61. Geometric Measure Theory: A Beginner's Guide This third edition of Geometric Measure Theory A Beginner's Guide presents, forthe first time in print, the proofs of the double bubble conjecture and the http://www.uni-protokolle.de/buecher/isbn/0125068514/

Forum Chat Newsletter Nachrichten ... Suche Specials Eignungstest Kreditkarte Geometric Measure Theory: A Beginner's Guide von Frank Morgan Kategorie: Mengenlehre ISBN: 0125068514 Kurzbeschreibung Geometric measure theory is the mathematical framework for the study of crystal growth, clusters of soap bubbles, and similar structures involving minimization of energy. It has contributed over the past thirty years to major advances in geometry and analysis, including, for example, the original proof of the positive mass conjecture in cosmology. The subject is receiving more and more attention. This third edition of Geometric Measure Theory: A Beginner's Guide presents, for the first time in print, the proofs of the Double Bubble Conjecture and the Hexagonal Honeycomb Conjecture. Within four new chapters, readers are also led through treatments of the Weaire-Phelan counterexample to Kelvin's conjecture, Almgren's optimal isoperimetric inequality, and immiscible fluids and crystals. The abundant illustrations, examples, exercises, and solutions in this book will enhance its reputation as the most accessible introduction to the subject. Synopsis Geometric measure theory is the mathematical framework for the study of crystal growth, clusters of soap bubbles, and similar structures involving minimization of energy. It has contributed to major advances in geometry and analysis, including, for example, the original proof of the positive mass conjecture in cosmology. This third edition of "Geometric Measure Theory: A Beginner's Guide" presents the proofs of the Double Bubble Conjecture and the Hexagonal Honeycomb Conjecture. Within four new...

62. Note That If You See A Reference To The Heart Of Math Book, Then General double bubble conjecture in R^3 Solved, Focus The Newsletter of the MathematicalAssociation of America, May/June 2000, Volume 20, Number 5, p. 45. http://www.cs.appstate.edu/~sjg/class/1010/mathematician/mathematicianreferences

Note that if you see a reference to the Heart of Math book, then you should look there - For the 2pm class it is on reserve in the library. Thomas Fuller One of Fuller's Calculations A History of Computers the history of computer speed MAD page http://www.math.buffalo.edu/mad/special/fuller_thomas_1710-1790.html African Slave and Calculating Prodigy: Bicentenary of the Death of Thomas Fuller, by Fauvel and Gerdes, Historia Mathematics 17 (1990), 141-151. The Great Mental Calculators: The Psychology, Methods, and Lives of Calculating Prodigies, Past and Present, by Steven B. Smith, 1983, On Mathematics in the History of Sub-Saharan Africa, by Paulus Gerdes, Historia Mathematica 21 (1994), 345-376, p. 345, 361-2, 366, 373. Maria Agnesi Maria Gaetana Agnesi The Living Witch of Agnesi http://www.astr.ua.edu/4000ws/witch-of-agnesi.html Why bother to learn Calculus - http://www.karlscalculus.org/why.html Definition of Calculus Women of mathematics : a biobibliographic sourcebook edited by Louise S. Grinstein p. 1-5. The Witch of Agnesi: A Lasting Contribution from the First Surviving Mathematical Work Written by a Woman - A commemoritive on the 200th anniversary of her death, by S. I. B. Gray and Tagui Malakyan, The College Mathematics Journal, Vol 30, No 4, September 1999, p. 258-268.

63. Dr. Joel Foisy's Home Page The trial of the semester. double bubble conjecture Proved! To see the pageof my graduate advisor, John Harer. My high school alma mater, SLC. http://www2.potsdam.edu/MATH/foisyjs/

Joel Foisy's Home Page Department of Mathematics, SUNY Potsdam. Office: MacVicar 204. Phone: (315)267-2084. Students should feel welcome to drop by at any time. I am thrilled that SUNY Potsdam had REU (Research Experience for Undergraduates) for the summers in 1997, 1999-2002 and we will have another in 2003. REU information and links. Calculus Resources and links. SUNY Potsdam Mathematics Departement homepage. Links of interest to me: The best band in the region. The trial of the semester. Double Bubble Conjecture Proved! To see the page of my graduate advisor, John Harer. ... Would you like to purchase a Klein Bottle?!. I am also a fellow in Project NExT. Here's a link: Project NExT. If you like mapping class groups, email me and I will send you a reprint of the paper: The second homology group of the level two mapping class group and extended Torelli group of an orientable surface. foisyjs@potsdam.edu

64. ADM Seminar for Primality Testing. Sept. 19, none, Sept. 26, none, Frank Morgan Proof of the double bubble conjecture at Furman University. Oct.3, http://www.math.clemson.edu/~gmatthe/admF02.html

