Preface IX Their generalization s involvin g infinite-dimensiona l space s o f path s (or solution s o f othe r PDEs ) hav e ha d a profoun d influenc e o n 20t h century mathematics . On e her e encounter s critica l path s tha t ma y not b e globall y o r eve n locall y lengt h minimizing . Fo r example , th e "ridge trail " ove r a mountai n rang e i s a length-critica l pat h tha t i s unstable. Som e slight variatio n ma y giv e a (mor e dangerous ) pat h of shorter length . Physicists an d mathematician s hav e lon g bee n intereste d i n un - derstanding an d modelin g vibratin g strings , a s i n bowe d o r plucke d instruments. Stev e Co x discusse s i n Chapte r 3 the caus e o f th e ob - served deca y o f th e amplitude . Suc h deca y i s usuall y neglecte d i n introductory treatment s i n physic s courses . Hi s chapte r wel l illus - trates th e ful l rang e an d difficult y o f scientific inquir y fro m acquirin g experimental data , t o synthesizin g data , t o mathematica l modeling , to finding actua l o r approximat e solutions . Th e discussio n her e in - cludes a useful introductio n an d illustration o f the classical "Principl e of Stationar y Action" . The isoperimetri c proble m tha t a surfac e o f leas t are a i n spac e enclosing a single given volume must b e an ordinary round spher e was solved rigorousl y ove r 10 0 years ago . I t wa s claime d b y Archimede s and Zenodoru s i n antiquity , bu t prove d b y H . Schwar z i n 1884 . A t the Ric e undergraduat e conference , Fran k Morga n discusse d th e im - portant Double Bubble Conjecture tha t a surface o f least are a enclos - ing tw o fixed volume s consist s simpl y o f two adjoine d spherica l cap s joined b y a thir d spherica l interfac e (wit h radi i determine d b y th e given volumes) . I n 1998 , thi s conjectur e wa s prove n b y M . Hutch - ings, F. Morgan, M . Ritore, an d A . Ros. Fo r Chapte r 4 of the presen t volume, Fran k Morgan' s origina l tal k ha s bee n replace d b y a reprin t of hi s excellen t 200 1 MAA articl e exposin g thi s result . Minimal surface s occu r i n th e calculu s o f variation s a s critica l points o f the are a functiona l an d provid e model s fo r som e soap films. K. Weierstras s (1815-1897 ) showe d tha t the y als o enjoy a mathemat - ical representatio n i n term s o f complex-value d functions . Chapte r 5 by Mik e Wol f explain s carefull y thi s connectio n an d give s a related , recently discovered representatio n tha t allow s the construction o f sev- eral ric h ne w familie s o f minima l surfaces . Se e th e man y beautifu l
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