A BRIE F DESCRIPTIO N O F MAS S PROGRA M 3 fall. Thi s makes it possible to offer researc h projects tha t requir e more than 7 weeks (the lengt h o f REU program ) fo r completion . Second, th e structur e o f th e RE U progra m reflect s ou r emphasi s o n advanc e intensive learnin g a s a bridg e t o creativ e research . Mathematica l researc h usuall y includes thre e components : stud y o f the subject, solvin g o f a problem, an d presen - tation o f th e result . Thes e thre e component s ar e presen t i n th e RE U program : i n addition t o th e traditiona l individual/smal l grou p researc h project s supervise d b y faculty mentors , the program include s two short courses , a weekly seminar, an d th e MASS Fest . Th e RE U course s ar e liste d i n Appendi x 1. Mor e abou t th e students ' research project s i s said i n the introductio n t o th e respectiv e par t o f this volume . MASS Fes t i s a 3-da y conferenc e a t th e en d o f REU a t whic h th e participant s present thei r research . Thi s i s als o a MAS S alumn i reunion . Interactio n betwee n the RE U participant s (som e o f whic h ar e futur e MAS S students ) an d th e MAS S alumni i s ver y beneficial , th e latte r ofte n providin g natura l role-model s fo r th e former. Alon g wit h th e RE U students , a numbe r o f gues t speakers , mostl y Pen n State faculty , giv e expository talk s a t th e conference . MASS courses . Th e cor e course s ar e custo m designe d fo r th e progra m an d ar e available onl y t o it s participants . A brie f loo k a t th e lis t o f course s reveal s th e main feature : eac h addresse s a topic whic h i s not likel y t o b e covered i n the usua l undergraduate (and , i n man y cases , even graduate ) curriculu m bu t alway s relate d with fundamental concept s of modern mathematics . Designin g and teachin g such a course, a n instructo r i s challenged t o reach a delicate balanc e betwee n coverin g th e basics, with whic h th e student s migh t b e unfamiliar , an d introducin g a n advance d material typicall y taugh t i n topic s courses . Se e th e articl e b y G . Andrew s i n thi s volume i n which h e describe s hi s MAS S teaching experience . Core MASS courses are represented i n this volume by detailed lectur e notes "p - Adic analysis in comparison wit h real" b y S. Katok (MAS S 2000), and by material s to two other courses: "Geometri c structures, symmetry an d elements of Lie groups" by A . Kato k an d "Geometrica l method s o f mechanics " b y M . Lev i (bot h MAS S 1999). Without repeatin g S . Katok' s introductio n t o he r cours e notes , th e choic e o f the topi c i s rathe r characteristi c fo r a MAS S course . O n th e on e hand , th e sub - ject i s quite unusua l fo r a n undergraduat e mathematic s curriculum . O n th e othe r hand, p-adic analysis i s built o n the same general principles as the real analysis an d involves fundamenta l concept s o f metri c topolog y (suc h a s completio n o f metri c spaces, Archimedea n an d non-Archimedea n norm , etc) , an d o f course involves fun - damental concept s o f abstrac t algebra : fields, rings , ideals , group s o f units . Thu s the course which officially wa s listed in the "Analysis " slo t included interaction of all three main branches of core mathematics: analysis , algebra and geometry/topology . Needless t o say , there wa s an additiona l benefi t t o th e student s wh o mastered thi s course of p-adic analysis: a deeper understandin g o f its real counterpart. S . Katok' s notes contain a comprehensive exposition of the subject, alon g with numerous exer - cises, and can serve as a textbook fo r a semester course for interested an d motivate d undergraduates. The note s b y A. Kato k an d M . Levi constitute wha t on e may cal l the skeleto n of a course . The y provid e a wealt h o f material s fo r a prospectiv e instructor : a detailed cours e syllabus, a collection o f home problems, examinatio n question s an d problems, a lis t o f researc h projects , reference s t o th e literature . Th e materia l
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