DEPARTMENT  OF  MATHEMATICS  
 
 
 
GUIDE  TO  OPTIONAL  MODULES  
 
 
 
FOURTH/FINAL  YEAR  (MSci)  
2018-­2019  
 
 
 
Notes   and   syllabus   details   on   Fourth   Year   modules   for   students  in  their  Fourth/Final  
Year  
 
 
 
For  degree  codings:  
 
G103   MATHEMATICS  (BSc,  MSci)  
G104   MATHEMATICS  WITH  A  YEAR  ABROAD  (MSci)  
 
NOTE   that   GG41,   IG11   and   GI43   MATHEMATICS   AND   COMPUTER   SCIENCE   are   administered   by   the  
Department  of  Computing.  
 
 
 
Professor  David  Evans  
Director  of  Undergraduate  Studies  
 
 
 
4  June  2018  
 
 
 
 
 
TO  BE  READ  IN  CONJUNCTION  WITH  THE  UNDERGRADUATE  HANDBOOK.  
 
 
This  information  WILL  be  subject  to  alteration.    Updated  programmes  can  be  viewed  on  the  MathsCentral  
Blackboard  site  and  online  at:      
https://www.imperial.ac.uk/natural-­
sciences/departments/mathematics/study/students/undergraduate/programme-­information/  
 
  
2  
FOURTH  YEAR  OVERVIEW  
ADVICE  ON  THE  CHOICE  OF  OPTIONS   2  
M4R  PROJECT   2  
NON-­MATHEMATICS  MODULES   3  
GRADUATION   3  
MARKS,  YEAR  TOTALS  AND  YEAR  WEIGHTINGS   4  
ECTS   4  
MODULE  ASSESSMENT  AND  EXAMINATIONS   5  
FOURTH  YEAR  MODULE  LIST   5  
FOURTH  YEAR  MATHEMATICS  SYLLABUSES:   6  
APPLIED  MATHEMATICS/MATHEMATICAL  PHYSICS/NUMERICAL  ANALYSIS   6  
PURE  MATHEMATICS   22  
STATISTICS   30  
OTHER  “NON-­MATHEMATICAL”  MATHEMATICS  MODULES   33  
CLCC/BUSINESS  SCHOOL   34  
IMPERIAL  HORIZONS   35  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

1  
 
FOURTH  YEAR  OVERVIEW  
 
The  MSci  Fourth  Year  is  available  to  those  on  the  G103  and  G104  codings  who  perform  to  a  satisfactory  
standard  in   their  Third  Year,  here  or  abroad.   There  is  considerable  overlap  with  the  taught  postgraduate  MSc  
programmes  in  Pure  and  Applied  Mathematics,  but  the  MSci  is  a  separate  degree.  
 
The  MSci  programme  is  designed  to  provide  a  breadth  and  depth  in  mathematics  to  a  level  of  attainment  broadly  
equivalent  to  that  of  an  MSc  degree  and  takes  place  over  three  terms  –  Term  1  (also  known  as  Autumn  Term),  
Term  2  (also  known  as  Spring  Term)  and  Term  3  (also  known  as  Summer  Term).  
 
Students   choose   six  lectured   modules  from  those   made   available   to   them   in   the   Department   and   from  
certain   modules  elsewhere.  Students  also  take  the  compulsory  M4R  project,  which  is  equivalent  to  two  
lecture  modules.  
 
Most,  but  not  all,  of  the  M4  modules  are  also  available  in  M3  form  and  4th  year  students  take  the  M4  version.  
Fourth  Year  examinations  normally  consist  of  5  questions  and  are  2.5  hours  long,  whereas  the  
corresponding  exams  for  3 rd  year  students  (if  any)  contain  4  questions  in  2  hours.    Students  may  not  
take  an  M4  module  if  they  have   already  taken  the  M3  version.  
 
Lecturing   will   take   place   during   Term   1   and   Term   2   with   three   hours   per   week,   which   usually   includes   some  
classes.   The  normal  expectation  is  that  there  should  be  a  'lecture'/'class'  balance  of  about  5/1.   The  identification  
of  particular  class  times  within  the  timetabled  periods  is  at  the  discretion  of  the  lecturer,  in  consultation  with  the  
class  and  as  appropriate  for  the  module  material.  
 
 
ADVICE  ON  THE  CHOICE  OF  OPTIONS  
 
Students   are   advised   to   read   these   notes   carefully   and   to   discuss   their   option   selections   with   their   Personal  
Tutor.  An  `Options  Fair’  will  take  place  after  exams  in  the  Summer  Term,  where  staff  will  answer  questions  on  all  
available  options.  Some  staff  from  the  Pure,  Statistics  and  AMMP  sections  will  hold  ‘office  hours’  towards  the  
end  of  the  3rd  term,  during  which  they  can  be  consulted  about  optional  modules.  
 
It   is   anticipated   that   lecturers   will   give   advice   on   suitable   books   at   the   start   of   each   module.   Students   should  
contact   the   proposed   lecturers   if   they   desire   any   further   details   about   module   content   in   order   to   make  
their   choice  of  course  options.   Students  should  also  feel  free  to  seek  advice  from  Year  Level  Tutors  and  the  
Senior   Tutor,  and  the  Director  of  Undergraduate  Studies.  
 
You  will  not  be  committed  to  your  choice  of  most  optional  modules  until  the  completion  of  your  examination  entry  
at  the  beginning  of  Term  2.  The  exception  to  this  is  that  students  do  become  committed  to  the  completion  of  
certain  modules  examined  only  by  project  at  some  stage  during  the  module,  as  will  be  made  clear  by  the  
lecturer.  
 
M4R  PROJECT  
 
M4R                ADVANCED  RESEARCH  PROJECT  IN  MATHEMATICS  
Compulsory  
 
Supervised  by  Various  Academic  Staff  
Co-­ordinator:   Dr  J.  Britnell  
(Terms  1,  2  &  3)  
 
A  fundamental  part  of  the  MSci  degree  is  a  substantial  compulsory  project  equivalent  to  two  lecture  modules.  
The   main   aim   of   this   module   is   to   give   a   deep   understanding   of   a   particular   area/topic   by   means   of   a  
supervised  project  in  some  area  of  mathematics.   The  project  may  be  theoretical  and/or  computational  and  the  
area/topic  for   each  student  is  chosen  in  consultation  with  the  Department.  
 
The  project  provides  an  excellent  ‘apprenticeship  in  research’  and  is  therefore  of  particular  value  to  students  who  

2  
 
are  considering  postgraduate  study  leading  to  a  PhD.  
 
Arrangements  for  this  project  will  be  set  in  motion  after  the  Third  Year  examinations.    Students  should  
approach  potential  supervisors  in  an  area  of  interest  before  the  end  of  their  Third  Year  and  some  
preparatory  work  should  be  performed  over  the  vacation  between  the  Third  and  Fourth  Years.    Work  on  the  
project  should  continue  throughout  all  three  terms  of  the  Fourth  Year  and  submitted  shortly  after  the  Fourth  Year  
examinations.    
 
G104:  For  those  on  a  Maths  with  a  year  abroad  coding,  the  third  year  is  spent  abroad  at  another  university.  
G104  students  should  ideally  negotiate  with  possible  M4R  supervisors  by  e-­mail  during  their  abroad,  but  this  is  
not  always  possible.  On  return  to  Imperial,  students  take  the  regular  Year  4  MSci  programme.  On  the  rare  
occasion  that  a  G104  student  performs  very  poorly  in  their  year  away  they  may,  at  the  discretion  of  the  Senior  
Tutor,  be  transferred  to  the  BSc  G100  Mathematics  Degree  and  take  M3  subjects  in  their  Final  Year.  
 
 
NON-­MATHEMATICS  MODULES  
 
The  Department  offers  a  few  options  which  are  deemed  to  be  ‘less-­Mathematical’.  
 
For  2018-­19,  these  options  are:  
 
M3C  High  Performance  Computing  
M3B  Mathematics  of  Business  
M3T  Communicating  Mathematics  (for  returning  G104  students  only)  
 
There  is  also  an  approved  list  of  Centre  for  Co-­Curricular  Studies/Business  School  non-­Mathematical  options  
which  may  be  taken  by  Mathematics  students  (see  later  in  this  guide).  
 
MSci  students  may  take  at  most  one  option  from  the  combined  list  of  ‘less  Mathematical’  and  CCS/  Business  
School  options  in  their  Fourth  Year.  
 
Subject  to  the  Department’s  approval,  students  may  take  a  module  given  outside  the  Department,  e.g.  in  the  
Departments  of  Physics  or  Computing.  Students  must  obtain  permission  from  the  Director  of  Undergraduate  
Studies  if  they  wish  to  consider  such  an  option.  The  DUGS  will  determine  whether  the  module  can  be  
substituted  for  a  Mathematics  option,  or  whether  it  will  count  as  one  of  the  less  (or  non-­)  Mathematical  options.  
 
 
.  
GRADUATION    
 
Students  graduating  will  receive  an  MSci  degree  that  explicitly  incorporates  a  BSc.  
 
It  is  normally  required  that  MSci  students  pass  all  course  components  in  order  to  graduate.    However,  the  
College  may  compensate  a  narrowly  failed  module  in  the  Final  Year  of  study.  The  Examination  Board  may  also  
graduate  students  who  have  one  or  more  badly  failed  module,  provided  the  overall  average  mark  is  high  
enough.      
   
The  total  of  marks  for  examinations,  assessed  coursework,  progress  tests,  assignments  and  projects,  with  the  
appropriate  year  weightings,  is  calculated  and  recommendations  are  made  to  the  Examiners’  Meeting  (normally  
held  at  the  end  of  June)  for  consideration  by  the  Academic  Staff  and  External  Examiners.    Degrees  are  formally  
decided  at  this  meeting.  
 
Students  at  graduation  may  be  awarded  Honours  degrees  classified  as  follows:    First,  Second  (upper  and  lower  
divisions)  and  Third,  with  a  good  Final  Year  and  project  being  viewed  favourably  by  the  External  Examiners  for  
borderline  cases.  
 
Rarely,  circumstances  may  require  the  Department  to  graduate  an  MSci  student  with  a  BSc.  
 

3  
 
Further  information  on  degree  classes  can  be  found  in  the  Scheme  for  the  Award  of  Honours  online  at:  
https://www.imperial.ac.uk/natural-­
sciences/departments/mathematics/study/students/undergraduate/programme-­information/  
 
 
In  general,  applications  for  postponement  of  consideration  for  Honours  will  NOT  be  granted  by  the  Department  
except  in  special  cases,  such  as  absence  through  illness.  
 
Information  about  Commemoration  (Graduation)  ceremonies  can  be  found  online  at:  
http://www3.imperial.ac.uk/graduation  
 
MARKS,  YEAR  TOTALS  AND  YEAR  WEIGHTINGS  
 
What  follows  is  a  brief  summary  –  more  details  of  these  topics  can  be  found  online  at:    
https://www.imperial.ac.uk/natural-­
sciences/departments/mathematics/study/students/undergraduate/programme-­information/  
(information  for  2018-­2019  will  be  updated  over  the  summer  of  2018).  
 
Within  the  Department  each  total  module  assessment  is  rescaled  so  that  overall  performances  in  different  
modules  may  be  compared.  From  2017-­18,  all  marks  will  be  computed  on  the  College  scale,  rather  than  the  
Mathematics  scale  used  in  previous  years.  The  rescaling  onto  the  scale  0  –  100  marks  is  such  that  50  then  
corresponds  to  the  lowest  Pass  Honours  mark  for  a  Masters  level  module  and  70  corresponds  to  the  lowest  First  
Class  performance.  
 
Marks  from  the  modules  taken  in  the  fourth  year  are  combined  into  a  year  total  expressed  as  a  percentage.    
 
Further  information  can  found  in  the  Scheme  for  the  Award  of  Honours.  
 
For  the  four  year  MSci  codings  G103  and  G104  the  year  weightings  are  1  :  3  :  4  :  5.  
 
(For  G104  students  who  first  enrol  in  the  Department  from  2017-­18  onwards,    
the  year  weightings  will  be  1  :  3:  3  :  5.)  
 
The  differences  in  year  weighting  reflect  the  increasing  level  of  mathematical  complexity.  
 
ECTS  
 
To  comply  with  the  European  ‘Bologna  Process’,  degree  programmes  are  required  to  be  rated  via  the  ECTS  
(European  Credit  Transfer  System)  –  which  is  based  notionally  on  hour  counts  for  elements  within  the  degree.  In  
principle,  1  ECTS  should  equate  to  around  25  hours  of  study  (including  examinations  and  private  study).  
 
 
As   in   Third   Year,   each   Fourth   Year   mathematics   module,   including   M3B,  M3T,  M3C  and  other  mathematical  
optional  modules,   has  an  ECTS  value  of  8  except  for  M4R  which  has  an  ECTS  value  of  16.  Centre  for  Co-­
Curricular   Studies/Business   School   modules   have   an   ECTS   value   of   6.   Each   Second   Year   mathematics  
module   has   an   ECTS  value  of  7  with  M2R  having  an  ECTS  value  of  5.    First  Year  mathematics  modules  have  
an  ECTS  value  of  6.5  except  for  M1R  which  has  an  ECTS  value  of  4.5  and  M1C  which  has  an  ECTS  value  of  
4.  Language  modules,  taken  by  G104  Mathematics  with  a  Year  in  Europe  students,  have  an  ECTS  value  of  6.  
   
MSci  students  who  wish  to  increase  their  ECTS  counts  from  roughly  240  to  270  must  undertake  
additional  study  over  the  summer  vacations  of  their  Second  and  Third  Years.    Contact  the  Director  of  
Undergraduate  Studies  for  further  information.  
 
 
 
Details  can  be  viewed  online  at:  
http://www.imperial.ac.uk/natural-­sciences/departments/mathematics/study/students/undergraduate/programme-­

4  
 
information/    
 
MODULE  ASSESSMENT  AND  EXAMINATIONS  
 
Most  M4  modules  are  examined  by  one  written  examination  of    2.5  hours  in  length.  Written  examinations  for  M3  
modules  are  2  hours  in  length.  
 
Some  of  the  modules  may  have  an  assessed  coursework/progress  test  element,  limited  in  most  cases  to  10%  of  
overall  module  assessment.    Some  modules  have  a  more  substantial  coursework  component  (for  example,  25  
percent)  and  others  are  assessed  entirely  by  coursework.  Details  can  be  found  in  the  tables  below.  Precise  
details  of  the  number  and  nature  of  coursework  assignments  will  be  provided  at  the  start  of  each  module.  
 
