61. Streaming Video - Spring 2001 June 25 July 6, 2001. Lecture Series Frank Morgan Geometric MeasureTheory and the Proof of the double bubble conjecture, Lecture 1; http://elasmo.kaist.ac.kr/video/index02.html

MSRI Home Page MSRI Publications Streaming Video / Streaming Video - Spring 2001 Streaming Video - Spring 2001 Introduction Hints for Viewing Other SV Sites CDROMs and Video Tapes for sale ... Mirror Sites Welcome to the MSRI Video Archive, Click on the triangle icons to expand/contract the list. MSRI: Spring 2001 Free Probability and Non-commutative Banach Spaces January 22 - 27, 2001 L. Ge Applications of free entropy T. Cabanal-Duvillard Large deviations for Gaussian matrices and free entropies M. Stefan Some properties of free group subfactors Z-J. Ruan R. Smith Approximation properties for crossed products G. Pisier Multipliers of H1 and power bounded operators E. Christensen The length of a Gamma factor W.B. Arveson Curvature and Dirac operator of a commuting d-tuple N. Randrianantoanina Subspaces of non-commutative Lorentz-spaces D. Voiculescu Free probability analogue of the Wasserstein distance on trace state space P. Zinn-Justin The enumeration of planar maps, links and knots A. Nica R-cyclic families of matrices in free probability P. Biane Free cummulants , matrix orbits and characters of symmetric groups

62. CIM Bulletin #9 17, 2000 Science is a piece by Barry Cipra Why Double Bubbles Form the Way TheyDo, and reporting on the recent solution of the double bubble conjecture. http://at.yorku.ca/i/a/a/h/13.htm

Topology Atlas Document # iaah-13 from CIM Bulletin #9 What's New in Mathematics Race to settle Catalan conjecture: it's people vs. computers Ivars Peterson reports in the December 2, 2000 Science on recent progress towards the resolution of this 150-year-old conjecture. Catalan noted that 8 = 2 and 9 = 3 are consecutive integers and conjectured that they were the only set of consecutive whole powers. This translates to the Fermat-like statement that the equation xp - yq = 1 has no whole-number solutions other than 3 Incompressible is incomprehensible New encription algorithm A new Federal encryption algorithm was reported in the October 20, 2000 Chronicle of Higher Education. The article, by Florence Olsen, relates how the Commerce Department, after a 4-year search, has declared the new federal standard for protecting sensitive information to be Rijndael, an algorithm named after its inventors Vincent Rijmen and Joan Daemen. The two Belgians beat out 20 other entries, including teams from IBM and RSA. The new encryption algorithm, of which no mathematical details were given, can be made stronger as more powerful computer processors are developed. This was an entry requirement for the competition. According to Raymond G. Kammer of NIST, which managed the selection process, it should be good for about 30 years, "that is, if quantum computing doesn't manifest itself in five or six years." Updating Ramanujan Double Bubbles Squeeze in a few more?

63. Notes From The Lab FOURDIMENSIONAL PROOF. Yvonne Lai, a junior in mathematics, has helped extend arecent mathematical proof of the double bubble conjecture to four dimensions. http://web.mit.edu/newsoffice/tt/2000/apr26/labnotes.html

Published by the MIT News Office at the Massachusetts Institute of Technology, Cambridge, Mass. April 26 Tech Talk Search MIT News ... MIT WEDNESDAY, APRIL 26, 2000 Notes from the Lab FOUR-DIMENSIONAL PROOF Yvonne Lai, a junior in mathematics, has helped extend a recent mathematical proof of the "double bubble conjecture" to four dimensions. In a March address to the Undergraduate Mathematics Conference at the Rose-Hulman Institute of Technology, mathematicians from Williams College, Stanford University and the University of Granada announced their proof that the familiar double soap bubble is indeed the optimal shape for enclosing and separating two chambers of air. In a postscript, a group of undergraduates from Stanford, Williams and MIT including Ms. Lai extended the theorem to four-dimensional bubbles. Working last summer at Williams, they found a way to extend the proof to 4-space and certain cases in 5-space and above. Their work was part of the Research Experiences for Undergraduates sponsored by the National Science Foundation and Williams College. The group's paper on their work is awaiting publication. METALS FOUND IN BOSTON HARBOR Caroline Tuit, a graduate student in the MIT/Woods Hole Oceanographic Institution Joint Program, is co-author of a study that reveals high levels of platinum and palladium in Boston Harbor surface sediments. The researchers say the most likely source of these metals is the use of catalytic converters in cars, as well as industrial waste entering the harbor through the sewage system.