Fall 2002 Aug. 29 Robert E. Jamison Clemson University Rank Tolerance Graph Classes Sept. 5 Gretchen L. Matthews Clemson University Semigroups of Triples of Points on Hermitian Curves Sept. 12 Kevin James Clemson University A New Algorithm for Primality Testing Sept. 19 none Sept. 26 none Frank Morgan "Proof of the Double Bubble Conjecture" at Furman University Oct. 3 Kevin James Neil Calkin Clemson University REU in Computational Number Theory and Combinatorics Oct. 10 Clemson Miniconference Oct. 17 Jirapha Limbupasiriporn Clemson University Small Sets of Even Type in Finite Projective Planes of Even Order Oct. 25 Gary Salazar* Trinity University An Examination of Affine Variety Codes with an Improved Minimum Distance Bound Nov. 1 Shuhong Gao* Clemson University Multivariate polynomial interpolation and decoding of algebraic geometric codes Nov. 7 Doug Shier Stochastic Minimum Spanning Trees Nov. 14 Wayne Goddard Computer Science, Clemson University Self-Stabilizing Graph Algorithms Nov. 21 Shuhong Gao Clemson University Multivariate polynomial interpolation and decoding of algebraic geometric codes II Nov. 28 Thanksgiving Dec. 5

65. 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.

66. 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?

67. 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.

68. Nonius Translate this page Millennium Prize Problems Goldbach's Conjecture $1,000,000 challengedouble bubble conjecture Proved. A semana de chamada de atenção http://www.mat.uc.pt/~jaimecs/

window.open ('index-t.html', 'newwindow', config='height=300,width=305,toolbar=no,menubar=no,scrollbars=yes,resizable=no,location=no,directories=no,status=no') O NONIUS em geral e em Portugal Arquivo completo do "NONIUS" (Universidade Aberta) Pesquise em toda a rede EscolaNet Portugal 20 Valores Selo de qualidade Educativa NET 2002: 500 anos do nascimento de Pedro Nunes - : recomendados anteriormente Mocho NONIUS apanhadas ao correr dos bytes na Internet Novo recorde: 39th Known Mersenne Prime Found!! Millennium Prize Problems Goldbach's Conjecture: $1,000,000 challenge Double Bubble Conjecture Proved Mathematics Awareness Week 1996 April 21-27, Mathematics and Decision Making Mathematics Awareness Week 1997 April 20-26, Mathematics and the Internet Mathematics Awareness Week 1998 April 26-May 2, Mathematics and Imaging Mathematics Awareness Month 1999 April 1999 Mathematics and Biology MATHEMATICS AWARENESS MONTH 2000 April 2000 Math spans all dimensions Mathematics Awareness Month 2001 April 2001 Mathematics and the Ocean Mathematics Awareness Month 2002 April 2002 Mathematics and the Genome Mathematics Awareness Month 2003 April 2003 Mathematics and Art Mathematical Quotations Server @ Furman University Numberland @ Count On Mathematicians of the day @ University of St Andrews, Scotland

69. 12 It makes you think of the double bubble conjecture, doesn't it? Well, this isn'tapure double, but it is close enough to view the 120 degree meeting. http://www.allie.com/171.htm

12/11/2002 Will it pop? While doing my dishes today, I noticed a double bubble formation. It makes you think of the double bubble conjecture, doesn't it? Well, this isn't a pure double, but it is close enough to view the 120 degree meeting. The little bubbles of soap might affect the stability of the two larger ones. Also, it is hard to take a picture of something transparent.

70. EXN.ca | Discovery Now, mathematicians at Williams College have proved the double bubble conjecture. Using a relatively simple mathematical formula, they showed this is indeed http://www.exn.ca/Stories/2000/03/20/54.asp

EXN Science Wire: Daily news from the world of science By: EXN Staff , March 20, 2000 In today's stories: A new theory on climate change Bubbling with answers Making phone calls thanks to mirrors A new theory on climate change On Monday, a new take on the causes of global climate change emerged from the Proceedings of the National Academy of Sciences. Researchers studying ice and sedimentary core records say they've identified a new climate cycle - one that repeats every 1,800 years. They say it's related to the lunar tides. It's thought that tidal forces cause cooling at the sea surface by vertically mixing the oceans. If the idea is correct, it would mean that global warming predicted over the next few decades would be the result of a natural warming trend that began at the end of the last Ice Age. The warming trend would continue into the 32nd century. And, according to the research, it would produce warmer conditions than the Earth has seen in the past millennium. Bubbling with answers A lot toil and trouble has gone into understanding the double bubble and now there are some answers. For years, mathematicians have wondered why ordinary soap bubbles act the way they do. When two round bubbles come together, they form a double bubble, with the surface between them bowing a bit into the larger bubble. The angle between the two surfaces is always 120 degrees. The assumption has been that this is the optimum shape for the bubbles to take. Now, mathematicians at Williams College have proved the "double bubble conjecture." Using a relatively simple mathematical formula, they showed this is indeed the most efficient shape for enclosing two chambers of air. They also proved why other bubble formations don't appear in nature.