Students  should  bear  in  mind  that  single-­term  modules  assessed  by  projects  usually  require  extra  time-­
commitment  during  that  term.  Thus,  the  Department  generally  advises  that  students  should  not  take  more  than  
one  such  module  in  a  term.  Students  wishing  to  take  more  than  one  such  module  in  a  term  will  be  required  to  
discuss  this  with  the  Senior  Tutor.  
 
 
The  module  M4R  is  examined  by  a  research  project;;  an  oral  element  forms  part  of  the  assessment.  
 
See  module  description  for  assessment  of  the  M4S18++  modules.  
 
The  module  M3T  is  examined  by  a  journal  of  teaching  activity,  teacher’s  assessment,  oral  presentation,  and  end  
of  module  report.  
 
Note:    Students  who  take  modules  which  are  wholly  assessed  by  project  will  be  deemed  to  be  officially  
registered  on  the  module  through  the  submission  of  a  specified  number  of  pieces  of  assessed  work  for  
that  module.    Thus,  once  a  certain  point  is  reached  in  these  modules,  a  student  will  be  committed  to  
completing  it.  In  contrast,  students  only  become  committed  to  modules  with  summer  examinations  
when  they  enter  for  the  examinations  in  February.  
 
Students  who  do  not  obtain  Passes  in  examinations  at  the  first  attempt  may  be  expected  to  attend  resit  
examinations   the   following   May/June   (NOT   normally  in  September)  spending  a  year  not  in  attendance.   Two  
resit  attempts  are  normally  available  to  students.  However,  the  Examinations  Board  has  the  power  to  
compensate    not-­too-­serious  fails  in  final  year  modules  and  permit  graduation.  Note  that  it  is  very  rare  for  a  4th  
year  student  to  fail  any  module,  because  of  the  high  selection  standards  for  the  MSci.  
 
Note:   Resits  may  not  be  offered  for  modules  assessed  solely  by  project.  
 
Resit   examinations   are   for   Pass   credit   only   –   a   maximum   mark   of   the  pass  mark  (50  percent  for  
Masters  level  modules)   will   be   credited.   Once   a   Pass   is   achieved,  no  further  attempts  are  permitted.  
 
 
FOURTH  YEAR  MODULE  LIST  
 
Note   that   not   all   of   the   individual   modules   listed   below   are   offered   every   session   and   the  
Department   reserves   the   right   to   cancel   a   particular   module   if,   for   example,   the   number   of   students  
attending   that   module  does  not  make  it  viable.   Similarly,  some  modules  are  occasionally  run  as  
‘Reading/Seminar  Courses’.  
 
Modules  marked  below  with  a  *  are  also  available  in  M3  form  for  Third  Year  undergraduates  students  (who  
typically  take  a  shorter  examination).    When  a  module  is  offered  it  is  usually,  but  not  always,  available  in  both  
forms.    No  student  may  take  both  the  M3  and  M4  forms  of  a  module.    In  the  rare  event  that  the  M4  version  
of  a  module  is  not  available,  the  Department  may  permit   one  M3  module  to  be  taken.  
 
M3B  and  M3C  are  also  available  to  Fourth  Year  students  but  function  like  a  Centre  for  Co-­Curricular  
Studies/Business  School  option,  except  that  their  ECTS  value  is  8.  The  module  M3T  may  only  be  taken  in  year  
4  by  returning  G104  students.  

5  
 
  
All  M3  and  M4  modules  except  M4R  are  equally  weighted and are worth 8  ECTS  points  unless  otherwise  
specified.  The  M4R  project  is  double-­weighted  and  is  worth  16  ECTS  points.  The  module  M4R18++  is  weighted  
the  same  as  a  standard  M4  module  but  is  worth  10  ECTS  and  students  require  permission  from  DUGS  to  take  
this  module.  
 
In  the  tables  below:  
 
Column  on  %  Exam  –  this  indicates  a  standard  closed-­book  written  exam,  unless  otherwise  indicated.  
 
Column  on  %  CW  –  this  indicates  any  coursework  that  is  completed  for  the  module.  This  may  include  in-­class  
tests,  projects,  or  problem  sets  to  be  turned  in.  
 
The  groupings  of  modules  below  have  been  organised  to  indicate  some  natural  affinities  and  connections.  
 
APPLIED  MATHEMATICS/MATHEMATICAL  PHYSICS/NUMERICAL  ANALYSIS  

FLUIDS  
Module   Lecturer   %  exam   %  CW  
Module  Titles   Terms  
Codes    
M4A2*   Fluid  Dynamics  1     1   Professor  A.  Ruban   100   0  
M4A10*   Fluid  Dynamics  2     2   Professor  A.  Ruban   100   0  
Introduction  to  Geophysical  Fluid   Dr  P.  Berloff   100   0  
M4A28*   2  
Dynamics  
M4A30   Hydrodynamic  Stability   2   Dr  M.S.  Mughal   90   10  
M4A32   Vortex  Dynamics   2   Professor  D.  Crowdy   100   0  
M4M7*   Asymptotic  Analysis   1   Professor  X.  Wu   100   0  
DYNAMICS  
Professor  S.  van   40  (Oral)   60  
M4PA48*   Dynamics  of  Games     1  
Strien  
M4PA23*   Dynamical  Systems   1   Professor  J.  Lamb   100   0  
M4PA24*   Bifurcation  Theory   2   Professor  D.  Turaev   100   0  
M4PA38   Advanced  Dynamical  Systems   2   Professor  J.  Lamb   40  (Oral)   60  
Random  Dynamical  Systems  and   Professor  J.  Lamb   40   60  
M4PA40   2  
Ergodic  Theory:  Seminar  Course   (oral)  
M4PA16*   Geometric  Mechanics   1   Professor  D.  Holm   90   10  
Dynamics,  Symmetry  and   Professor  D.  Holm   90   10  
M4PA34*   2  
Integrability  
FINANCE  
Mathematical  Finance:    An   Dr  P.  Siorpaes   90   10  
M4F22*   1  
Introduction  to  Option  Pricing  
BIOLOGY  
M4A49*   Mathematical  Biology   1   Dr  N.  Jones     90   10  
M4A50*   Methods  for  Data  Science   1   Dr  D.  Oyarzun   0   100  
MATHEMATICAL  PHYSICS  
Mathematical  Physics  1:    Quantum   Dr  E-­M  Graefe   90   10  
M4A4*   1  
Mechanics  
Special  Relativity  and   Dr  G.  Pruessner   90   10  
M4A6*   1  
Electromagnetism  
Tensor  Calculus  and  General   Dr  C.  Ford   90   10  
M4A7*   2  
Relativity  
M4A29*   Theory  of  Complex  Systems   2   Professor  H.  Jensen   0   100  
M4A52*   Quantum  Mechanics  II   2   Dr  R.  Barnett   90   10  
M4M6*   Methods  of  Mathematical  Physics   1   Dr  S.  Olver   90   10  
 
 

6  
 
  
APPLIED  PDEs,  NUMERICAL  ANALYSIS  and  COMPUTATION  
M4A51   Stochastic  Differential  Equations   1   Dr  M.  Coti  Zelati   100   0  
Introduction  to  Partial  Differential   Dr  M.  Delgadino   90   10  
M4M3*   1  
Equations    
Professor  Pierre   90   10  
M4M11*   Function  Spaces  and  Applications   1  
Degond  
Advanced  Topics  in  Partial   Professor  P.  Degond   90   10  
M4M12*   2  
Differential  equations  
Finite  Elements:  Numerical   Dr  C.  Cotter  &  Dr  D.   50   50  
M4A47*   2  
Analysis  and  Implementation   Ham  
Numerical  Solution  of  Ordinary   Dr  I.  Shevchenko   0   100  
M4N7*   1  
Differential  Equations  
M4N9*   Computational  Linear  Algebra   1   Dr  E.  Keaveny   0   100  
Computational  Partial  Differential   Professor  J.  Mestel   0   100  
M4N10*   2  
Equations    
M4SC*   Scientific  Computation   2   Dr  P.  Ray   0   100  
 
PURE  MATHEMATICS  
 
Module   Lecturer   %  exam   %  CW  
Module  Titles   Terms  
Codes    
Not  for   Various   -­   -­  
Pure  Mathematics  Study  Group   1+2  
Credit  
 
ANALYSIS  
Module   Lecturer   %  exam   %  CW  
Module  Titles   Terms  
Codes    
Professor  B.   100   0  
M4P6*   Probability   2  
Zegarlinski  
Professor  B.   90   10  
M4P7*   Functional  Analysis   2  
Zegarlinski  
Fourier  Analysis  and  Theory  of   Dr  S.  Boegli   90   10  
M4P18*   2  
Distributions  
M4P19*   Measure  and  Integration   1   Dr  I.  Krasovsky   90   10  
M4P60*   Geometric  Complex  Analysis   2   Dr  A.  De  Zotti   90   10  
Analytic  Methods  in  Partial   Dr  G.  Holzegel   90   10  
M4P41   2  
Differential  Equations  
Stochastic  Calculus  with   Professor  D.  Crisan     100   0  
M4P67   1  
Applications  to  non-­Linear  Filtering  
M4P70*   Markov  Processes   1   Professor  X-­M.Li   90   10  
GEOMETRY  
M4P5*   Geometry  of  Curves  and  Surfaces   1   Professor  T.  Coates   90   10  
M4P20*   Geometry  1:    Algebraic  Curves   1   Dr  M.  Talpo   90   10  
M4P21*   Geometry  2:    Algebraic  Topology   2   Dr  C.  Urech   90   10  
Dr  G.  Fernandez  da   90   10  
M4P33   Algebraic  Geometry   2  
Silva  
M4P51   Riemannian  Geometry   2   Dr  H.  Argüz   90   10  
M4P52   Manifolds   1   Dr  S.  Feyzbakhsh   90   10  
M4P54   Differential  Topology   2   Dr  S.A.  Filippini   90   10  
M4P57   Complex  Manifolds   2   Dr  C.  Spicer   90   10  
 
 
ALGEBRA  AND  DISCRETE  MATHEMATICS    
M4P8*   Algebra  3   1   Dr  D.  Helm   90   10  

7  
 
M4P10*   Group  Theory   1   Professor  A.  Ivanov   90   10  
M4P11*   Galois  Theory   2   Professor  A  Corti   90   10  
M4P12*   Group  Representation  Theory   2   Dr  T.  Schedler     90   10  
M4P17*   Algebraic  Combinatorics   1   Dr  J.  Fawcett   90   10  
M4P46   Lie  Algebras   2   Dr  A.  Pal   90   10  
Professor  A.   90   10  
M4P55   Commutative  Algebra   1  
Skorobogatov  
M4P61   Infinite  Groups   1   Dr  J.  Britnell     90   10  
M4P63   Algebra  4   2   Dr  N.  Arbesfeld   90   10  
M4P65*   Mathematical  Logic   1   Professor  D.  Evans   90   10  
M4P72   Modular  Representation  Theory   2   Professor  M.  Liebeck   90   10  
NUMBER  THEORY  
M4P14*   Number  Theory   1   Professor  T.  Gee   100   0  
M4P15*   Algebraic  Number  Theory   2   Dr  A.  Caraiani   90   10  
M4P32   Number  Theory:    Elliptic  Curves   1   Professor  T.  Gee   90   10  
M4P58   Modular  Forms   1   Dr  D.  Helm   100   0  
STATISTICS  
M4S1*   Statistical  Theory     2   TBC   90   10  
M4S2*   Statistical  Modelling  2   2   Dr  C.  Hallsworth   75   25  
M4S4*   Applied  Probability   1   Dr  A.  Abu-­Khazneh   90     10  
M4S8*   Time  Series   1   Dr  E.  Cohen   90   10  
M4S9*   Stochastic  Simulation   1   Dr  S.  Virtanen   75   25  
Survival  Models  and  Actuarial   Professor  A.  Gandy   90   10  
M4S14*   2  
Applications  
M3S16*   Credit  Scoring     1   Dr  A.  Bellotti   75   25  
Quantitative  Methods  in  Retail   Dr  A.  Bellotti   75   25  
M3S17*   2  
Finance  
Topics  in  Advanced  Statistics        
(choose  one  of  each  of  the  A/B  
M4S18   2  
options  below;;  requires  
permission  from  DUGS)  
M4S18A1   Dr  G.  Priyadarshini     100   0  
Multivariate  Analysis   2  
(5  ECTS)    
M4S18A2     Dr  B.  Calderhead     0   100  
Machine  Learning   2  
(5  ECTS)  
M4S18B1   Dr  T.  Bedhiafi     100   0  
Graphical  Models   2  
(5  ECTS)    
M4S18B2   Dr  D.  Mortlock   80   20  
Bayesian  Methods   2  
(5  ECTS)    
 
PROJECT  (Compulsory)  
 
Module   Coordinator   %  exam   %  CW  
Module  Title   Terms  
Codes    
M4R   Research  Project  in  Mathematics   1,  2  +  3   Dr  J.  Britnell   0   100  
 
 
OTHER  MATHEMATICAL  OPTIONS  
 
Module   Lecturer   %  exam   %  CW  
Module  Titles   Terms  
Codes    
2  +  3   Professor  E.J.   0   100  
M3T   Communicating  Mathematics  
McCoy,  Dr  L.V.  White    
Mathematics  of  Business  &   1   Dr  T.  Fissler   90   10  
M3B  
Economics  
M3C   High  Performance  Computing   1   Dr  P.  Ray   0   100  

8  
 
  
 
FOURTH  YEAR  MATHEMATICS  SYLLABUSES    
 
Most  modules  running  in  2018-­2019  will  also  be  available  in  2019-­2020,  although  there  can  be  no  absolute  
guarantees.      
 
APPLIED  MATHEMATICS/MATHEMATICAL  PHYSICS/NUMERICAL  ANALYSIS  
 
FLUIDS  
 
M4A2*   FLUID  DYNAMICS  1    
 
Term  1  
 
This  module  is  an  introduction  to  the  Fluid  Dynamics.    It  will  be  followed  by  Fluid  Dynamics  2  in  Term  2.  
 
Fluid  Dynamics  deals  with  the  motion  of  liquids  and  gases.    Being  a  subdivision  of  Continuum  Mechanics  the  
fluid  dynamics  does  not  deal  with  individual  molecules.    Instead  an  ‘averaged’  motion  of  the  medium  is  of  
interest.    Fluid  dynamics  is  aimed  at  predicting  the  velocity,  pressure  and  temperature  fields  in  flows  past  rigid  
bodies.    A  theoretician  achieves  this  goal  by  solving  the  governing  Navier-­Stokes  equations.    In  this  module  a  
derivation  of  the  Navier-­Stokes  equations  will  be  presented,  followed  by  description  of  various  techniques  to  
simplify  and  solve  the  equation  with  the  purpose  of  describing  the  motion  of  fluids  at  different  conditions.  
 