64. HRUMC'00 - Session III Details Exciting New Minimal Surfaces! Joshua White, Williams College; Thedouble bubble conjecture - Frank Morgan, Williams College. Room 301. http://www.skidmore.edu/academics/mcs/00sess3.htm

HRUMC '00 Session III Lecture Times: First talk: 3:30 PM Second talk: 3:50 PM Third talk: 4:10 PM Fourth talk: 4:30 PM (An asterisk (*) indicates a level-II talk) The title of each session appears on a button. Press the button to see the abstracts of the talks from that session. All talks are in Rockefeller Hall Room: 101 Chair: Gary Krahn, United States Military Academy AN Awesome Sequence I - Michele Kelley, Colgate University AN Awesome Sequence II - Nathan Bailey, Colgate University Enumeration of Kirkman Triple Systems of Order 21 - Lee Ann Ives, University of Vermont Investigation of the Proof by N. G. de Bruijn of the Number of de Bruijn Cycles - Phillip M. LaCasse, United States Military Academy Room: 201 Chair: Phil Beaver, United States Military Academy Java Servlets for Data Access Solutions Across the Internet - Jason Miele and Paul Evans, Siena College A Web Interface for MIDI Transformations - Aileen Ang, St. Lawrence University TCP/IP Subnetting - Doug Chimento, St. Lawrence University Chaos and Complex Adaptive Systems - Mary Oliastro, United States Military Academy

65. Archived Front Pages 21/4/2000 Onwards 28/4/00. . .The double bubble conjecture.. On February 25, 2000 itwas announced that the double bubble conjecture had been proved. http://www.madras.fife.sch.uk/maths/ArchHomePages9.html

Welcome to our P7 visitors today! Today we see the start of a week of visits from our future S1 students. They will work through a timetable of sample lessons given by the various subject departments. At Mathematics some may even be reading this sentence at this precise moment and are about to be shown other SENTENCES created by our present S1 students. Their homework will be to create their own sentences, send them to us and if they stretch our minds into complicated enough logical knots then we will publish them here. May at the Nrich Site Try some excellent sets of problems for and . They have a deadline of 22nd May for sending in solutions. In this month's Interact Magazine our S3 students: Sam Larg, Dave Stewart, Richard Mason, Joe Neilson, Matthew Broadbent and Ross Craig have had their work published on the problem Never Prime Sue Liu in S5 had an excellent month with solutions published to By the quad - quick solve Shape and territory and Napoleon's Hat Congratulations to all these students for their excellent effort and results.

66. Book Detail Information Flat Chains Modulo v, Varifolds, Minimal Sets; Miscellaneous Useful Results; SoapBubble Clusters; Proof of double bubble conjecture; Hexagonal Honeycomb http://opamp.com/cf/details.cfm?ISBN=0125068514

67. SMALL Research 2000. With Hutchings, Ritore, and Ros, Proof of the double bubble conjecture, Annals of Math. 155 (March, 2002), 459489. Geometric http://www.williams.edu/Mathematics/SMALL2004Research.html

