256 MASTER S PROGRAMME IN MATHEMATICAL SCIENCES 2014/15 INTRODUCTION The Master s prgramme in mathematics is stipulated t take tw years and builds n the three-year bachelr prgramme. The aim f the prgramme is t give the students a deep understanding f a specific field within mathematics r statistics, as well as practice in wrking independently. The prgramme is suitable fr students with a genuine interest fr mathematics, it being the theretical aspect f the subject r its vast ptential in applicatins. LEARNING OBJECTIVES FOR THE MASTER S PROGRAMMES The Master s prgrammes in mathematics prvide the students with an in-depth knwledge f a chsen mathematical area. The prgramme cmbines research based teaching in clse cllabratin with the civil engineering prgrammes and independent wrk with the help f mdern mathematical/statistical literature and sftware. The degree prvides the student with a strng backgrund fr future wrk in teaching and research, as well as in the public and private sectr where a slid educatin within mathematics is required. Knwledge Upn graduatin, the candidate has: - a brad cmpetence in mathematics, including mathematical analyses, algebra, numerical methds, tplgy, prbability and statistics - a slid knwledge in a chsen area within mathematics (crrespnding t the student s specializatin) - depth knwledge in a specific field f mathematics, related t active research, including ability t understand and cnvey new results f research. Skills Upn graduatin, the candidate is able t: - use frmal and stringent mathematical language in bth theretical and applied prblem slving - cnstruct, analyse and cmmunicate mathematical methds, mdels and arguments - cnduct independent research prjects and present research results bth verbally and in writing - participate in interdisciplinary teamwrk and implement relevant mathematical methds and mdels t prblem slving
257 - asses their wn adequacy, seek new surces f mathematical knwledge and t renew and develp their mathematical skills. General cmpetence Upn graduatin, the candidate is able t: - fllw the prfessinal develpment within a mathematical field and is prepared t cntinuusly strengthen their prfessinal cmpetency - make infrmed chices in frming their wn educatin WORK OPPORTUNITIES With a Master s degree in mathematics r statistics yu are qualified t wrk in a variety f interesting jbs and sectrs. Sme f the sectrs where yur cmpetence is needed include, amng thers, the il industry, medicine, finance, research and teaching, bth in the public and private sectr. One can als chse t d a PhD in mathematics r statistics. After a Master s degree in mathematics it is als pssible t take ne year f practical pedaggy studies, which allws yu t teach at secndary level schls (prvided yu als have teaching cmpetence crrespnding t 60 ECTS in a secnd subject). There is a high demand fr teachers with cmpetence in mathematics in the secndary schl, and the demand will cntinue t rise in the years t fllw. ADMISSION The general rules regulating the admissin t the Master s prgramme is described here: http://www.ntnu.n/dcuments/314472/0/degreeprgr13-14.pdf/d3b5c8da-b205-4c84-8b49-2bc083a2349a The specific criteria fr admissin t the Master s prgramme in mathematics are described belw. Bth internal and external applicants must fulfil the fllwing criteria fr admissin t the Master s prgramme: A Bachelr degree, cand.mag. degree r similar Fulfil criteria fr a specializatin (at least 80 ECTS in mathematics), where curses equivalent t the fllwing must be included: MA1101 Basic Calculus I (7.5 ECTS)