Aims  of  this  module:  
To  introduce  students  to  fundamental  concepts  and  notions  used  in  fluid  dynamics.    To  demonstrate  how  the  
governing  equations  of  fluid  motion  are  deduced,  paying  attention  to  the  restriction  on  their  applicability  to  real  
flows.    Then  a  class  of  exact  solutions  to  the  Navier-­Stokes  equations  will  be  presented.    This  will  follow  by  a  
discussion  of  possible  simplifications  of  the  Navier-­Stokes  equations.  The  main  attention  will  be  a  wide  class  of  
flows  that  may  be  treated  as  inviscid.    To  this  category  belong,  for  example,  aerodynamic  flows.    Students  will  
be  introduced  to  theoretical  methods  to  calculate  inviscid  flows  past  aerofoils  and  other  aerodynamic  bodies.    
They  will  be  shown  how  the  lift  force  produced  by  an  aircraft  wing  may  be  calculated.  
 
Content:  
Introduction:    The  continuum  hypothesis.    Knudsen  number.    The  notion  of  fluid  particle.    Kinematics  of  the  flow  
field.    Lagrangian  and  Eulerian  variables.    Streamlines  and  pathlines.    Vorticity  and  circulation.    The  continuity  
equation.    Streamfunction  and  calculation  of  the  mass  flux  in  2D  flows.  
Governing  Equations:    First  Helmholtz’s  theorem.    Constitutive  equation.    The  Navier-­Stokes  equations.  
Exact  Solutions  of  the  Navier-­Stokes  Equations:    Couette  and  Poiseuille  flows.    The  flow  between  two  coaxial  
cylinders.    The  flow  above  an  impulsively  started  plate.    Diffusion  of  a  potential  vortex.  
Inviscid  Flow  Theory:    Integrals  of  motion.    Kelvin’s  circulation  theorem.    Potential  flows.    Bernoulli’s  equation.    
Cauchy-­Bernoulli  integral  for  unsteady  flows.    Two-­dimensional  flows.    Complex  potential.    Vortex,  source,  
dipole  and  the  flow  past  a  circular  cylinder.    Adjoint  mass.    Conformal  mapping.    Joukovskii  transformation.    
Flows  past  aerofoils.    Lift  force.    The  theory  of  separated  flows.    Kirchhoff  and  Chaplygin  models.  
 
 
M4A10*     FLUID  DYNAMICS  2    
 
Term  2  
   
Prerequisites:    Fluid  Dynamics  2  is  a  continuation  of  the  module  Fluid  Dynamics  1  given  in  Term  1.  
 
In  Fluid  Dynamics  1  the  main  attention  was  with  exact  solutions  of  the  Navier-­Stokes  equations  governing  
viscous  fluid  motion.    The  exact  solutions  are  only  possible  in  a  limited  number  of  situations  when  the  shape  of  
the  body  is  rather  simple.    A  traditional  way  of  dealing  with  more  realistic  shapes  (like  aircraft  wings)  is  to  seek  
possible  simplifications  in  the  Navier-­Stokes  formulation.    We  shall  start  with  the  case  when  the  internal  viscosity  
of  the  fluid  is  very  large,  and  the  Navier-­Stokes  equations  may  be  substituted  by  the  Stokes  equations.    The  
latter  are  linear  and  allow  for  simple  solutions  in  various  situations.    Then  we  shall  consider  the  opposite  limit  of  

9  
 
very  small  viscosity,  which  is  characteristic,  for  example,  of  aerodynamic  flows.    In  this  cast  the  analysis  of  the  
flow  past  a  rigid  body  (say,  an  aircraft  wing)  requires  Prandtl’s  boundary-­layer  equations  to  be  solved.    These  
equations  are  parabolic,  and  in  many  situations  may  be  reduced  to  ordinary  differential  equations.    Solving  the  
Prandtl  equations  allows  us  to  calculate  the  viscous  drag  experienced  by  the  bodies.    The  final  part  of  the  
module  will  be  devoted  to  the  theory  of  separation  of  the  boundary  layer,  known  as  Triple-­Deck  theory.  
 
Aims  of  the  module:  
To  introduce  the  students  to  various  aspects  of  Viscous  Fluid  Dynamics,  and  to  demonstrate  the  power  (and  
beauty)  of  modern  mathematical  methods  employed  when  analysing  fluid  flows.    This  includes  the  Method  of  
Matched  Asymptotic  Expansions,  which  was  put  forward  by  Prandtl  for  the  purpose  of  mathematical  description  
of  flows  with  small  viscosity.    Now  this  method  is  used  in  all  branches  of  applied  mathematics.  
 
Content:  
Dynamic  and  Geometric  Similarity  of  fluid  flows.    Reynolds  Number  and  Strouhal  Number.  
Fluid  Flows  at  Low  Values  of  The  Reynolds  Number:    Stokes  equations.    Stokes  flow  past  a  sphere.    Stokes  
flow  past  a  circular  cylinder.    Skokes  paradox.  
Large  Reynolds  Number  Flows:    the  notion  of  singular  perturbations.    Method  of  matched  asymptotic  
expansions.    Prandtl’s  boundary-­layer  equations.    Prandtl’s  hierarchical  concept.    Displacement  thickness  of  the  
boundary  layer  and  its  influence  on  the  flow  outside  the  boundary  layer.  
Self-­Similar  Solutions  of  the  Boundary-­Layer  Equations:    Blasius  solution  for  the  boundary  layer  on  a  flat  plate  
surface.    Falkner-­Skan  solutions  for  the  flow  past  a  wedge.    Schlichting’s  jet  solution.    Tollmien’s  far  field  
solution.    Viscous  drag  of  a  body.    Shear  layers.    Prandtl  transposition  theorem.  
Triple-­Deck  Theory:    The  notion  of  boundary-­layer  separation.    Formulation  of  the  triple-­deck  equations  for  a  
flow  past  a  corner.    Solution  of  the  linearised  problem  (small  corner  angle  case).  
 
 
M4A28*   INTRODUCTION  TO  GEOPHYSICAL  FLUID  DYNAMICS  
 
Term  2  
 
This   is   an   advanced-­level   fluid-­dynamics   course   with   geophysical   flavours.   The   lectures   target   upper-­level  
undergraduate   and   graduate   students   interested   in   the   mathematics   of   planet   Earth,   and   in   the   variety   of  
motions  and  phenomena  occurring  in  planetary  atmospheres  and  oceans.  The  lectures  are  a  mix  of  theory  and  
applications.  Take  a  look  at  the  lecture  notes  to  get  some  idea  of  the  material:  
http://wwwf.imperial.ac.uk/˜pberloff/gfd  lectures.html  
 
Main  topics  
•  Introduction  and  basics;;  
•  Governing  equations  (continuity  of  mass,  material  tracer,  momentum  equations,  equation  of  state,  
thermodynamic  equation,  spherical  coordinates,  basic  approximations);;  
•  Geostrophic  dynamics  (shallow-­water  model,  potential  vorticity  conservation  law,  Rossby  number  expansion,  
geostrophic  and  hydrostatic  balances,  ageostrophic  continuity,  vorticity  equation);;  
•  Quasigeostrophic  theory  (two-­layer  model,  potential  vorticity  conservation,  continuous  stratification,  planetary  
geostrophy);;  
•  Ekman  layers  (boundary-­layer  analysis,  Ekman  pumping);;  
•  Rossby  waves  (general  properties  of  waves,  physical  mechanism,  energetics,  reflections,  mean-­flow  effect,  
two  layer  and  continuously  stratified  models);;  
•  Hydrodynamic  instabilities  (barotropic  and  baroclinic  instabilities,  necessary  conditions,  physical  mechanisms,  
energy  conversions,  Eady  and  Phillips  models);;  
•  Ageostrophic  motions  (linearized  shallow-­water  model,  Poincare  and  Kelvin  waves,  equatorial  waves,  ENSO  
“delayed  oscillator”,  geostrophic  adjustment,  deep-­water  and  stratified  gravity  waves);;  
•  Transport  phenomena  (Stokes  drift,  turbulent  diffusion);;  
•  Nonlinear  dynamics  and  wave-­mean  flow  interactions  (closure  problem  and  eddy  parameterization,  triad  
interactions,  Reynolds  decomposition,  integrals  of  motion,  enstrophy  equations,  classical  3D  turbulence,  2D  
turbulence,  transformed  Eulerian  mean,  Eliassen-­Palm  flux).  
 
Suggested  textbooks:  Introduction  to  geophysical  fluid  dynamics  (Cushman-­Roisin);;  Atmospheric  and  oceanic  
fluid  dynamics  (Vallis);;  Geophysical  fluid  dynamics  (Pedlosky);;  Fundamentals  of  geophysical  fluid  dynamics  

10  
 
(McWilliams).  
Prerequisites:  Introductory  fluid  mechanics.  
 
 
M4A30         HYDRODYNAMIC  STABILITY  
 
Term  2  
This  course  is  an  introduction  to  the  basic  concepts  and  techniques  of  modern  hydrodynamic  stability  theory.  
Many  physical  systems  can  become  unstable,  in  that  small  disturbances  to  the  basic  state  can  amplify  and  
significantly  alter  the  initial  state.  The  course  introduces  the  basic  theoretical  concepts  and  analysis  methods  
required  to  understand  and  predict  hydrodynamic  instabilities  in  a  variety  of  fluid  flows  ranging  from  thin  layers  
heated  from  below,  to  flow  between  rotating  cylinders  and  boundary  layers.  The  general  concepts  apply  in  many  
disparate  fields  a  few  of  which  are:  oceanography,  meteorology,  physics,  astrophysics,  aerodynamics,  
combustion,  laminar  to  turbulent  transition,  sand  dune  formation,  river  meandering.  

Classical  as  well  as  more  modern  ideas  and  techniques  will  be  covered  to  give  students  a  broad  background  in  
the  field.    Some  prior  experience  of  the  mathematical  modelling  of  fluid  problems  (from  e.g.  M2AM  or  M3A2)  is  
desirable.  

Numerical  methods  for  eigenvalue  problems  will  also  be  covered  and  students  expected  to  complete  an  
assessed  numerical  assignment  (involving  MATLAB).  

Topics  covered  will  be  a  selection  from  the  following  list:  

Basic  concepts  of  linear  stability  theory,  normal  modes,  dispersion  relations,  marginal  stability,  temporal/spatial  
instability.  

Rayleigh-­Benard  instability.  Formulation  of  the  linearised  stability  problem.  Boundary  conditions  for  free-­free,  
rigid-­free  and  rigid-­rigid  problems.  Some  exact  solutions  and  discussion  of  marginal  stability  properties  and  cell  
patterns.    

Interfacial  instabilities:  Rayleigh-­Taylor,  Kelvin  Helmholtz  instabilities.  Jet  break  up  and  capillary  instabilities.  

Inviscid/viscous  shear  flow  instabilities:  Couette  flow,  Poiseuille  flow,  pipe  flow,  Taylor-­Couette  flow.  Circular  
flows/Taylor  vortices  and  boundary-­layers.  

Stability  of  parallel  flows.  Inviscid  stability  theory  and  properties  of  Rayleigh  equation.  Rayleigh's  inflexion  point  
criteria,  Fjortoft's  theorem,  Howard's  semi-­circle  theorem,  Squire's  theorem.  Viscous/Tollmien-­Schlichting  
instability.  Orr-­Sommerfeld  equation.    Parallel  flow  approximation  and  application  to  boundary  layers.  Discussion  
of  e-­n  method,  Non-­parallel  stability  theories.  

Linear  and  nonlinear  Parabolised  Stability  Equations  and  Receptivity  modelling.  

Linear  instability  versus  nonlinear  instability.  Weakly  nonlinear  theory,  derivation  of  Ginzburg-­Landau  and  Stuart-­
Landau  equations.  Local  bifurcation  theory.    
Recommended  Background  Reading:  
Charru,  Hydrodynamic  Instabilities,  Cambridge  Univ.  Press  
 
Drazin,  Introduction  to  Hydrodynamic  Stability,  Cambridge  Univ.  Press  
 
Huerre  &  Rossi,  Hydrodynamic  Instabilities  in  Open  Flows,  Cambridge  Univ.  Press  
 
Schmid  &  Henningson,  Stability  and  Transition  in  Shear  Flows,  Springer  
 
 
 
M4A32         VORTEX  DYNAMICS  
 
Term  2  

11  
 
  
Prerequisites:    A  knowledge  of  basic  applied  mathematical  methods  is  the  only  prerequisite.  A  basic  knowledge  
of  inviscid  fluid   dynamics  (e.g.  M3/4A2)  is  desirable  but  not  required.  
 
The   module   will   focus   on   the   mathematical   study  of   the   dynamics   of   vorticity  in   an   ideal   fluid   in   two   and  
three   dimensions.  The  module  will  be  pitched  in  such  a  way  that  it  will  be  of  interest  both  to  fluid  dynamicists  
and  as  an   application  of  various  techniques  in  dynamical  systems  theory.  
 
Fundamental  properties  of  vorticity.  
Helmholtz  Laws  and  Kelvin's  circulation  theorem.   Singular  distributions  of  vorticity;;  Biot-­Savart  law.  
Dynamics  of  line  vortices  in  2d  and  other  geometries;;  dynamics  of  2d  vortex  patches,  contour  dynamics.  
Axisymmetric  vortex  rings.   Dynamics  of  vortex  filaments.  
Stability  problems.  
Miscellaneous  topics  (effects  of  viscosity,  applications  to  turbulence,  applications  in  aerodynamics).    
 
M4M7*   ASYMPTOTIC  ANALYSIS  
 
Term  1  
 
Asymptotic  series  and  expansions.  Asymptotic  expansion  of  integrals,  Laplace  method,  Watson's  lemma,  
stationary  phase  and  steepest  descent.  Singular  perturbations,  matched  asymptotic  expansions:  inner/outer  
expansions  and  the  matching  principle,  boundary  layers  and  interior  layers.  Multiple-­scale  method,  Poincaré-­
Lindstedt  method,  method  of  strained  co-­ordinate,  averaging  method,  non-­linear  oscillations.  Differential  
equations  with  a  large  parameter  -­  the  WKBJ  method,  turning  point  problems,  caustics.  
 