71. Math Coffees Frank Morgan of Williams College talk on The Proof of the double bubble conjectureand The Soap Bubble Geometry Contest at 4 pm and 730 pm, respectively, on http://www.davidson.edu/math/frontpage/Math_Coffees.htm

Faculty Courses Programs Math Center ... Student Job Opportunities Math Coffees Bernard Review Bernard Lecture [ Math Coffees ] Problem Contest Math Coffees Math Coffees are weekly meetings of students and faculty to hear a local or visiting speaker. Check here frequently for updates! Rob McLean , DC'03 Paths that Turn at a Constant Rate: Special Curves in the Hyperbolic Plane 4 p.m.5 p.m., Thursday, March 20, Chambers 125 In the Euclidean plane there are only two curves of constant curvature, the line and the circle. However, in the hyperbolic plane the curves of constant curvature are more abundant. I will introduce hyperbolic geometry and summarize two methods that classify the curves of constant curvature in the hyperbolic plane. In one method I will introduce a model for hyperbolic geometry and give an argument that classifies the curves of constant curvature based solely on the structure of the model. In the other method I will introduce the notion of an orbit of a point and use a more algebraic argument to classify the curves of constant curvature. Spring 2003 Math Coffee Schedule (click on an image at the left to view the Math Coffee announcement) January 23 CANCELLED , Wake Forest University A Liberal Arts Approach to Mathematical Modeling January 30 45 p.m.

72. 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

73. GoSeekIt And MosaicFX Directory - Multilingual Directory Of The 5. Proof of the double bubble conjecture proof that the standard double bubble providesthe leastarea way to enclose and separate two regions of prescribed http://uk.dir.mosaicfx.com/serf.pl?data=9293

74. Page014 March 2001 The double bubble conjecture is a true statement in dimension 4 Thiswas proved by the undergraduate students Ben Reichardt, Cory Heilmann, Yvonne http://www.math.utoledo.edu/~jevard/Page014.htm

Mathematical events and history Page maintained by Jean-Claude Evard. Last update: March 16, 2003. Content of this page: 1. News and events in mathematics Link 2. List of Web pages about events in mathematics Link 3. List of Web pages on history of mathematics Link 4. List of Web pages on meetings and conferences Link News and events in mathematics (In reverse chronological order) Dates: It is not easy to obtain information about the exact date of events, like the date when a proof of a conjecture was officially certified as correct by referees. The dates that I provide on this page are not official. They are only the best I have obtained so far. Any additional information you could provide will be very welcome. Mathematical Review Copies of reviews from Mathematical Review cannot be posted on Web pages, but they can be seen through links to MathSciNet. These links work only in the networks of institutions or on the computers of users who are current subscribers to MathSciNet. February 21, 2003:

75. ISAMA Eversions. 1100 1145, Colin Adams, Why Knot? 1145 - 1230, FrankMorgan, The double bubble conjecture, Illustrated. 1230 - 200, Lunch.2 http://math.albany.edu/isama/2000/prog.html

ISAMA 2000: Program The Second Annual Conference of the International Society of the Arts, Mathematics and Architecture University at Albany Albany, New York June 24 - 28, 2000 The Program Saturday, June 24, Lecture Center 2 Welcome. Christopher D'Elia , Vice Present for Research, University at Albany Nat Friedman , Director, ISAMA Ivars Peterson Moebius Fantasies and Other Excursions Into Mathematical Art George Hart Sculpture Based on Propellorized Polyhedra Douglas Dunham A Unified Classification For Repeating Patterns Lunch Thom O'Connor Geometry and Light Stephanie Strickland Enumeration, Constraint, and Other Mathematical/Literary Delights Robert Krawczyk The Art of Spirolateral Reversals Eva Knoll and Simon Morgan Decomposing Deltahedra Douglas Peden Gridfield Form and Patterning Sunday, June 25, Lecture Center 2 Larry Kagan New Sculptural Form in Steel and Light, and a Speculative Meditation on the Margins of Dimensionality Robert Longhurst Poetry in Wood Thomas Sakoulas Antistrophe Lunch Edward Mayer Architectural Sculpture Charles Ginnever Rashomon Allen Linder Divisions of Space Susan Happersett Mathematical Meditations Hanna Bizek Non-Cubical Designs Made From Rubik's Cubes Monday, June 26, Lecture Center 2