SMALL 2004 Research Descriptions Computational Cartography Advisor: Satyan Devadoss Project Description: Data for analyzing the Earth's surface is obtained as a set of discrete 3D points from Global Positioning Systems (GPS), radar measurements and ground surveys. These point-sets are then used by computer algorithms and mesh-building graphics software to reconstruct the surfaces. This is a beautiful subject with a tremendous amount of active research, relating powerful ideas from mathematics, elegant tools from computer science, and concrete data from geodetic science. Our research group will work on numerous unsolved problems in reconstructing surfaces, in the field of computational geometry. Ideas such as curvature, polyhedral geometry, triangulations, and combinatorial topology will be blended together. We will work from a mathematical perspective using theoretical tools as well as actual data sets using ground-breaking software. No computer science experience is needed (no programming will be done). Students should have had Linear Algebra. Some background in either topology and/or geometry would be helpful. Geometry Advisor: Frank Morgan Project Description: The Double Bubble Theorem (2002) says that the familiar double soap bubble is the least-area way to enclose and separate two given volumes of air (see references below). The "SMALL" Geometry Group, since its 1990 proof of the

68. Research Supervision Spielman). Generalized the recent proof of the double bubble conjecturefrom R 3 to R 4 and certain higher dimensional cases. Papers 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: Julian Lander, 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. Benny Cheng, 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: Gary Lawlor, A sufficient criterion for a cone to be area-minimizing, Memoirs of the AMS 91, No. 464 (1991) 1-111.

69. Nonius Conjecture $1,000,000 challengedouble bubble conjecture Proved. A semana de chamada de atenção http://www.mat.uc.pt/~jaimecs/

70. Clay Research Institute On The Global Theory Of Minimal Surfaces Frank Morgan will give nine lectures on the subject of Geometric Measure Theoryand the Proof of the double bubble conjecture Last year Hutchings, Morgan http://www.msri.org/ext/minimal2.html

Clay Research Institute on The Global Theory of Minimal Surfaces Program for weeks one and two The first two weeks of the 2001 Clay Mathematics Institute will include a graduate level introduction to the theory of minimal surfaces. There will be two or three lecture series and additional events and activities, including homework sessions, open problem discussions, demonstrations and instruction on computer graphics techniques. Attending will be graduate students from the MSRI sponsoring institutions and additional graduate students and researchers sponsored by the Clay Institute. Attending students are nominated by an MSRI sponsor or nominated as a Clay Mathematics Institute participant via the methods indicated in the CMI Workshop Page at the Clay Research Institute on The Global Theory of Minimal Surfaces. For the main program see the MSRI Workshop Page for the Clay Research Institute on The Global Theory of Minimal Surfaces Program for weeks one and two Main Lecture Series - Topics Frank Morgan will give nine lectures on the subject of Geometric Measure Theory and the Proof of the Double Bubble Conjecture: Last year Hutchings, Morgan, Ritore and Ros announced a proof of the Double Bubble Conjecture, which says that the familiar standard double soap bubble provides the least-area way to enclose and separate two given volumes of air. It was only with the advent of geometric measure theory in the 1960s that mathematicians were ready to deal with such problems involving surfaces meeting along singularities in unpredictable ways. The lectures will discuss modern, measure-theoretic definitions of "surface," compactness of spaces of surfaces, and finally the proof of the double bubble conjecture. Homework will vary from basic exercises to open problems. The text

71. Science News: Trailing After Double Bubbles - Research On Geometry Of Bubble Con Now, a proof of the doublebubble conjecture for the case where the two volumesare unequal appears within reach, says Frank Morgan of Williams College in http://www.findarticles.com/cf_dls/m1200/5_157/59607799/p1/article.jhtml

@import url(/css/us/style.css); @import url(/css/us/searchResult1.css); @import url(/css/us/articles.css); Advanced Search Home Help IN all publications this publication Reference Automotive Business Computing Entertainment Health News Reference Sports YOU ARE HERE Articles Science News Jan 29, 2000 Content provided in partnership with Print friendly Tell a friend Find subscription deals Trailing after double bubbles - research on geometry of bubble configurations - Brief Article Science News Jan 29, 2000 by I.P. From Washington, D.C., at the Joint Mathematics Meetings Predicting the geometric shapes of bubble clusters can lead to surprisingly difficult problems. In 1995, mathematicians finally proved that the so-called standard double bubble, familiar to any soap-bubble enthusiast, represents the least surface area when the two bubble volumes are equal (SN: 8/12/95, p. 101). Such a two-chambered structure triumphs over any other possible geometric form as the most efficient way of enclosing and separating two equal volumes of space. Now, a proof of the double-bubble conjecture for the case where the two volumes are unequal appears within reach, says Frank Morgan of Williams College in Williamstown, Mass.