DYNAMICS  
 
M4PA48*  DYNAMICS  OF  GAMES  
 
Term  1  
 
Recently  there  has  been  quite  a  lot  of  interest  in  modeling  learning  through  studying  the  dynamics  of  games.    
The  settings  to  which  these  models  may  be  applied  is  wide-­ranging,  from  ecology  and  sociology  to  business,    as  
actively  pursued  by  companies  like  Google.  Examples  include
  
(i)                  optimization  of  strategies  of  populations  in  ecology  and  biology
  
(ii)                strategies  of  people  in  a  competitive  environment,  like  online  auctions  or  (financial)  markets.  
 (iii)      learning  models  used  by  technology  companies  
This  module  is  aimed  at  discussing  a  number  of  dynamical  models  in  which  learning  evolves  over  time,  and  
which  have  a  game  theoretic  background.  
 The  module  will  take  a  dynamical  systems  perspective.  Topics  will  
include  replicator  dynamics  and  best  response  dynamics.  
 
M4PA23*   DYNAMICAL  SYSTEMS  
 
Term  1  
 
The  theory  of  Dynamical  Systems  is  an  important  area  of  mathematics  which  aims  at  describing  objects  whose  
state  changes  over  time.  For  instance,  the  solar  system  comprising  the  sun  and  all  planets  is  a  dynamical  
system,  and  dynamical  systems  can  be  found  in  many  other  areas  such  as  finance,  physics,  biology  and  social  
sciences.  This  course  provides  a  rigorous  treatment  of  the  foundations  of  discrete-­time  dynamical  systems,  
which  includes  the  following  subjects:  
   
-­  Periodic  orbits  
-­  Topological  and  symbolic  dynamics  
-­  Chaos  theory  
-­  Invariant  manifolds  
-­  Statistical  properties  of  dynamical  systems  
 
M4PA38  ADVANCED  DYNAMICAL  SYSTEMS  

12  
 
  
Term  2  
This  reading  course  deals  with  topics  in  dynamical  systems  at  an  advanced  level,  touching  upon  current  frontline  
research.  Each  year  a  selection  will  be  made  of  material  from  the  area  of  local  bifurcation  theory,  global  
bifurcation  theory,  ergodic  theory  of  dynamical  systems  or  dynamical  systems  methods  for  PDEs/FDEs.  The  
selection  of  reading  material  will  be  detailed  at  the  beginning  of  the  academic  year.  
Assessment:    Students  taking  the  course  for  credit  are  to  prepare  an  essay  (counting  for  60%)  and  give  an  oral  
presentation  about  their  work  (counting  for  40%).  
 
 
M4PA40  RANDOM  DYNAMICAL  SYSTEMS  AND  ERGODIC  THEORY  
(Replaces  M4PA36)  
 
Term  2  
 
This  is  an  introductory  course  on  the  theory  and  applications  of  random  dynamical  systems  and  ergodic  theory.  
Random  dynamical  systems  are  (deterministic)  dynamical  systems  driven  by  a  random  input.  The  goal  will  be  to  
present  a  solid  introduction  and,  time  permitting,  touch  upon  several  more  advanced  developments  in  this  field.  
The  contents  of  the  module  are:  
 
1)   Random  dynamical  systems;;  definition  in  terms  of  skew  products  and  elementary  examples  (including  
iterated  function  systems,  discrete  time  dynamical  systems  with  bounded  noise  and  stochastic  differential  
equations).  
2)   Introduction  to  random  dynamical  systems  theory  in  iterated  function  systems  context.  
3)   Background  on  measure  theory  and  probability  theory.  
4)   Introduction  to  Ergodic  Theory:  Birkhoff  Ergodic  Theorem  and  Oseledets  Ergodic  Theorem.  
5)   Dynamics  of  random  circle  maps:  synchronisation.  
6)   Chaos  in  random  dynamical  systems.  
 
 
M4PA16*   GEOMETRIC  MECHANICS    
 
Term  1  
 
This  module  on  geometric  mechanics  starts  with  Fermat’s  principle,  that  light  rays  follow  geodesics  determined  
from  a  least  action  variational  principle.  It  then  treats  subsequent  developments  in  mechanics  by  Newton,  Euler,  
Lagrange,  Hamilton,  Lie,  Poincaré,  Noether,  and  Cartan,  who  all  dealt  with  geometric  optics.    
 
The  module  will  explicitly  illustrate  the  following  concepts  of  geometric  mechanics:    
*  Configuration  space,  variational  principles,  Euler-­Lagrange  equations,  geodesic  curves,  
*  Legendre  transformation,  phase  space,  Hamilton’s  canonical  equations,  
*  Poisson  brackets,  Hamiltonian  vector  fields,  symplectic  transformations,  
*  Lie  group  symmetries,  conservation  laws,  Lie  algebras  and  their  dual  spaces,    
*  Divergence  free  vector  fields,  momentum  maps  and  coadjoint  motion.    
 
All  of  these  concepts  from  geometric  mechanics  will  be  illustrated  with  examples,  first  for  Fermat’s  principle  and  
then  again  for  three  primary  examples  in  classical  mechanics:  (1)  motion  on  the  sphere,  (2)  the  rigid  body  and  
(3)  pairs  of  n  :m  resonant  oscillators.    
 
Lectures  will  be  given  from  the  textbook,  Geometric  Mechanics  I:  Dynamics  and  Symmetry,  by  DD  Holm,  World  
Scientific:  Imperial  College  Press,  Singapore,  2nd  edition  (2011)..  ISBN  978-­1-­84816-­195-­5.  
 
M4PA34*   DYNAMICS,  SYMMETRY  AND  INTEGRABILITY  
 
Term  2  
 
The  following  topics  will  be  covered:    
 

13  
 
*  Introduction  to  smooth  manifolds  as  configuration  spaces  for  dynamics.    
*  Transformations  of  smooth  manifolds  as  flows  of  smooth  vector  fields.  
*  Introduction  to  differential  forms,  wedge  products  and  Lie  derivatives.  
*  Adjoint  and  coadjoint  actions  of  matrix  Lie  groups  and  matrix  Lie  algebras    
*  Action  principles  on  matrix  Lie  algebras,  their  corresponding  Euler-­Poincaré  ordinary  differential  equations  and  
the  Lie-­Poisson  Hamiltonian  formulations  of  these  equations.    
*  EPDiff:  the  Euler-­Poincaré  partial  differential  equation  for  smooth  vector  fields  acting  on  smooth  manifolds  
*  The  Hamiltonian  formulation  of  EPDiff:  Its  momentum  maps  and  soliton  solutions    
^Integrability  of  EPDiff:  Its  bi-­Hamiltonian  structure,  Lax  pair  and  isospectral  problem,  as  well  as  the  
relationships  of  these  features  to  the  corresponding  properties  of  KdV.  
 
 
 
FINANCE  
 
M4F22*    MATHEMATICAL  FINANCE:    AN  INTRODUCTION  TO  OPTION  PRICING  
 
Term  1  
 
Prerequisites:    Differential  Equations  (M2AA1),  Multivariable  Calculus  (M2AA2),  Real  Analysis  (M2PM1)  and  
Probability  and  Statistics  2  (M2S1).  
   
The  mathematical  modeling  of  derivatives  securities,  initiated  by  Bachelier  in  1900  and  developed  by  Black,  
Scholes  and  Merton  in  the  1970s,  focuses  on  the  pricing  and  hedging  of  options,  futures  and  other  derivatives,  
using  a  probabilistic  representation  of  market  uncertainty.  This  module  is  a  mathematical  introduction  to  this  
theory,  in  a  discrete-­time  setting.  We  will  mostly  focus  on  the  no-­arbitrage  theory  in  market  models  described  by  
trees;;  eventually  we  will  take  the  continuous-­time  limit  of  a  binomial  tree  to  obtain  the  celebrated  Black-­Scholes  
model  and  pricing  formula.  
 
We  will  cover  and  apply  mathematical  concepts  -­such  as  conditional  expectation,  filtrations,  Markov  processes,  
martingales  and  martingale  transforms,  the  separation  theorem,  and  change  of  measure-­  and  financial  concepts  
such  as  self-­financing  portfolios,  replication  and  delta  hedging,  risk-­neutral  probability,  complete  markets,  non-­
anticipative  strategies,  and  the  fundamental  theorem  of  asset  pricing.  
 
 
BIOLOGY  
 
M4A49*   MATHEMATICAL  BIOLOGY    
 
Term  1  
 
The  aim  of  the  module  is  to  describe  the  application  of  mathematical  models  to  biological  phenomena.  A  variety  
of  contexts  in  human  biology  and  diseases  are  considered,  as  well  as  problems  typical  of  particular  organisms  
and  environments.  

The  syllabus  includes  topics  from:    

1.  Population  dynamics.  Growth  and  spatial  spread  of  organisms.  Fisher's  equation.  
2.  Epidemiology  -­  the  spread  of  plagues.  

3.  Reaction-­Diffusion  models:  Turing  mechanism  for  pattern  formation.    

4.  Enzyme  Kinetics  and  chemical  reactions:  Michaelis-­Menten  theory.    

5.  Mass  transport;;  Taylor  dispersion.  

14  
 
6.  Biomechanics:  Blood  circulation,  animal  locomotion:  swimming,  flight.    
Effects  of  scale  and  size.  
7.  Other  particular  problems  from  biology.      
 
M4A50*     METHODS  FOR  DATA  SCIENCE  
 
Term  1  
 
This  course  is  in  two  halves:  machine  learning  and  complex  networks.  We  will  begin  with  an  introduction  to  the  R  
language  and  to  visualisation  and  exploratory  data  analysis.  We  will  describe  the  mathematical  challenges  and  
ideas  in  learning  from  data.  We  will  introduce  unsupervised  and  supervised  learning  through  theory  and  through  
application  of  commonly  used  methods  (such  as  principle  components  analysis,  k-­nearest  neighbours,  support  
vector  machines  and  others).  Moving  to  complex  networks,  we  will  introduce  key  concepts  of  graph  theory  and  
discuss  model  graphs  used  to  describe  social  and  biological  phenomena  (including  Erdos-­Renyi  graphs,  small-­
world  and  scale-­free  networks).  We  will  define  basic  metrics  to  characterise  data-­derived  networks,  and  illustrate  
how  networks  can  be  a  useful  way  to  interpret  data.  
 
 
MATHEMATICAL  PHYSICS  
 
M4A4*   MATHEMATICAL  PHYSICS  1:    QUANTUM  MECHANICS  
 
Term  1      
 
Quantum  mechanics  is  one  of  the  most  successful  theories  in  modern  physics  and  has  an  exceptionally  
beautiful  underlying  mathematical  structure.  It  provides  the  basis  for  many  areas  of  contemporary  physics,  
including  atomic  and  molecular,  condensed  matter,  high-­energy  particle  physics,  quantum  information  theory,  
and  quantum  cosmology,  and  has  led  to  countless  technological  applications.  
 
This  module  aims  to  provide  an  introduction  to  quantum  phenomena  and  their  mathematical  description.  
Quantum  theory  combines  tools  and  concepts  from  various  areas  of  mathematics  and  physics,  such  as  classical  
mechanics,  linear  algebra,  probability  theory,  numerical  methods,  analysis  and  even  geometry.  However,  most  
of  the  concepts  are  basic,  and  little  background  knowledge  is  required  before  we  can  put  them  to  practical  use.  
 
Core  topics:  Hamiltonian  dynamics;;  Schrödinger  equation  and  wave  functions;;  stationary  states  of  one-­
dimensional  systems;;  mathematical  foundations  of  quantum  mechanics;;  quantum  dynamics;;  angular  momentum  
Additional  optional  topics  may  include:  Approximation  techniques;;  explicitly  time-­dependent  systems;;  geometric  
phases;;  numerical  techniques;;  many-­particle  systems;;  cold  atoms;;  entanglement  and  quantum  information.  
 
M4A6*   SPECIAL  RELATIVITY  AND  ELECTROMAGNETISM  
 
Term  1  
 
This  module  presents  a  beautiful  mathematical  description  of  a  physical  theory  of  great  historical,  theoretical  
and  technological  importance.  It  demonstrates  how  advances  in  modern  theoretical  physics  are  being  made  and  
gives  a  glimpse  of  how  other  theories  (say  quantum  chromodynamics)  proceed.  
 
At  the  beginning  of  special  relativity  stands  an  experimental  observation  and  thus  the  insight  that  all  physical  
theories  ought  to  be  invariant  under  Lorentz  transformations.  Casting  this  in  the  language  of  Lagrangian  
mechanics  induces  a  new  description  of  the  world  around  us.  After  some  mathematical  work,  but  also  by  
interpreting  the  newly  derived  objects,  Maxwell’s  equations  follow,  which  are  truly  fundamental  to  all  our  every-­
day  interaction  with  the  world.  In  particular,  Maxwell’s  equations  can  be  used  to  characterise  the  behaviour  of  
charges  in  electromagnetic  fields,  which  is  rich  and  beautiful.  
 
This  module  does  not  follow  the  classical  presentation  of  special  relativity  by  following  its  historical  development,  
but  takes  the  field  theoretic  route  of  postulating  an  action  and  determining  the  consequences.  The  lectures  
follow  closely  the  famous  textbook  on  the  classical  theory  of  fields  by  Landau  and  Lifshitz.  

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Special  relativity:  Einstein’s  postulates,  Lorentz  transformation  and  its  consequences,  four  vectors,  dynamics  
of  a  particle,  mass-­energy  equivalence,  collisions,  conserved  quantities.  
Electromagnetism:  Magnetic  and  electric  fields,  their  transformations  and  invariants,  Maxwell’s  equations,  
conserved  quantities,  wave  equation.  
 
M4A7*   TENSOR  CALCULUS  AND  GENERAL  RELATIVITY  
 
Term  2      
 
The  mathematical  description  of  a  theory,  which  is  fundamental  to  gravitation  and  to  behaviour  of  systems  at  
large  scales.  
 
Tensor  calculus  including  Riemannian  geometry;;  principle  of  equivalence  for  gravitational  fields;;  Einstein’s  field  
equations  and  the  Newtonian  approximation;;  Schwarzschild’s  solution  for  static  spherically  symmetric  systems;;  
the  observational  tests;;  significance  of  the  Schwarzschild  radius;;  black  holes;;  cosmological  models  and  ‘big  
bang’  origin  of  the  universe.  
 
Variational  principles.  
 
 
M4A29*   THEORY  OF  COMPLEX  SYSTEMS  
 
Term  2  
 
Objective:  To  become  familiar  with  the  subject  matter  of  Complexity  Sciences,  its  methodology  and  
mathematical  tools.  
 
