUNDARY (CALLED "CIRCUMFERENCE", COMPARABLE TO THE "PERIMETER" OF A POLYGON) ACHIEVING MAXIMIN BEAUTY THE GOLDEN MEAN IS THAT DIVISION (BOUNDARY) OF LINE SEGMENT WHICH FORMS THE GREATEST INTERRELATIONSHIP BY THE LEAST EXTENSION ACHIEVING MAXIMIN BEAUTY NOW HEAR THIS!!! All three of these achieve MAXIMIN BEAUTY via IRRATIONAL METRICS: THE DIAGONAL OF A SQUARE RELATES TO THE SQUARE AS THE DIAMETER OF A CIRCLE TO THE CIRCLE. THE DIAGONAL OF A SQUARE IS INCOMMENSURABLE WITH ITS SIDES (IN THE RATIO, for any side of length S

59. Bubble space. For , the problems are called the double bubble conjecture andthe solution to both problems is known to be the Double Bubble. http://mathworld.pdox.net/math/b/b428.htm

Bubble A bubble is a Minimal Surface of the type that is formed by soap film. The simplest bubble is a single Sphere . More complicated forms occur when multiple bubbles are joined together. Two outstanding problems involving bubbles are to find the arrangements with the smallest Perimeter (planar problem) or Surface Area Area problem) which enclose and separate given unit areas or volumes in the plane or in space. For , the problems are called the Double Bubble Conjecture and the solution to both problems is known to be the Double Bubble See also Double Bubble Minimal Surface Plateau's Laws Plateau's Problem References Morgan, F. ``Mathematicians, Including Undergraduates, Look at Soap Bubbles.'' Amer. Math. Monthly The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 219, 1989. Eric W. Weisstein

60. Stanford University Geometric Analysis Seminar 1999-2000 April 19 Michael Hutchings, Stanford University Title The double bubble conjectureAbstract The double bubble conjecture states that the leastarea way to http://math.stanford.edu/~moore/ga-sem99-00.html

Stanford University Geometric Analysis Seminar 1999-2000 Unless otherwise noted, all seminars begin at 4:00 pm in room 380-381T, and last approximately one hour. There is tea at 3:30 in the lounge area on the third floor of the Math Department Building (Building 380). Spring 2000 Schedule March 29 Tobias Ekholm, Stanford University Title: Total curvatures of holonomic links Abstract: April 5 Jingyi Chen, University of British Columbia, Vancouver, Canada Title: Holomorphic connections and lagrangian cycles Dinner: We will be taking the speaker out to dinner at a nearby restaurant after the seminar. April 12 Dan Ostrov, Santa Clara University Title: Determining shapes from discontinuous shading data and solving related first order PDEs where the flux function is discontinuous Abstract: Shape from shading is the study of how to determine a 3-D surface from a 2-D picture of the surface (plus as minimal an amount of additional information as possible.) When the picture has discontinuities (i.e., a bright part of the picture borders a darker part), difficulties arise in determining existence, uniqueness, and a method of computation for the solution of the underlying PDE describing the surface. We will explain a method of resolving these questions involving a control theory representation for the PDE, which will also allow us to answer larger questions about much more general first order PDEs with discontinuous flux/Hamiltonian functions. Dinner: We will be taking the speaker out to dinner at a nearby restaurant after the seminar.

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