258 MA1102 Basic Calculus 2 (7.5 ECTS) MA1103 Vectr Calculus (7.5 ECTS) MA1201 Linear Algebra and Gemetry (7.5 ECTS) MA1202 Linear Algebra with Applicatins (7.5 ECTS) Fr the specializatins in Algebra, Analysis and Tplgy, additinal curses must be included: TMA4120 Calculus 4K (7.5 ECTS) MA2201 Algebra (7.5 ECTS) 2 curses in pure mathematics, where at least ne must be n an intermediate level Fr the specializatin in Applied Mathematics, additinal curses must be included: TMA4120 Calculus 4K (7.5 ECTS) MA2501 Numerical Methds (7.5 ECTS) r TMA4215 Numerical Mathematics (7.5 ECTS). The latter is specifically relevant fr this specializatin. Fr this specializatin it is als recmmended with sme knwledge in physics and infrmatics (prgramming) Fr the specializatin in Statistics at least 30 ECTS must be statistics curses, where the fllwing curses must be included: ST1101 Prbability and Statistics (7.5 ECTS) ST1201 Statistical Methds (7.5 ECTS) In additin t ne f the fllwing curses: TMA4267 Linear Statistical Mdels (7.5 ECTS) TMA4265 Stchastic Prcesses (7.5 ECTS) r a different statistics curse n the same level The applicant must have a grade average f C in the curses required fr admissin. The grade average is calculated frm bth the bligatry and ptinal curses, in ttal 11 curses. In the case where the number f qualified applicants exceeds the number f spts available, the applicants will be ranked accrding t the rules determined by the NTNU. PROGRAMME COMPONENTS The Master s prgramme in mathematics is stipulated t take tw years (120 ECTS). The prgramme cnsists f a Master s thesis f 45 ECTS and curses crrespnding t a ttal f
259 75 ECTS. The duratin f each curse is usually ne semester (7.5 ECTS) with 4 hurs f lecture a week. Depending n capacity and need, there will be arranged reading curses, seminars and cllquies in rder t aid students in their wrk with the Master s thesis. Activities and examinatin prcedures fr each curse is described n the curse s infrmatin page, and may cnsist f a final exam (ral r written), midterm exam, exercises and/r prject wrk. When the Master s thesis has been submitted and all the curses included in the degree plan have been passed, the candidate must pass a final ral examinatin based n their Master s thesis (cf. 23 in Utfyllende regler fr realfagsstudiene ). The presentatin is public and shuld last fr apprximately 30 minutes. After the presentatin, the candidate will receive their grade. The Master s prgram ffers specializatin in five different fields: Algebra, Analysis, Tplgy, Applied Mathematics and Statistics. Within each field f study the student may chse between studies in applied mathematics r pure mathematics f this field. Cmmn fr all the specializatins are requirements that the student chses certain curses that are in the given specializatin, but als that the student chses a certain amunt f curses/credits frm ther fields f mathematics in rder t achieve an in depth expertise in the field f specializatin and a wide knwledge f mathematics in general. The requirement t have a certain amunt f curses/credits utside f the chsen specializatin f the Master s prgram applies t the Bachelr prgram and Master s prgram cmbined. It s required t have fur curses relevant t the specializatin, each f 7.5 ECTS. These curses are specified fr each specializatin. The curses shuld be n a master level (higher level). In additin, the curses TMA4145 Linear Methds, TMA4212 Numerical Slutin f Differential Equatins by Difference Methds, TMA4165 Differential Equatins and Dynamical Systems, TMA4275 Lifetime Analysis, TMA4265 Stchastic Prcesses and TMA4267 Linear Statistical Methds can be included. TMA4155 Cryptgraphy, Intrductin is n a master level, but cannt be included in the Master s degree. It s pssible, and ften advantageus, t take ne r mre f the curses in the field f specializatin during the bachelr study. The requirement fr curses frm ther mathematical fields than the field f specializatin applies t the Master s prgram in itself, and states that at least tw mathematical curses n a master level (each f 7.5 ECTS) have t be utside f the specializatin. In additin the curse Experts in Team (7.5 ECTS) is mandatry and has t be taken within the first year f the master study. The remaining curses are chsen frm the nes ffered by the department f mathematical sciences (prvided they are n a master level), including curses frm the technlgy and PhD prgrams. Curses ffered by ther departments may als be included in the Master s degree prvided they are apprved by the faculty. The students shuld cntact the faculty at an early stage fr apprval f their degree plan. Curses passed befre admissin t the master prgramme can nly be included in the Master s degree upn apprval f the faculty.