Prerequisites:  Curiosity  and  an  interest  in  being  able  to  understand  the  complex  world  surrounding  us.  Standard  
undergraduate  mathematics  (such  as  calculus,  linear  algebra).  Some  familiarity  with  computing  (e.g.  matlab  or  
other  programming  language).  A  little  familiarity  with  statistical  mechanics  may  be  helpful.  
 
This  module  will  provide  the  basic  foundation  in  terms  of  concepts  and  mathematical  methodology  needed  to  
analyse  and  model  complex  systems.  
 
1)   Simple  functional  integration:  to  discuss  the  emergent  vortex  solutions  in  terms  extremal  configurations  for  
the  partition  integral  of  the  2D  XY  model.    
2)   Record  statistics  and  record  dynamics:  to  discuss  the  statistics  of  intermittent  slowly  decelerating  dynamics  
as  observed  in  models  of  evolution  and  many  other  complex  systems.  Relations  to  extreme  value  statistics.  
3)   Branching  processes:  to  present  a  mean  field  discussion  of  avalanche  dynamics  in  models  of  complex  
systems  such  as  the  sand  pile,  forest  fires  and  more  recent  models  of  fusions  of  banks.    
4)   The  Kuramoto  transition  to  synchronisation  as  an  example  of  collective  cooperative  dynamical  behaviour  of  
potential  relevance  to  brain  dynamics.  
5)   Intermittency  in  low  (non-­linear  maps)  and  high  dimensional  systems  (e.g.  Tangled  Nature  model)  and  
relation  to  renormalisation  theory  (low  dim.)  and  mean  field  stability  analysis  (high  dim).  
 
Assessment:  Two  mini  projects.    
 
M4A52*  QUANTUM  MECHANICS  II  
 
Term  2  
 
Quantum  mechanics  is  one  of  the  most  successful  theories  in  modern  physics  and  has  an  exceptionally  
beautiful  underlying  mathematical  structure.  It  provides  the  basis  for  many  areas  of  contemporary  physics,  
including  atomic  and  molecular,  condensed  matter,  high-­energy  particle  physics,  quantum  information  theory,  
and  quantum  cosmology,  and  has  led  to  countless  technological  applications.  Quantum  theory  combines  tools  
and  concepts  from  various  areas  of  mathematics  and  physics,  such  as  classical  mechanics,  linear  algebra,  
probability  theory,  numerical  methods,  analysis  and  even  geometry.  However,  most  of  the  concepts  are  basic,  

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and  little  background  knowledge  is  required  before  we  can  put  them  to  practical  use.    
 
This  module  is  intended  to  be  a  second  course  in  quantum  mechanics  and  will  build  on  topics  covered  in  
Quantum  Mechanics  I.  
 
In  addition  to  the  material  below,  this  level  7  (Masters)  version  of  the  module  will  have  additional  extension  
material  for  self-­study.  This  will  require  a  deeper  understanding  of  the  subject  than  the  corresponding  level  6  
(Bachelors)  module.  
 
Core  topics:  Quantum  mechanics  in  three  spatial  dimensions,  the  Heisenberg  picture,  perturbation  theory,  
addition  of  spin,  adiabatic  processes  and  the  geometric  phase,  Floquet-­Bloch  theory,  second  quantization  and  
introduction  to  many-­particle  systems,  Fermi  and  Bose  statistics,  quantum  magnetism.    Additional  topics  may  
include  WKB  theory  and  the  Feynman  path  integral.  
 
 
 
M4M6*   METHODS  OF  MATHEMATICAL  PHYSICS  
 
Term  1  
 
The  aim  of  this  module  is  to  learn  tools  and  techniques  from  complex  analysis  and  orthogonal  polynomials  that  
are  used  in  mathematical  physics.  The  course  will  focus  on  mathematical  techniques,  though  will  
discuss  relevant  physical  applications,  such  as  electrostatic  potential  theory.  The  course  also  incorporates  
computational  techniques  in  the  lectures.  
 
Prerequisites:  Complex  analysis.  
Topics:  
 
1.   Revision  of  complex  analysis:  Complex  integration,  Cauchy’s  theorem  and  residue  calculus  [Revision]  
2.   Singular  integrals:  Cauchy,  Hilbert,  and  log  kernel  transforms  
3.   Potential  theory:  Laplace’s  equation,  electrostatic  potentials,  distribution  of  charges  in  a  well  
4.   Riemann–Hilbert  problems:  Plemelj  formulae,  additive  and  multiplicative  Riemann–Hilbert  problems  
5.   Orthogonal  polynomials:  recurrence  relationships,  solving  differential  equations,  calculating  singular  
integrals  
6.   Integral  equations:  integral  equations  on  the  whole  and  half  line,  Fourier  transforms,  Laplace  transforms  
7.   Wiener–Hopf  method:  direct  solution,  solution  via  Riemann–Hilbert  methods  
8.   Singularities  of  differential  equations:  analyticity  of  solutions,  regular  singular  points,  Hypergeometric  
functions  
 
APPLIED  PDEs,  NUMERICAL  METHODS  and  COMPUTATION  
 
 
 
M4M3*   INTRODUCTION  TO  PARTIAL  DIFFERENTIAL  EQUATIONS  
 
Term  1  
 
1.   Basic  concepts:    PDEs,  linearity,  superposition  principle.    Boundary  and  Initial  value  problems.  
2.   Gauss  Theorem:    gradient,  divergence  and  rotational.  Main  actors:  continuity,  heat  or  diffusion,  Poisson-­
Laplace,  and  the  wave  equations.  
3.   Linear  and  Qasilinear  first  order  PDEs  in  two  independent  variables.    Well-­posedness  for  the  Cauchy  
problem.  The  linear  transport  equation.    Upwinding  scheme  for  the  discretization  of  the  advection  equation.  
4.   A  brief  introduction  to  conservation  laws:    The  traffic  equation  and  the  Burgers  equation.  Singularities.  
5.   Derivation  of  the  heat  equation.    The  boundary  value  problem:    separation  of  variables.    Fourier  Series.  
Explicit  Euler  scheme  for  the  1d  heat  equation:    stability.  
6.   The  Cauchy  problem  for  the  heat  equation:    Poisson’s  Formula.    Uniqueness  by  maximum  principle.  
7.   The  ID  wave  equation.    D’Alembert  Formula.  The  boundary  value  problem  by  Fourier  Series.    Explicit  finite  
difference  scheme  for  the  1d  wave  equation:    stability.  

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8.   2D  and  3D  waves.    Casuality  and  Energy  conservation:    Huygens  principle.  
9.   Green’s  functions:    Newtonian  potentials.    Dirichlet  and  Neumann  problems.  
10.  Harmonic  functions.    Uniqueness:    mean  property  and  maximum  principles.  
 
M4M11*   FUNCTION  SPACES  AND  APPLICATIONS    
 
Term  1  
 
The  purpose  of  this  course  is  to  introduce  the  basic  function  spaces  and  to  train  the  student  into  the  basic  
methodologies  needed  to  undertake  the  analysis  of  Partial  Differential  Equations  and  to  prepare  them  for  the  
course  ‘Advanced  topics  in  Partial  Differential  Equations’’  where  this  framework  will  be  applied.  The  course  is  
designed  as  a  stand-­alone  course.  No  background  in  topology  or  measure  theory  is  needed  as  these  concepts  
will  be  reviewed  at  the  beginning  of  the  course.  
 
The  course  will  span  the  basic  aspects  of  modern  functional  spaces:  integration  theory,  Banach  spaces,  spaces  
of  differentiable  functions  and  of  integrable  functions,  convolution  and  regularization,  compactness  and  Hilbert  
spaces.  The  concepts  of  Distributions,  compact  operators  and  Sobolev  spaces  will  be  taught  in  the  follow-­up  
course  ‘’Advanced  topics  in  Partial  Differential  Equations’’  as  they  are  tightly  connected  to  the  resolution  of  
elliptic  PDE’s  and  the  material  taught  in  the  present  course  is  already  significant.  
 
The  syllabus  of  the  course  is  as  follows:  
 
1)  Elements  of  metric  topology  
 
2)  Elements  of  Lebesgue’s  integration  theory.  
 
3)  Normed  vector  spaces.  Banach  spaces.  Continuous  linear  maps.  Dual  of  a  Banach  space.  Examples  of  
function  spaces:  continuously  differentiable  function  spaces  and  Lebesgue  spaces.  Hölder  and  Minkowski’s  
inequalities.  Support  of  a  function;;  Convolution.  Young’s  inequality  for  the  convolution.  Mollifiers.  Approximation  
of  continuous  or  Lebesgue  integrable  functions  by  infinitely  differentiable  functions  with  compact  support.  
 
4)  Hilbert  spaces.  The  projection  theorem.  The  Riesz  representation  theorem.  The  Lax-­Milgram  theorem.  Hilbert  
bases  and  Parseval’s  identity.  Application  to  Fourier  series.    
 
 
M4M12*   ADVANCED  TOPICS  IN  PARTIAL  DIFFERENTIAL  EQUATIONS    
 
Term  2  
 
This  course  develops  the  analysis  of  boundary  value  problems  for  elliptic  and  parabolic  PDE’s  using  the  
variational  approach.  It  is  a  follow-­up  of  ‘Function  spaces  and  applications’  but  is  open  to  other  students  as  well  
provided  they  have  sufficient  command  of  analysis.  An  introductory  Partial  Differential  Equation  course  is  not  
needed  either,  although  certainly  useful.  
 
The  course  consists  of  three  parts.  The  first  part  (divided  in  two  chapters)  develops  further  tools  needed  for  the  
study  of  boundary  value  problem,  namely  distributions  and  Sobolev  spaces.  The  following  two  parts  are  devoted  
to  elliptic  and  parabolic  equations  on  bounded  domains.  They  present  the  variational  approach  and  spectral  
theory  of  elliptic  operators  as  well  as  their  use  in  the  existence  theory  for  parabolic  problems.  The  aim  of  the  
course  is  to  expose  the  students  some  important  aspects  of  Partial  Differential  Equation  theory,  aspects  that  will  
be  most  useful  to  those  who  will  further  work  with  Partial  Differential  Equations  be  it  on  the  Theoretical  side  or  
on  the  Numerical  one.  
 
The  syllabus  of  the  course  is  as  follows:  
 
1.  Distributions.The  space  of  test  functions.  Definition  and  examples  of  distributions.  Differentiation.  Convolution.  
Convergence  of  distributions.    
 
2.  Sobolev  spaces:  The  space  H1.  Density  of  smooth  functions.  Extension  lemma.  Trace  theorem.  The  space  

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H10.  Poincare  inequality.  The  Rellich-­Kondrachov  compactness  theorem  (without  proof).  Sobolev  imbedding  (in  
the  simple  case  of  an  interval  of  R).  The  space  Hm.  Compactness  and  Sobolev  
imbedding  for  arbitrary  dimension  (statement  without  proof).  
 
3.  Linear  elliptic  boundary  value  problems:  Dirichlet  and  Neumann  boundary  value  problems  via  the  Lax-­Milgram  
theorem.  The  maximum  principle.  Regularity  (stated  without  proofs).  Classical  examples:  
elasticity  system,  Stokes  system.  
 
4)  Spectral  Theory  :  compact  operators  in  Hilbert  spaces.  The  Fredholm  alternative.  Spectral  decomposition  of  
compact  self-­adjoint  operators  in  Hilbert  spaces.  Spectral  theory  of  linear  elliptic  boundary  value  problems.    
 
5.  Linear  parabolic  initial-­boundary  value  problems.  Existence  and  uniqueness  by  spectral  decomposition  on  the  
eigenbasis  of  the  associated  elliptic  operator.  Classical  examples  (Navier-­Stokes  equation).  
 
 
M4A51   STOCHASTIC  DIFFERENTIAL  EQUATIONS  (Replaces  A42)  
 
Term  1  
 
This  is  a  basic  introductory  course  on  the  theory  and  applications  of  stochastic  differential  equations.  The  goal  
will  be  to  present  the  basic  theory  of  SDEs  and,  time  permitting,  to  also  present  some  specific  applications  such  
as  stochastic  optimal  control,  applications  of  SDEs  to  Partial  differential  equations  etc.  
 
The  contents  of  the  module  are:  
 
1)   Modelling  with  SDEs,  overview  of  applications.  
2)   Background  on  probability  theory  and  the  theory  of  stochastic  processes.  
3)   Ito’s  theory  of  stochastic  integration  and  Ito’s  formula.  
4)   Stochatic  differential  equations,  basic  theory  including  existence  and  uniqueness  of  solutions.  
5)   Ergodic  properties  of  SDEs.  
6)   Connection  between  SDEs  and  the  forward  and  backward  Kolmogorov  equations.  
7)   Applications  including  stochastic  optimal  control,  Feynman-­Kac  formulas,  etc.  
 
 
M4A47*  FINITE  ELEMENTS:  NUMERICAL  ANALYSIS  AND  IMPLEMENTATION.  
 
Term  2  
 
Finite  element  methods  form  a  flexible  class  of  techniques  for  numerical  solution  of  PDEs  that  are  both  
accurate  and  efficient.    
The  finite  element  method  is  a  core  mathematical  technique  underpinning  much  of  the  development  of  
simulation  science.  Applications  are  as  diverse  as  the  structural  mechanics  of  buildings,  the  weather  forecast,  
and  pricing  financial  instruments.  Finite  element  methods  have  a  powerful  mathematical  abstraction  based  on  
the  language  of  function  spaces,  inner  products,  norms  and  operators.    
 
This  module  aims  to  develop  a  deep  understanding  of  the  finite  element  method  by  spanning  both  its  analysis  
and  implementation.  in  the  analysis  part  of  the  module  you  will  employ  the  mathematical  abstractions  of  the  finite  
element  method  to  analyse  the  existence,  stability,  and  accuracy  of  numerical  solutions  to  PDEs.  At  the  same  
time,  in  the  implementation  part  of  the  module  you  will  combine  these  abstractions  with  modern  software  
engineering  tools  to  create  and  understand  a  computer  implementation  of  the  finite  element  method.  
 