76. Non-determinate Polynomial Time Complete Calculations Using Sythetic Bubble Emul 1992. 193223. Haas, J. General double bubble conjecture in Solved. FocusThe Newsletter of the Math. Assoc. Amer. No. 5. May/June 2000. http://452.microgeek.org/tsp/sts.html

Non-deterministic Polynomial Time Complete Calculations Using Synthetic Bubble Emulation Daniel Kluesing Leigh High School San Jose, CA USA November 13, 2002 SUMMARY ABSTRACT n cities, a Voronoi diagram is generated. Voronoi edge lengths are used to identify city poor regions of the graph and place bubble centers. Cities are associated with a minimum of one bubble and a maximum of three in compliance with Plateau's laws, and bubble arcs are flattened to lines giving a graph G (V,E) with all vertices lying at the intersection of a minimum of two edges. A start city is defined and a Hamiltonian cycle traced in a clockwise direction. The algorithm correctly predicts the location of bubbles and the cities bound to each bubble. The predicted bubbles are verifiable on a physical soap bubble array as the bubble configuration needed to solve a given graph, and the algorithm attempts to extract the shortest path from the bubble configuration. The algorithm is guaranteed to always produce a Hamiltonian cycle for a subset of G in an amount of time bounded by a fourth order polynomial and the runtime of the algorithm does not appear to be influenced by the arrangement of the cities on the graph. Due to weak edge detection methods, the algorithm excludes fringe cities from the Hamiltonian cycle for graphs of large n during the tracing phase. The number of cities included in the Hamiltonian cycle is based on the arrangement of cities in the graph. The author is working to implement more robust edge detection methods capable of including fringe cities.

77. Proving Conjectures By Use Of Interval Arithmetic - Frommer (ResearchIndex) Appel, Haken 1977 10 Every planar graph is four colorable (context) - Appel, Haken- 1977 7 The double bubble conjecture (context) - Hass, Hutchings et al. http://citeseer.nj.nec.com/516547.html

Proving Conjectures by Use of Interval Arithmetic (2001) (Make Corrections) (2 citations) Andreas Frommer Home/Search Context Related View or download: math.uniwuppertal.de/or SC0101.ps.gz Cached: PS.gz PS PDF DjVu ... Help From: math.uniwuppertal (more) Homepages: A.Frommer HPSearch (Update Links) Rate this article: (best) Comment on this article (Enter summary) Abstract: Machine interval arithmetic has become an important tool in computer assisted proofs in analysis. Usually, an interval arithmetic computation is just one of many ingredients in such a proof. The purpose of this contribution is to highlight and to summarize the role of interval arithmetic in some outstanding results obtained in computer assisted analysis. 'Outstanding' is defined through the observation that the importance of a mathematical result is at least to some extent indicated by the fact ... (Update) Context of citations to this paper: More ...deserve success. And als long as we are not willing to do it, the mathematical successes of interval methods (for a review see Frommer ) will be like in the past achieved by people outside the interval camp (Eckmann et al. 3] Feffermann Seco [4] Hales [9] Hass et

78. Guardian | Blowing Out The Bubble Reputation Hutchins of Stanford university, and Manuel Ritori and Antonio Ros of the universityof Granada, in Spain, have, they say, proved the doublebubble conjecture. http://www.guardian.co.uk/Print/0,3858,3977398,00.html

Blowing out the bubble reputation Four mathematicians have just cleaned up a long-standing conundrum set by soapy water, writes Keith Devlin Thursday March 23, 2000 The Guardian Last Saturday, at a mathematics meeting in Indiana, four mathematicians announced that they had solved a long-standing conundrum about the shape of soap bubbles. Frank Morgan, of Williams college, Michael Hutchins of Stanford university, and Manuel Ritori and Antonio Ros of the university of Granada, in Spain, have, they say, proved the double-bubble conjecture. This says that the two-chambered soap bubble you can see in your bath is the one that has the least surface area compared with any other double-bubble combination enclosing the same two volumes of air. (A computer-generated picture of this double-bubble is shown in the illustration. The conjecture describes its shape precisely.) The conjecture goes back to work of the French mathematician Plateau in 1873, although it was not formulated precisely until a student at Williams College in Massachusetts - Joel Foisy - wrote an undergraduate thesis about soap bubbles in 1991. Since thin soap films adopt the shape that uses the minimum surface area to enclose a given volume of air, the question about the possible shapes of soap bubbles is of interest to both mathematicians and physicists. The first mathematician to think about the geometry of soap bubbles was Archimedes - he of the famous bath story. He claimed that the sphere was the most efficient way to enclose a given volume, but he did not have the mathematics to prove this. That had to wait until a German mathematician called Schwarz found a proof in 1884.

79. 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.

80. Math Department Colloquia Series The double bubble Theorem, proved in 2000, says that the double soap bubble, formedwhen They proved this conjecture for the case when the image group is 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.

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