4 SVETLANA KATO K an d SERG E TABACHNIKO V presented i n a flexible form , an d a professor wh o choose s t o teac h a cours e base d on thes e note s wil l hav e a n ampl e roo m fo r innovatio n an d improvisation . Unlik e the developed detaile d note s the material fo r thes e two course also give a very close portrait o f how the classe s were actually run . Summer RE U course s hav e a format , simila r t o MAS S cor e courses , bu t o n a smalle r scale : typically , suc h a cours e i s taugh t fo r tw o week s fo r th e tota l o f eight hours . RE U course s ar e represente d i n thi s volum e b y detaile d lectur e note s "Continued fractions, hyperboli c geometry and quadratic forms" b y S. Katok taugh t in summer 2001. A number of research projects by the REU participants was related to th e materia l o f this course . A fe w word s abou t th e MAS S semina r whic h ha s a variet y o f functions i n th e program. On e of them is to introduce the students to the topics that, otherwise , are likely t o "fal l betwee n crack s i n th e floor" (suc h as , fo r example , th e classica l con - figuration theorem s o f projective geometry : Pappus , Desargues , Pascal , Briancho n and Poncelet) . Anothe r purpos e o f th e semina r i s t o prepar e th e student s fo r th e upcoming MASS colloquium talks. A colloquium speaker is asked whether a certain material shoul d b e covere d i n advanc e s o that th e student s ge t th e mos t fro m th e talk. Stil l another functio n o f the seminar i s to rehearse the students 5 presentation s of the researc h project s o n the final exa m an d alon g the wa y to expos e student s t o the additiona l materia l related to the courses this usually occupies the last quarte r of th e semester . Probabl y th e mos t importan t functio n o f th e semina r i s t o brin g about element s o f unity o f mathematics. Ofte n identica l o r simila r notion s appea r in differen t course s i n variou s guises , an d th e semina r i s th e plac e t o explor e an d clarify thes e connections . Participants. MAS S participant s ar e selecte d fro m applicant s currentl y enrolle d in U S college s o r universitie s wh o ar e juniors , senior s o r sometime s sophomores . They ar e expected t o have demonstrated a sustained interes t i n mathematics an d a high level of mathematical ability . Th e required backgroun d include s a full calculu s sequence, basi c linea r algebra , th e first mathematica l cours e o n proof s (typically , discrete mathematics ) an d advance d calculu s o r basi c rea l analysis . Th e searc h o f the participant s i s nationwide . Participant s ar e selecte d base d o n thei r academi c record, recommendatio n letter s fro m facult y an d a n essay . The numbe r o f MAS S participant s varie s fro m yea r t o yea r wit h a n averag e of 15 pe r semester th e number s wer e somewha t highe r i n recen t years . A par t of the MAS S clas s consist s o f Pen n Stat e student s bu t th e majorit y ar e outsiders . It i s interestin g t o analyz e wher e the y com e from . Fo r thi s purpos e on e divide s American universitie s int o fou r categories : 1) small , mostl y libera l arts , schools 2) stat e universities , mostl y large 3 ) elit e privat e universities 4 ) Pen n State . Th e statistics ove r seve n year s of the progra m existenc e i s as follows: abou t 22 % of th e participants belon g t o th e first category , abou t 40 % to th e second , onl y 3 % to th e third an d 35 % to th e fourth . Anothe r piec e of statistics: th e percentag e o f wome n was about a quarter (wit h considerable deviations and the general tendency t o slow increase i n recen t years) . Som e Pen n Stat e student s ar e part-tim e participant s (they tak e on e o r tw o courses ) bu t a fe w o f the m participat e i n MAS S mor e tha n once. Recentl y mos t Pen n Stat e participant s cam e fro m th e elit e Schreyer s Honor s College. MAS S is beginning to play the role as the heart of developing departmenta l honors program in mathematics. A complete list of participants with their affiliatio n is given in Appendi x 3 .

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