Syllabus:  
•  Basic  concepts:  Weak  formulation  of  boundary  value  problems,  Ritz-­Galerkin  approximation,  error  
estimates,  piecewise  polynomial  spaces,  local  estimates.  
•  Efficient  construction  of  finite  element  spaces  in  one  dimension,  1D  quadrature,  global  assembly  of  

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 mass  matrix  and  Laplace  matrix.  
•  Construction  of  a  finite  element  space:  Ciarlet’s  finite  element,  various  element  types,  finite  element  
interpolants.  
•  Construction  of  local  bases  for  finite  elements,  efficient  local  assembly.  
•  Sobolev  Spaces:  generalised  derivatives,  Sobolev  norms  and  spaces,  Sobolev’s  inequality.  
•  Numerical  quadrature  on  simplices.  Employing  the  pullback  to  integrate  on  a  reference  element.  
•  Variational  formulation  of  elliptic  boundary  value  problems:  Riesz  representation  theorem,  symmetric  
and  nonsymmetric  variational  problems,  Lax-­Milgram  theorem,  finite  element  approximation  estimates.  
•  Computational  meshes:  meshes  as  graphs  of  topological  entities.  Discrete  function  spaces  on  meshes,  
local  and  global  numbering.  
•  Global  assembly  for  Poisson  equation,  implementation  of  boundary  conditions.  General  approach  for  
nonlinear  elliptic  PDEs.  
•  Variational  problems:  Poisson’s  equation,  variational  approximation  of  Poisson’s  equation,  elliptic  
regularity  estimates,  general  second-­order  elliptic  operators  and  their  variational  approximation.  
•  Residual  form,  the  Gâteaux  derivative  and  techniques  for  nonlinear  problems.  
The  course  is  assessed  50%  by  examination  and  50%  by  coursework  (implementation  exercise  in  Python).  
 
M4N7*   NUMERICAL  SOLUTION  OF  ORDINARY  DIFFERENTIAL  EQUATIONS  
 
Term  1  
 
An  analysis  of  methods  for  solving  ordinary  differential  equations.    Totally  examined  by  project.  
 
Runge-­Kutta,  extrapolation  and  linear  multistep  methods.    Analysis  of  stability  and  convergence.  
Error  estimation  and  automatic  step  control.    Introduction  to  stiffness.  
Boundary  and  eigenvalue  problems.    Solution  by  shooting  and  finite  difference  methods.  
Introduction  to  deferred  and  defect  correction.  
 
M4N10*   COMPUTATIONAL  PARTIAL  DIFFERENTIAL  EQUATIONS    
 
Term  2  
 
The  module  will  introduce  a  variety  of  computational  approaches  for  solving  partial  differential  equations,  
focusing  mostly  on  finite  difference  methods,  but  also  touching  on  finite  volume  and  spectral  methods.  Students  
will  gain  experience  implementing  the  methods  and  writing/modifying  short  programs  in  Matlab  or  other  
programming  language  of  their  choice.  Applications  will  be  drawn  from  problems  arising  in  Mathematical  Biology,  
Fluid  Dynamics,  etc.  At  the  end  of  the  module,  students  should  be  able  to  solve  research-­level  problems  by  
combining  various  techniques.  Assessment  will  be  by  projects,  probably  3  in  total.  The  first  project  will  only  count  
for  10-­20%  and  will  be  returned  quickly  with  comments,  before  students  become  committed  to  completing  the  
module.  Typically,  the  projects  will  build  upon  each  other,  so  that  by  the  end  of  the  module  a  research  level  
problem  may  be  tackled.  Matlab  codes  will  be  provided  to  illustrate  similar  problems  and  techniques,  but  these  
will  require  modification  before  they  can  be  applied  to  the  projects.  The  use  of  any  reasonable  computer  
language  is  permitted.  
 
Topics  (as  time  permits).  
 
-­  Finite  difference  methods  for  linear  problems:  order  of  accuracy,  consistency,  stability  and  convergence,  CFL  
condition,  von  Neumann  stability  analysis,  stability  regions;;  multi-­step  formula  and  multi-­stage  techniques.  
 
-­  Solvers  for  elliptic  problems:  direct  and  iterative  solvers,  Jacobi  and  Gauss-­Seidel  method  and  convergence  
analysis;;  geometric  multigrid  method.  
 
-­  Methods  for  the  heat  equation:  explicit  versus  implicit  schemes;;  stiffness.  

20  
 
  
-­  Techniques  for  the  wave  equation:  finite-­difference  solution,  characteristic  formulation,  non-­reflecting  boundary  
conditions,  one-­way  wave  equations,  perfectly  matched  layers.  Lax-­Friedrichs,  Lax-­Wendroff,  upwind  and  semi-­
Lagrangian  advection  schemes.  
 
-­  Domain  decomposition  for  elliptic  equations:  overlapping  alternating  Schwarz  method  and  convergence  
analysis,  non-­overlapping  methods.  
 
M4SC*   SCIENTIFIC  COMPUTATION  
 
Term  2  
 
Scientific  computing  is  an  important  skill  for  any  mathematician.  It  requires  both  knowledge  of  algorithms  and  
proficiency  in  a  scientific  programming  language.  The  aim  of  this  module  is  to  expose  students  from  a  varied  
mathematical  background  to  efficient  algorithms  to  solve  mathematical  problems  using  computation.  
 
The  objectives  are  that  by  the  end  of  the  module  all  students  should  have  a  good  familiarity  with  the  essential  
elements  of  the  Python  programming  language,  and  be  able  to  undertake  programming  tasks  in  a  range  of  
common  areas  (see  below).  
 
There  will  be  four  sub-­modules:  1.  A  PDE-­module  covering  elementary  methods  for  the  solution  of  time-­
dependent  problems.  2.  An  optimization-­module  covering  discrete  and  derivative-­free  algorithms.  3.  A  pattern-­
recognition-­module  covering  searching  and  matching  methods.  4.  A  statistics-­module  covering,  e.g.,  Monte-­
Carlo  techniques.  
 
Each  module  will  consist  of  a  brief  introduction  to  the  underlying  algorithm,  its  implementation  in  the  python  
programming  language,  and  an  application  to  real-­life  situations.  
 
M4N9*   COMPUTATIONAL  LINEAR  ALGEBRA  
 
Term  1  
 
Examined  solely  by  project.  Computational  aspects  of  the  projects  will  require  programming  in  Matlab  and/  or  
Python.  
 
Whether  it  be  statistics,  mathematical  finance,  or  applied  mathematics,  the  numerical  implementation  
of  many  of  the  theories  arising  in  these  fields  relies  on  solving  a  system  of  linear  equations,  and  often  
doing  so  as  quickly  as  possible  to  obtain  a  useful  result  in  a  reasonable  time.  This  course  explores  
the  different  methods  used  to  solve  linear  systems  (as  well  as  perform  other  linear  algebra  
computations)  and  has  equal  emphasis  on  mathematical  analysis  and  practical  applications.  
 
Topics  include:  
1.  Direct  methods:  Triangular  and  banded  matrices,  Gauss  elimination,  LU-­decomposition,  conditioning  and  
finite-­precision  
arithmetic,  pivoting,  Cholesky  factorisation,  QR  factorisation.  
2.  Symmetric  eigenvalue  problem:  power  method  and  variants,  Jacobi's  method,  Householder  
reduction  to  tridiagonal  form,  eigenvalues  of  tridiagonal  matrices,  the  QR  method  
3.  Iterative  methods:    
(a)  Classic  iterative  methods:  Richardson,  Jacobi,  Gauss  -­  Seidel,  SOR  
(b)  Krylov  subspace  methods:  Lanczos  method  and  Arnoldi  iteration,  conjugate  gradient  method,  GMRES,  
preconditioning.  
 
 
 

21  
 
  
 
 
 
 
PURE  MATHEMATICS  
 
PURE  MATHEMATICS  STUDY  GROUP  
 
Various  lecturers  
Terms  1  and  2  
 
This  is  a  non-­examined,  not-­for-­credit  module.  It  will  consist  of  a  mixture  of  independent  study  and  discussion  
groups,  together  with  lectures  delivered  by  students  or  staff.  The  choice  of  topics  will  complement  that  available  
in  taught  modules  and  will  be  determined  by  students  in  discussion  with  and  under  the  guidance  of  a  member  of  
staff.    
 
ANALYSIS  
 
M4P6*   PROBABILITY  THEORY  
 
Term  2  
 
Prerequisites:  Measure  and  Integration  (M3/4P19,  Term  1)  
 
A  rigorous  approach  to  the  fundamental  properties  of  probability.  
 
Probability  measures.    Random  variables    Independence.    Sums  of  independent  random  variables;;  weak  and  
strong  laws  of  large  numbers.    Weak  convergence,  characteristic  functions,  central  limit  theorem.    Elements  of  
Brownian  motion.  Martingales.  
 
M4P7*   FUNCTIONAL  ANALYSIS  
 
Term  2    
 
This  module  brings  together  ideas  of  continuity  and  linear  algebra.    It  concerns  vector  spaces  with  a  distance,  
and  involves  linear  maps;;  the  vector  spaces  are  often  spaces  of  functions.  
 
Vector  spaces.    Existence  of  a  Hamel  basis.    Normed  vector  spaces.    Banach  spaces.    Finite  dimensional  
spaces.    Isomorphism.    Separability.    The  Hilbert  space.    The  Riesz-­Fisher  Theorem.  The  Hahn-­Banach  
Theorem.    Principle  of  Uniform  Boundedness.    Dual  spaces.    Operators,  compact  operators.    Hermitian  
operators  and  the  Spectral  Theorem.  
 
M4P18*    FOURIER  ANALYSIS  AND  THEORY  OF  DISTRIBUTIONS  
 
Term  2  
 
Spaces  of  test  functions  and  distributions,  Fourier  Transform  (discrete  and  continuous),  Bessel’s,  Parseval’s  
Theorems,  Laplace  transform  of  a  distribution,  Solution  of  classical  PDE’s  via  Fourier  transform,  Basic  Sobolev  
Inequalities,  Sobolev  spaces.  
 
 
 
M4P19*     MEASURE  AND  INTEGRATION  
 
Term  1  
 

22  
 
Rings  and  algebras  of  sets,  construction  of  a  measure.  Mesurable  functions  and  their  properties,  Egorov's  
theorem,  convergence  in  measure.  Lebesgue  integral,  its  elementary  properties,  integral  and  sequences,  Fubini  
theorem.  Differentiation  and  integration:  
monotone  functions,  functions  of  bounded  variation,  absolutely  continuous  functions,  signed  measures.  
Lebesgue-­Stiltjes  measures.  Lp  spaces.  
 
M4P60*  GEOMETRIC  COMPLEX  ANALYSIS  
 
Term  2  
 
Complex  analysis  is  the  study  of  functions  of  complex  numbers.  It  is  employed  in  a  wide  range  of  topics,  
including  dynamical  systems,  algebraic  geometry,  number  theory,  and  quantum  field  theory,  to  name  a  few.    
 
While  you  become  familiar  with  basics  of  functions  of  a  complex  variable  in  the  complex  analysis  module,  here  
we  look  at  the  subject  from  a  more  geometric  viewpoint.  We  shall  look  at  geometric  notions  associated  with  
domains  in  the  plane  and  their  boundaries,  and  how  they  are  transformed  under  holomorphic  mappings.  In  turn,  
the  behaviour  of  conformal  maps  is  highly  dependent  on  the  shape  of  their  domain  of  definition.    
 
The  module  will  discuss  the  following  topics:  Schwarz  lemma,  automorphisms  of  the  disk,  the  Riemann  sphere  
and  the  rational  maps,  hyperbolic  geometry  on  the  disk,  conformal  mappings,  normal  families  and  Montel's  
theorem,  Riemann  mapping  theorem,  distortion  theorems,  quasiconformal  mappings,  Beltrami  equation.  
 
 
 
M4P41         ANALYTIC  METHODS  IN  PARTIAL  DIFFERENTIAL  EQUATIONS  
 
Term  2  
 
This  module  introduces  some  of  the  partial  differential  equations  appearing  in  physics  and  geometry,  as  well  as  
a  number  of  classical  techniques  to  study  them  analytically.  Topics  include  
 
—  Review  of  ODE  Theory  (Picard’s  Theorem,  Gronwall’s  inequality)  
—  Theory  of  first  order  quasilinear  PDE  (Methods  of  Characteristics)  
—  Cauchy-­Kovalevskaya  Theorem  (with  sketch  of  the  proof)  
—  Holmgren’s  uniqueness  theorem  (with  proof  via  Cauchy-­Kovalevskaya,  examples)  
—  Laplace’s  equation  (fundamental  solution,  regularity  of  harmonic  functions,  maximum  principle,  Green’s  
function  for  a  ball)  
—  General  second  order  elliptic  equations  (Existence  and  Regularity  Theory,  Fredholm  Alternative)  
—  Discussion  of  Schroedinger  and  Heat  Equation  (Schwartz  space,  Fourier  techniques)  
—  Wave  Equation  (Energy  estimate,  domain  of  dependence,  domain  of  influence,  fundamental  solution,  solution  
via  Fourier  techniques,  Duhamel’s  principle)  
 
Prerequisites:  It  will  be  helpful  if  you  have  taken  one  or  more  of  the  following  courses:    
Functional  Analysis,  Measure  and  Integration,  Fourier  Analysis  and  Distributions.  
 
 
M4P67  Stochastic  Calculus  with  Applications  to  Nonlinear  Filtering    
 
Term  1  
 
Prerequisites:  Ordinary  differential  equations,  partial  differential  equations,  real  analysis,  probability  theory.  
 
The  course  offers  a  bespoke  introduction  to  stochastic  calculus  required  to  cover  the  classical  theoretical  results  
of  nonlinear  filtering  as  well  as  some  modern  numerical  methods  for  solving  the  filtering  problem.  The  first  part  
of  the  course  will  equip  the  students  with  the  necessary  knowledge  (e.g.,  Ito  Calculus,  Stochastic  Integration  by  
Parts,  Girsanov’s    theorem)    and  skills  (solving  linear  stochastic  differential  equation,  analysing  continuous  
martingales,  etc)  to  handle  a  variety  of  applications.  The  focus  will  be  on  the  use  of  stochastic  calculus  to  the  
theory  and  numerical  solution  of  nonlinear  filtering.        

23  
 
  
1.  Martingales  on  Continuous  Time  (Doob  Meyer  decomposition,  L_p  bounds,  Brownian  motion,  exponential  
martingales,  semi-­martingales,  local  martingales,  Novikov’s  condition)      
2.  Stochastic  Calculus  (Ito’s  isometry,  chain  rule,  integration  by  parts)    
3.  Stochastic  Differential  Equations  (well  posedness,  linear  SDEs,  the  Ornstein-­Uhlenbeck  process,  Girsanov's  
Theorem)  
4.  Stochastic  Filtering  (definition,  mathematical  model  for  the  signal  process  and  the  observation  process)    
5.  The  Filtering  Equations  (well-­posedness,  the  innovation  process,  the  Kalman-­Bucy  filter)  
6.  Numerical  Methods  (the  Extended  Kalman-­filter,  Sequential  Monte-­Carlo  methods).  
 