72. ¾ÊÇ·. is the unique stable double bubble in R^2. , File, 2, Proof of the planar triple bubble conjecture. http://webhost.wu.ac.th/dpst/searchInside0.php?id=301008

73. ¾ÊÇ·. The standard double bubble is the unique stable double bubble in R^2.Proof of the planar triple bubble conjecture. . ? http://webhost.wu.ac.th/dpst/listArticleInside.php?xlimit=90&crow=208&no=30&page

74. Math Department Colloquia Series Theorem, proved in 2000, says that the double soap bubble, formed when two bubblescome together They proved this conjecture for the case when the image http://www.math.unt.edu/colloquiaarchive.htm

Math Department Colloquia Series Archive September, 18, 2002 Nicholas Alikakos, University of North Texas Title: "The Normalized Mean Curvature Flow for a Small Bubble in a Riemannian Manifold" April 12, 2002 Frank Morgan, Williams College Title: "2000 Proof of the Double Bubble Theorem" Abstract: A round soap bubble is the most efficient, least area shape for enclosing a given volume of air. The Double Bubble Theorem, proved in 2000, says that the double soap bubble, formed when two bubbles come together, provides the least area shape for enclosing and separating two given volumes of air. Undergraduate students contributed to the proof of this theorem. June 6, 2002 Miklós Laczkovich, Eötvös Loránd University, Budapest, Hungary Title: "Squaring the Circle" Abstract: In 1925, A.Tarski asked whether or not the disc can be decomposed into finitely many pieces and the pieces can be moved to get a decomposition of a square. We outline the history and the solution of Tarski’s problem, and give a review of some recent developments and of some open questions.

75. [math/0208120] Double Bubbles In The 3-torus We present a conjecture, based on computational results, on the area minimizing comparablesmall volumes, we prove that an area minimizing double bubble in the http://arxiv.org/abs/math.DG/0208120

Mathematics, abstract math.DG/0208120 From: Genevieve Walsh [ view email ] Date: Thu, 15 Aug 2002 00:33:49 GMT (636kb) Double bubbles in the 3-torus Authors: (UC Riverside), Joseph Corneli (New College of Florida), Genevieve Walsh (UC Davis), Shabnam Beheshti (Texas Tech University) Comments: 13 pages, 4 figures. Prepared on behalf of the participants in the Clay Mathematics Institute Summer School on the Global Theory of Minimal Surfaces, held at the Mathematical Sciences Research Institute in Berkeley, California, Summer 2001 Subj-class: Differential Geometry; Geometric Topology MSC-class: We present a conjecture, based on computational results, on the area minimizing way to enclose and separate two arbitrary volumes in the flat cubic 3-torus. For comparable small volumes, we prove that an area minimizing double bubble in the 3-torus is the standard double bubble from R^3. Full-text: PostScript PDF , or Other formats References and citations for this submission: CiteBase (autonomous citation navigation and analysis) Which authors of this paper are endorsers?

76. Bubbles For 2 areas, a standard double bubble is shortest (early 90’s). Will we live longenough to see the proof of the planar triple bubble conjecture? , Time. http://www.math.uiuc.edu/~wichiram/maths/bubbles.html

The Planar Soap bubble Problem: to find the least perimeter way to enclose and separate m regions of m given areas on the plane. For 1 area, a circle is shortest (late 19th century). For 2 areas, a standard double bubble is shortest (early 90’s). For 3 areas, it is still unproven but we believe that a standard triple bubble is shortest while these triple bubbles are not minimizing. Worldwide responses: "...even the case of triple bubble in the plane is still open...", CNN. "Will we live long enough to see the proof of the planar triple bubble conjecture?", Time. "How many more mathematicians have to die to get this done?", People. "Closing triple, hidden dragon", SONY classic picture. "...so frustrated. Do we really have to prove it?", Hustler. Results that actually happened: '75: Almgren: well defined the problem. '87: Bleicher: some sufficient conditions on geometry of minimizing bubbles. '92: Morgan: existence of minimizing bubbles. '93: Foisy,Alfaro,Brock,Hodges,Zimba: the standard doubble is uniquely minimizing. '94: Cox,Hutchings,Kim,Light,Mauer,Tilton: the standard triple bubble is best among those with connected regions.