260 All students must include in their degree(s) (including credits taken during the bachelr prgram) at least 30 credits chsen frm the fllwing list f curses: MA3201 Rings and mdules MA3202 Galis thery TMA4145 Linear Methds TMA4225 Fundatins f Analysis TMA4190 Maniflds MA3402 Analysis n Maniflds TMA4195 Mathematical Mdelling TMA4212 Numerical Slutin f Differential Equatins by Difference Methds TMA 4295 Statistical Inference TMA4300 Cmputer Intensive Statistical Methds The wrklad f the prgramme will usually be distributed as fllws: Year Semester 2 4 spring Thesis 3 fall Thesis Elective curse Elective curse 1 2 spring Experts in team Elective curse Elective curse Elective curse 1 fall Elective curse Elective curse Elective curse Elective curse Credits/curse 7,5 ECTS 7,5 ECTS 7,5 ECTS 7,5 ECTS Depending upn the chsen specializatin, the elective curses in the table, have t satisfy guidelines fr mandatry curses (see list fr each specializatin). Nte that in sme cases it may be advisable t start wrk n the thesis already in the secnd semester, in rder t make rm fr a curse that nly can be taken during the third semester f the prgram.
261 SPECIALIZATIONS Algebra Within the field f algebra, there are tw available specializatins, Algebraic Structures and Applied Algebra. The curses MA3201 Rings and Mdules and MA3202 Galis Thery are mandatry fr bth specializatins. Fr specializatins in algebraic structures it is als mandatry t take the curse MA3203 Ring Thery and at least ne ther curse in algebra, it is recmmended that the furth curse is MA3204 Hmlgical Algebra. Fr specializatin in applied algebra it is mandatry t take curses TMA 4185 Cding Thery and TMA4160 Cryptgraphy. It is an advantage t have a backgrund in infrmatics when chsing this specializatin. Analysis In the field f analysis there are three fields f specializatins available: Differential Equatins, Functinal Analysis and Cmplex/Harmnic Analysis. Within all these three it is als pssible t wrk with either applied prblems fr a master s degree, r prblems in pure mathematics. The curses TMA4145 Linear Methds and TMA4225 Fundatins f Analysis are bth mandatry fr all three specializatins, it is t the students advantage t d the curse TMA4145 already in the Bachelr prgram. Specializatin in differential equatins als requires the curse TMA4305 Partial Differential Equatins t be taken, and at least ne mre curse in analysis. It is recmmended fr the student t chse at least ne f the fllwing three curses: TMA4195 Mathematical Mdelling, MA8103 Nn-linear Partial Differential Equatins r TMA4170 Furier Analysis. T specialize in the field f functinal analysis it is mandatry t take the curse TMA4230 Functinal Analysis, and at least ne mre curse in analysis. Fr the third specializatin, Cmplex/Harmnic Analysis it is mandatry t take TMA4175 Cmplex Analysis and at least ne mre curse in analysis. It is recmmended t take ne f the three fllwing curses: TMA4170 Furier Analysis, TMA4195 Mathematical Mdelling and MA3105 Advanced Real Analysis. Tplgy Within tplgy it is pssible t specialize in tplgy r gemetry. Bth applied and pure prblems are pssibilities fr students wrking n their Master s thesis. Fr bth specializatins mandatry curses are TMA4190 Maniflds, MA3402 Analysis n Maniflds and MA3403 Algebraic tplgy. It is als recmmended t chse either MA3405 Algebraic Tplgy 2 r MA8402 Lie Grups and Lie Algebras, depending n specializatin.
262 The curses MA3002 General Tplgy and TMA4165 Differential Equatins and Dynamical Systems shuld be taken during the Bachelr degree. Applied mathematics Mandatry curses fr this specializatin are TMA4145Linear Methds and TMA4212 Numerical Slutin f Differential Equatins by Difference Methds. In additin t these, the student is advised t als chse at least three f the fllwing curses: TMA4165 Differential Equatins and Dynamical Systems, TMA4180 Optimizatin Thery, TMA4195 Mathematical Mdelling, TMA4220 Numerical Slutin f Partial Differential Equatins Using Element Methds, TMA4205 Numerical Linear Algebra and TMA4305 Partial Differential Equatins. It is t the students advantage if the curses TMA4145 and TMA4212 are taken during the Bachelr degree. Statistics The curses TMA4295 Statistical Inference and TMA4300 Cmputer Intensive Statistical Methds are mandatry fr the master s prgramme in statistics. These curses can als be taken as part f the bachelr degree, in rder t facilitate a gradual transitin int higher level curses. In rder t attain a master s degree in statistics, the student must have dne curses f at least 82,5 ECTS in statistics, f which the curses dne in the bachelr degree are als included. Other curses relevant t the study f statistics may als be included if preapprved fr this purpse.