M4P70*  Markov  Processes  
 
Term  1  
 
Markov  processes  are  widely  used  to  model  random  evolutions  with  the  Markov  property  `given  the  present,  the  
future  is  independent  of  the  past’.  The  theory  connects  with  many  other  subjects  in  mathematics  and  has  vast  
applications.  This  course  is  an  introduction  to  Markov  processes.  We  aim  to  build  intuitions  and    foundations  for  
further  studies  in  stochastic  analysis  and  in  stochastic  modelling.  
 
The  module  is  largely  self-­contained,  but  it  would  be  useful  for  students  to  also  take  Measure  and  Integration  
(M345P19).  A  good  knowledge  of  real  analysis  would  be  helpful  (M2PM1).  
 
It  is  related  to:  
Applied  probability  (M345S4),  Random  Dynamical  Systems  and  Ergodic  Theory  (M4PA40),  Probability  theory  
(M345P6),  Stochastic  Calculus  with  Applications  to  non-­Linear  Filtering    (M45P67),  Stochastic  Differential  
Equations    (M45A51),  
Stochastic  simulation  (M4S9*),  Ergodic  Theory  (M4PA36),  Computational  Stochastic  Processes  (M4A44),  and  
many  Mathematical    Finance  modules.    
 
Contents:      1.  Discrete  time    and  finite  state  Markov  chains  :    Chapman-­  
Kolmogorov  equations,  
irreducible,  Perron-­Froebenius  theorem  for  stochastic  matrices,  recurrent  and  
transient.  
2.  Discrete  time  Markov  processes  on  general  state  space.    Conditional  
expectations,  Chapman-­Kolmogorov  equation,    Feller  property,  strong  Feller  
property,  Kolmogorov's  theorem,  stopping  times,  strong  Markov,  stationary  
process,  weak  convergence  and  Prohorov's  theorem,  Existence  of  invariant  
measures  :  Krylov-­Bogolubov  method,  Lyapunov  method.  Ergodicity  by  
contraction  method  and  Doeblin's  criterion.  Structures  of  invariant  measures,  
ergodic  theorems.  
 
 
GEOMETRY  
 
M4P5*   GEOMETRY  OF  CURVES  AND  SURFACES  
 
Term  1    
 
The  main  object  of  this  module  is  to  understand  what  is  the  curvature  of  a  surface  in  3-­dimensional  space.  
 
Topological  surfaces:    Defintion  of  an  atlas;;  the  prototype  definition  of  a  surface;;  examples.    The  topology  of  a  
surface;;  the  Hausdorff  condition,  the  genuine  definition  of  a  surface.  Orientability,  compactness.  
Subdivisions  and  the  Euler  characteristic.  
Cut-­and-­paste  technique,  the  classification  of  compact  surfaces.  Connected  sums  of  surfaces.  
Smooth  surfaces:  Definition  of  a  smooth  atlas,  a  smooth  surface  and  of  smooth  maps  into  and  out  of  smooth  
surfaces.  Surfaces  in  R3,  tangents,  normals  and  orientability.  
The  first  fundamental  form,  lengths  and  areas,  isometries.  
The  second  fundamental  form,  principal  curvatures  and  directions.  

24  
 
The  definition  of  a  geodesic,  existence  and  uniqueness,  geodesics  and  co-­ordinates.  
Gaussian  curvature,  definition  and  geometric  interpretation,  Gauss  curvature  is  intrinsic,  surfaces  with  constant  
Gauss  curvature.  
The  Gauss-­Bonnet  theorem.  
(Not  examinable  and  in  brief)  Abstract  Riemannian  surfaces,  metrics.  
 
Mean  curvature  and  minimal  surfaces,  including  the  definition  of  mean  curvature,  its  geometric  interpretation,  the  
definition  of  minimal  surfaces  and  some  examples.  
 
M4P20*    GEOMETRY  1:    ALGEBRAIC  CURVES  
 
Term  1    
 
Plane  algebraic  curves;;  Projective  spaces;;  Projective  curves;;  Smooth  cubics  and  the  group  
structure;;  Intersection  of  projective  curves.    
Genus  of  a  curve  (Riemann  surfaces);;  Meromorphic  differentials  and  Abel’s  theorem.  
 
 
M4P21*    GEOMETRY  2:    ALGEBRAIC  TOPOLOGY  
 
Term  2  
 
Homotopies  of  maps  and  spaces.  Fundamental  group.  Covering  spaces,  Van  Kampen  (only  sketch  of  proof).  
Homology:  singular  and  simplicial  (following  Hatcher’s  notion  of  Delta-­complex).  Mayer-­Vietoris  (sketch  proof)  
and  long  exact  sequence  of  a  pair.  Calculations  on  topological  surfaces.  Brouwer  fixed  point  theorem.  
 
 
 
M4P33  ALGEBRAIC  GEOMETRY  
 
Term  2  
 
Pre-­requisites:  M4P55  Commutative  Algebra  
 
Algebraic  geometry  is  the  study  of  the  space  of  solutions  to  polynomial  equations  in  several  variables.  In  this  
course,  you  will  learn  to  use  algebraic  and  geometric  ideas  together,  studying  some  of  the  basic  concepts  from  
both  perspectives  and  applying  them  to  numerous  examples.  
 
Affine  varieties,  projective  varieties.  The  Nullstellensatz.  
Regular  and  rational  maps  between  varieties.  Completeness  of  projective  varieties.  
Dimension.  Regular  and  singular  points.  
Examples  of  algebraic  varieties.  
 
M4P51         RIEMANNIAN  GEOMETRY  
 
Term  2  
 
Prerequisites:   Geometry  of  Curves  and  Surfaces  (M4/4P5)  and  Manifolds  (M4P52).  
 
The  main  aim  of  this  module  is  to  understand  geodesics  and  curvature  and  the  relationship  between  them.   Using  
these  ideas  we  will  show  how  local  geometric  conditions  can  lead  to  global  topological  constraints.  
 
Theory  of  (embedded)  surfaces:  Gauss  map,  second  fundamental  form,  curvature  and  Gauss  Theorem  
Egregium.  Riemannian  manifolds:  Levi-­Civita  connection,  geodesics,  (Riemann)  curvature,  Jacobi  fields.  
Isometric   immersions   and   second   fundamental   form.   Completeness:   Hopf-­Rinow   Theorem   and   Hadamard  
Theorem.   Constant  curvature.   Variations  of  energy:  Bonnet-­Myers  Theorem  and  Synge  Theorem.  
 

25  
 
M4P52       MANIFOLDS  
 
Term  1  
 
Smooth  manifolds,  quotients,  smooth  maps,  submanifolds,  rank  of  a  smooth  map,  tangent  spaces,  vector  fields,  
vector   bundles,   differential   forms,   the  exterior   derivative,  orientations,  integration  on  manifolds   (with  boundary)  
and  Stokes'  Theorem.  This  module  focuses  on  foundations  as  well  as  examples.  
 
 
 
M4P54         DIFFERENTIAL  TOPOLOGY  
 
Term  2  
 
Prerequisites:   Fundamental  group  and  covering  spaces  from  Algebraic  Topology  (M4P21)  and  vector  fields  and  
differential  forms,  derivatives  and  pull-­backs  of  smooth  maps,  exterior  differentiation  and  integration  from  
Manifolds  (M4P52).  
 
Differential  topology  is  concerned  with  the  topology  of  smooth  manifolds.  
The  first  part  of  the  module  deals  with  de  Rham  cohomology,  a  form  of  cohomology  defined  in  terms  of  
differential   forms.  We  will  prove  the  Mayer-­Vietoris   exact  sequence,  Künneth  formula  and  Poincaré  duality  in  
this  context,   and  discuss  degrees  of  maps  between  manifolds.  
The  second  part  of  the  module  introduces  singular  homology   and  cohomology,  the  relation  to  de  Rham  
cohomology  via  de  Rham's  theorem,  and  the  general  form  of  Poincaré  duality.  Time  permitting,  there  will  also  be  
a  brief  introduction  to  Morse  theory.  
 
M4P57    COMPLEX  MANIFOLDS  
 
Term  2  
 
Prerequisite:  Manifolds  (M4P52).  Some  useful  overlap  with  Differential  Topology  (M4P54).  
 
Complex  and  almost  complex  manifolds,  integrability.  Examples  such  as  the  Hopf  manifold,  projective  space,  
projective  varieties.  Hermitian  metrics,  Chern  connection.  Various  equivalent  formulations  of  the  Kaehler  
condition.  Hodge  decomposition  for  Kaehler  manifolds.  Line  bundles  and  Kodaira  embedding.  Statement  of  
GAGA.  Basic  Kodaira-­Spencer  deformation  theory.    
 
ALGEBRA  AND  DISCRETE  MATHEMATICS  
 
M4P8*   ALGEBRA  3  
 
Term  1  
 
Rings,  integral  domains,  unique  factorization  domains.  
Modules,  ideals  homomorphisms,  quotient  rings,  submodules  quotient  modules.  
Fields,  maximal  ideals,  prime  ideals,  principal  ideal  domains.  
Euclidean  domains,  rings  of  polynomials,  Gauss’s  lemma,  Eisenstein’s  criterion.  
Field  extensions.  
Noetherian  rings  and  Hilbert’s  basis  theorem.  
Dual  vector  space,  tensor  algebra  and  Hom.  
Basics  of  homological  algebra,  complexes  and  exact  sequences.  
 
 
M4P10*    GROUP  THEORY  
 
Term  1  
 
An  introduction  to  some  of  the  more  advanced  topics  in  the  theory  of  groups.  

26  
 
  
Composition  series,  Jordan-­Hölder  theorem,  Sylow’s  theorems,  nilpotent  and  soluble  groups.  
Permutation  groups.    Types  of  simple  groups.  
 
 
M4P11*    GALOIS  THEORY  
 
Term  2  
 
The  formula  for  the  solution  to  a  quadratic  equation  is  well-­known.  There  are  similar  formulae  for  cubic  and  
quartic  equations  but  no  formula  is  possible  for  quintics.  The  module  explains  why  this  happens.  
 
Irreducible  polynomials.  Field  extensions,  degrees  and  the  tower  law.  Extending  embeddings.  Normal  field  
extensions,  splitting  fields,  separable  extensions.  Groups  of  automorphisms,  fixed  fields.  The  fundamental  
theorem  of  Galois  theory.  Finite  fields,  cyclotomic  extensions.  Extensions  of  the  rationals  and  Frobenius  
elements.  The  solubility  of  polynomials  of  degree  at  most  4  and  the  insolubility  of  quintic  equations.  
 
M4P12*    GROUP  REPRESENTATION  THEORY  
 
   
Term  2  
 
Representations  of  groups:  definitions  and  basic  properties.  Maschke's  theorem,  Schur's  lemma.  
Representations  of  abelian  groups.  Tensor  products  of  representations.  
The  character  of  a  group  representation.  Class  functions.  Character  tables  and  orthogonality  relations.  
Finite-­dimensional  algebras  and  modules.  Group  algebras.  Matrix  algebras  and  semi-­simplicity.  
 
Representations  of  quivers.  
 
M4P17*    ALGEBRAIC  COMBINATORICS  
 
Term  1  
 
An  introduction  to  a  variety  of  combinatorial  techniques  that  have  wide  applications  to  other  areas  of  
mathematics.  
 
Elementary  coding  theory.    The  Hamming  metric,  linear  codes  and  Hamming  codes.  
Combinatorial  structures:  block  designs,  affine  and  projective  planes.  Construction  of  examples  using  finite  fields  
and  vector  spaces.    Steiner  systems  from  the  Golay  code.    Basic  theory  of  incidence  matrices.  
Strongly  regular  graphs:  examples,  basic  theory,  and  relationship  with  codes  and  designs.  
 
The  Mathieu  groups  and  their  relationship  with  codes  and  strongly  regular  graphs.  
 
 
M4P46         LIE  ALGEBRAS  
 
Term  2  
 
The  semisimple  complex  Lie  Algebras:  root  systems,  Weyl  groups,  Dynkin  diagrams,  classification.  
Cartan  and  Borel  subalgebras.   Classification  of  irreducible  representations.  
 
 
M4P55         COMMUTATIVE  ALGEBRA  
 
Term  1      
 
Prime  and  maximal  ideals,  nilradical,  Jacobson  radical,  localization.  Modules.  Primary  decomposition  of  ideals.  
Applications   to   rings   of   regular   functions   of   affine   algebraic   varieties.   Artinian   and   Noetherian   rings,  

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discrete   valuation   rings,   Dedekind   domains.   Krull   dimension,   transcendence   degree.   Completions   and  
local   rings.   Graded  rings  and  their  Poincaré  series.  
 
M4P61            INFINITE  GROUPS  
 
Term  1  
 
Free  groups.  Group  presentations,  Tietze  transformations,  the  word  problem.  Residually  finite  groups.  Cayley  
graphs,  actions  on  graphs,  the  Nielsen-­-­Schreier  Theorem.  Free  products,  the  Table-­Tennis  Lemma,  
amalgams.  HNN  extensions,  the  Higman  Embedding  Theorem,  the  Novikov-­-­Boone  Theorem.  Geometry  of  
groups,  hyperbolic  groups.  
 
M4P63  ALGEBRA  IV  
 
Term  2  
 
An  introduction  to  homological  algebra:  
Projective,  injective,  and  flat  modules;;  tensor  product;;  simple,  semisimple,  and  indecomposable  modules;;  
representations  of  quivers.    
Modules  over  principal  ideal  domains.  
Abelian  categories,  chain  complexes,  the  homotopy  category.  
Resolutions,  derived  functors,  Tor  and  Ext,  Koszul  complexes.  
Additional  topics  may  include:  group  homology  and  cohomology,  an  introduction  to  triangulated  and  derived  
categories,  Morita  equivalence.    
 
M4P65*  MATHEMATICAL  LOGIC  
 
Term  1  
 
The  module  is  concerned  with  some  of  the  foundational  issues  of  mathematics.  In  propositional  and  predicate  
logic,  we  analyse  the  way  in  which  we  reason  formally  about  mathematical  structures.  In  set  theory,  we  will  look  
at  the  ZFC  axioms  and  use  these  to  develop  the  notion  of  cardinality.  These  topics  have  applications  to  other  
areas  of  mathematics:  formal  logic  has  applications  via  model  theory  and  ZFC  provides  an  essential  toolkit  for  
handling  infinite  objects.    
 
In  addition  to  the  material  below,  this  M4  version  of  the  module  will  have  additional  extension  material  for  self-­
study.  This  will  require  a  deeper  understanding  of  the  subject  than  the  corresponding  M3  module.  
 
Propositional  logic:  Formulas  and  logical  validity;;  a  formal  system;;  soundness  and  completeness.  
 