77. Double Bubble From MathWorld double bubble from MathWorld A double bubble is pair of bubbles which intersect and are separated by a membrane bounded by the intersection. The usual double bubble is illustrated in the left http://rdre1.inktomi.com/click?u=http://mathworld.wolfram.com/DoubleBubble.html&

78. UIUC Dept. Of Mathematics Seminar Calendar early 90 s, Joel Foisy found the solution for the case of two areas (using standarddouble bubbles). Explicitly, the planar triple bubble conjecture states that http://torus.math.uiuc.edu/cal/math/cal?year=2002&month=12&day=10&interval=day

79. Geometric Measure Theory And The Proof Of The Double Bubble Conjecture, Lecture Calendar. Geometric measure theory and the proof of the double BubbleConjecture, Lecture 1. Frank Morgan (Scheduled Workshop Talk). http://zeta.msri.org/calendar/talks/TalkInfo/486/show_talk

Calendar Geometric measure theory and the proof of the Double Bubble Conjecture, Lecture 1 Frank Morgan (Scheduled Workshop Talk) Monday, Jun 25, 2001 9:30 am to 10:30 am at the MSRI Lecture Hall, Mathematical Sciences Research Institute, Berkeley, California Frank Morgan will give nine lectures on the subject of Geometric Measure Theory and the Proof of the Double Bubble Conjecture. Last year Hutchings, Morgan, Ritore and Ros announced a proof of the Double Bubble Conjecture, which says that the familiar standard double soap bubble provides the least-area way to enclose and separate two given volumes of air. It was only with the advent of geometric measure theory in the 1960s that mathematicians were ready to deal with such problems involving surfaces meeting along singularities in unpredictable ways. The lectures will discuss modern, measure-theoretic definitions of "surface," compactness of spaces of surfaces, and finally the proof of the double bubble conjecture. Homework will vary from basic exercises to open problems. The text Geometric Measure Theory: A Beginner's Guide (3rd edition) by Frank Morgan will be made available, as well as additional notes and materials. (Students nominated by MSRI sponsors will receive a copy of the book on arrival. Several copies will be available for use by other participants.) There will be sessions on exercises and on open problems.

80. Mathematicians Prove Double Soap Bubble Had It Right Four mathematicians have announced a mathematical proof of the double bubbleConjecture that the familiar double soap bubble is the optimal shape for http://www.globaltechnoscan.com/19april-25april/soap_bubble.htm

Please register here to Search or Submit B usiness O pportunities REGISTER HERE LOGIN Mathematicians Prove Double Soap Bubble Had It Right For Business Opportunities in Engineering Industry please click here Four mathematicians have announced a mathematical proof of the Double Bubble Conjecture: that the familiar double soap bubble is the optimal shape for enclosing and separating two chambers of air. In an address to the Undergraduate Mathematics Conference at the Rose-Hulman Institute of Technology in Indiana on Saturday (March 18), Frank Morgan of Williams College announced that he, Michael Hutchings of Stanford, and Manuel Ritori and Antonio Ros of Granada had finally proved that the double soap bubble had it right all along. When two round soap bubbles come together, they form a double bubble. Unless the two bubbles are the same size, the surface between them bows a bit into the larger bubble. The separating surface meets each of the two bubbles at 120 degrees. This precise shape is now known to have less area than any other way to enclose and separate the same two volumes of air, even wild possibilities, in which the second bubble wraps around the first, and a tiny separate part of the first wraps around the second. Such wild possibilities are shown to be unstable by a new argument which involves rotating different portions of the bubble around a carefully chosen axis at different rates. The breakthrough came while Morgan was visiting Ritori and Ros at the University of Granada last spring. Their work is supported by the National Science Foundation and the Spanish scientific research foundation DGICYT.

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