Predicate  logic:  First-­order  languages  and  structures;;  satisfaction  and  truth  of  formulas;;  the  formal  system;;  
Goedel’s  completeness  theorem;;  the  compactness  theorem;;  the  Loewenheim-­Skolem  theorem.  
 
Set  theory:  The  axioms  of  ZF  set  theory;;  ordinals;;  cardinality;;  the  Axiom  of  Choice.  

M4P72  MODULAR  REPRESENTATION  THEORY  

Term  2  
 
Modular  representation  theory  is  the  study  of  representations  of  finite  groups  over  fields  of  characteristic  other  
than  zero.  It  has  many  applications  and  connections  both  inside  and  outside  group  theory.  The  subject  was  
pioneered    
by  Richard  Brauer,  whose  methods  were  mainly  character-­theoretic.  Later  on,  the  theory  was  revolutionised  by  
J.  A.  Green,  whose  techniques  were  different  to  those  of  Brauer,  the  main  goal  being  to  understand  the  

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modules,  rather  than  just  their  characters.  This  course  will  combine  both  approaches.  The  material  covered  will  
include  Brauer  characters,  defect  groups,  blocks,  decomposition  numbers  as  well  as  projective  and  injective  
modules.  For  much  of  the  theory,  our  guiding  example  will  be  the  family  of  finite  simple  groups  PSL(2,p),  where  
p  is  a  prime  number.  
 
-­-­  Modular  character  theory:  p-­singular  and  p-­regular  elements,  Brauer  characters.    
-­-­  Decomposition  numbers  and  the  decomposition  matrix.  Counting  irreducible  modules.  
-­-­  Irreducible,  projective  and  injective  modules.  Projective  indecomposable  modules.  
-­-­  Group  algebras,  blocks  and  defect  groups.  
-­-­  Vertices  and  sources;;  the  Green  correspondence.  

 
NUMBER  THEORY  
 
M4P14*    NUMBER  THEORY  
 
Term  1  
   
The  module  is  concerned  with  properties  of  natural  numbers,  and  in  particular  of  prime  numbers,  which  can  be  
proved  by  elementary  methods.  
Fermat-­Euler  theorem,  Lagrange's  theorem.    Wilson's  theorem.  
Arithmetic  functions,  multiplicative  functions,  perfect  numbers,  Möbius  inversion,  Dirichlet  Convolution.  
Primitive  roots,  Gauss's  theorem,  indices.  
Quadratic  residues,  Euler's  criterion,  Gauss's  lemma,  law  of  quadratic  reciprocity,  Jacobi  symbol.  
Sums  of  squares.  Distribution  of  quadratic  residues  and  non-­residues.  
Irrationality,  Liouville's  theorem,  construction  of  a  transcendental  number.  
Diophantine  equations.  Pell's  equation,  Thue's  Theorem,  Mordell's  equation.  
 
 
 
M4P15*     ALGEBRAIC  NUMBER  THEORY  
 
Term  2      
 
An  introduction  to  algebraic  number  theory,  with  emphasis  on  quadratic  fields.    In  such  fields  the  familiar  unique  
factorisation  enjoyed  by  the  integers  may  fail,  but  the  extent  of  the  failure  is  measured  by  the  class  group.  

The  following  topics  will  be  treated  with  an  emphasis  on  quadratic  fields   .  
Field   extensions,   minimum   polynomial,   algebraic   numbers,   conjugates   and   discriminants,   Gaussian   integers,  
algebraic  integers,  integral  basis,  quadratic  fields,  cyclotomic  fields,  norm  of  an  algebraic  number,  existence  of  
factorisation.  

Factorisation  in      Ideals,   Z  -­basis,  maximal  ideals,  prime  ideals,  unique  factorisation  theorem  of  ideals  and  
consequences,  relationship  between  factorisation  of  numbers  and  of  ideals,  norm  of  an  ideal.   Ideal  classes,  
finiteness  of  class  number,  computations  of  class  number.  
Fractional   ideals,   Minkowski’s  theorem  on  linear  forms,  Ramification,  characterisation  of  units  of  cyclotomic  
fields,  a  special  case   of  Fermat’s  last  theorem.  
 
 
M4P32         NUMBER  THEORY:   ELLIPTIC  CURVES  
 
Term  1  
 
The  p  -­adic  numbers.   Curves  of  genus  0  over   Q  .   Cubic  curves  and  curves  of  genus  1.   The  group  law  on  a  
cubic  curve.   Elliptic  curves  over  p  -­adic  fields  and  over   Q  .   Torsion  points  and  reduction  mod  p  .   The  weak  
Mordell-­Weil  theorem.   Heights.   The  (full)  Mordell-­Weil  theorem.  

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M4P58         MODULAR  FORMS  
 
Term  1  
 
The  action  of  SL(2,R)  on  the  upper  half  plane.  Congruence  subgroups.  Fundamental  domains.  
 Modular  curves.  Cusps.  Modular  forms.  Expansions  at  cusps.  Examples:    
Eisenstein  series.  The  Delta  function.  Finite-­dimensionality  of   spaces  of  modular  forms.  
Hecke  operators.  Eigenforms.  Petersson  inner  product.  
Non-­examinable:  relation  to  elliptic  curves.  Fermat's  Last  Theorem.  
 
 
 
STATISTICS  
 
M4S1*   STATISTICAL  THEORY    
 
Term  2  
 
This  module  deals  with  the  criteria  and  the  theoretical  results  necessary  to  develop  and  evaluate  optimum  
statistical  procedures  in  hypothesis  testing,  point  and  interval  estimation.    
Theories  of  estimation  and  hypothesis  testing,  including  sufficiency,  completeness,  exponential  families,  
minimum  variance  unbiased  estimators,  Cramér-­Rao  lower  bound,  maximum  likelihood  estimation,  Rao-­
Blackwell  and  Neyman-­Pearson  results,  and  likelihood  ratio  tests  as  well  as  elementary  decision  theory  and  
Bayesian  estimation.  
 
M4S2*   STATISTICAL  MODELLING  2  
 
Term  2  
 
Prerequisites:    This  module  leads  on  from  the  linear  models  covered  in  M2S2  and  Probability  and  Statistics  2  
covered  in  M2S1.  
 
The  Generalised  Linear  Model  is  introduced  from  a  theoretical  and  practical  viewpoint  and  various  aspects  are  
explained.  
Generalised  Linear  Model,  as  a  unifying  statistical  framework  –  linear  models  and  quantitative  responses.    
Generalised  Additive  Models,  Kernel  and  non-­parametric  Regression.  Fixed  and  random  effect  models.  
 
The  R  statistical  package  will  be  used  to  expose  how  the  different  models  can  be  applied  on  example  data.  
 
 
 
M4S4*   APPLIED  PROBABILITY    
 
Term  1  
 
This   module   aims   to   give   students   an   understanding   of   the   basics   of   stochastic   processes.   The   theory   of  
different   kinds   of   processes   will   be   described,   and   will   be   illustrated   by   applications   in   several   areas.   The  
groundwork   will   be   laid   for   further   deep   work,   especially   in   such   areas   as   genetics,   finance,   industrial  
applications,  and  medicine.    
 
Revision   of   basic   ideas   of   probability.   Important   discrete   and   continuous   probability   distributions.   Random  
processes:   Bernoulli   processes,   point   processes.   Poisson   processes   and   their   properties;;   Superposition,  
thinning   of   Poisson   processes;;   Non-­homogeneous,   compound,   and   doubly   stochastic   Poisson   processes.  
Autocorrelation   functions.   Probability   generating   functions   and   how   to   use   them.   General   continuous-­time  
Markov   chains:   generator,   forward   and   backward   equations,   holding   times,   stationarity,   long-­term   behaviour,  
jump  chain,  explosion;;  birth,  death,  immigration,  emigration  processes.  Differential  and  difference  equations  and  
pgfs.   Finding   pgfs.   Embedded   processes.   Time   to   extinction.   Queues.   Brownian   motion   and   its   properties.  

30  
 
 Random  walks.  Gambler’s  ruin.  Branching  processes  and  their  properties.  Galton-­Watson  model.  Absorbing  and  
reflecting   barriers.   Markov   chains.   Chapman-­Kolmogorov   equations.   Recurrent,   transient,   periodic,   aperiodic  
chains.  Returning  probabilities  and  times.  Communicating  classes.  The  basic  limit  theorem.  Stationarity.  Ergodic  
Theorem.  Time-­reversibility.  Martingales.  
 
 
M4S8*   TIME  SERIES  
 
Term  1    
 
An  introduction  to  the  analysis  of  time  series  (series  of  observations,  usually  evolving  in  time)  is  given,  which  
gives  weight  to  both  the  time  domain  and  frequency  domain  viewpoints.    Important  structural  features  (e.g.  
reversibility)  are  discussed,  and  useful  computational  algorithms  and  approaches  are  introduced.    The  module  is  
self-­contained.  
 
Discrete  time  stochastic  processes  and  examples.    ARMA  processes.    Trend  removal  and  seasonal  adjustment.    
General  linear  process.    Invertibility.    Directionality  and  reversibility  in  time  series.    Spectral  representation.    
Aliasing.    Generating  functions.    Estimation  of  mean  and  autocovariance  sequence.    The  periodogram.    
Tapering  for  bias  reduction.    Parametric  model  fitting.    Forecasting.  
 
 
 
M4S9*   STOCHASTIC  SIMULATION  
     
Term  1  
 
Prerequisites:    Material  from  M2S1  would  form  a  firm  foundation.  
 
Computational  techniques  have  become  an  important  element  of  modern  statistics  (for  example  for  testing  new  
estimation  methods  and  with  notable  applications  in  biology  and  finance).  The  aim  of  this  module  is  to  provide  an  
up-­to-­date   view   of   such   simulation   methods,   covering   areas   from   basic   random   variate   generation   to   Monte  
Carlo  methodology.  The  implementation  of  stochastic  simulation  algorithms  will  be  carried  out  in  R,  a  language  
that  is  widely  used  for  statistical  computing  and  well  suited  to  scientific  programming.  
 
Pseudo-­random   number   generators.   Generalized   methods   for   random   variate   generation.   Monte   Carlo  
integration.   Variance   reduction   techniques.   Markov   chain   Monte   Carlo   methods   (including   Metropolis-­Hastings  
and   Gibbs   samplers).   Monitoring   and   optimisation   of   MCMC   methods.   Introduction   to   sequential   Monte   Carlo  
methods.  
 
 
M4S14*    SURVIVAL  MODELS  AND  ACTUARIAL  APPLICATIONS  
 
Term  2  
 
Survival  models  are  fundamental  to  actuarial  work,  as  well  as  being  a  key  concept  in  medical  statistics.    This  
module  will  introduce  the  ideas,  placing  particular  emphasis  on  actuarial  applications.  
 
Explain  concepts  of  survival  models,  right  and  left  censored  and  randomly  censored  data.  Introduce  life  table  
data  and  expectation  of  life.  
Describe  estimation  procedures  for  lifetime  distributions:  empirical  survival  functions,  Kaplan-­Meier  estimates,  
Cox  model.  Statistical  models  of  transfers  between  multiple  states,  maximum  likelihood  estimators.  
Binomial  model  of  mortality.  Counting  process  models  and  the  Poisson  model.  Estimation  of  transition  intensities  
that  depend  on  age.  
Graduation  and  testing  crude  and  smoothed  estimates  for  consistency.  
 
 
M4S16*    CREDIT  SCORING    
 

31  
 
 Term  1  
 
Prerequisites:    Statistical  Modelling  1  (M2S2)  with  some  dependency  on  Statistical  Modelling  2  (M3S2).  
 
This  course  introduces  the  fundamentals  of  credit  scoring  and  predictive  analystics.  We  cover  the  aims  and  
objectives  of  scoring,  along  with  legislative  and  commercial  aspects.  We  consider  issues  regarding  consumer  
credit  data:  characteristics,  transformations,  data  quality  and  transaction  types.  The  concept  of  a  statistical  
scorecard  is  introduced  and  models  developed  using  logistic  regression,  Naïve  Bayes  and  decision  tree  
methods.  Application  and  behavioural  model  types  and  characteristics,  including  segmented  models  are  
explored.    Basic  methods  of  model  selection,  estimation  and  testing  are  considered,  along  with  issues  of  
selection  bias  and  reject  inference.  Probability  of  default  (PD)  models  are  introduced,  along  with  probability  
calibration  and  cost-­basd  measures  for  model  assessment.  
The  R  statistical  package  will  be  used  to  explore  credit  scoring  models  on  example  data.  
 
M4S17*    QUANTITATIVE  METHODS  IN  RETAIL  FINANCE    
 
Term  2  
 
Prerequisites:    Essential  -­  Credit  Scoring  1  (M3/4S16).    Useful  –  Statistical  Modelling  2  (M3S2).  
This  course  explores  advanced  and  new  methods  in  retail  finance,  dealing  with  statistical  modelling  and  
optimization  problems.  Core  topics  will  be:  behavioural  models,  profitability,  fraud  detection  and  regulatory  
requirements.  
Specific  topic  areas  are:-­  
Survival  models  for  credit  scoring  to  determine  time  to  default  and  include  time  varying  information.  
Roll-­rate  and  Markov  transition  models  to  determine  patterns  of  missed  payments.  
Mover-­Stayer  models  of  behaviour.  
Profit  estimation:  concepts  and  use  of  behavioural  models.  
Setting  optimal  credit  limits.  
Fraud  detection  
Neural  networks  for  fraud  detection.  Back-­propagation  and  gradient  descent  methods.  
Cost  analysis  of  AUC  and  the  H  measure.  
Expected  Loss,  PD,  EAD  and  LGD  models  (using  beta  regression,  Tobit  and  classification  tree  structures).  
Regulation  and  portfolio-­level  analysis.  Capital  requirements.  One-­factor  Merton-­type  model.  
Asset  correlation  and  mixed  effects  panel  models.  
The  R  statistical  package  will  be  used  to  explore  a  topic  from  the  course  based  on  a  retail  finance  data  set.  
 
M4S18++  TOPICS  IN  ADVANCED  STATISTICS    
 
Term  2    
 
Students  need  explicit  permission  from  the  DUGS  to  take  this  module.  
This  is  a  demanding  10  ECTS  module  comprising  a  choice  of  two  from  four  already  existing  half-­modules  from  
the  MSc  in  Statistics,  one  from  pool  A  and  one  from  pool  B.  Each  half-­module  comprises  around  20  lectures.  
 
Note  for  example  that  the  choice  A1  &  B1  will  be  assessed  by  2  x  90  minute  exams,  and  A2  &  B1  by  a  90  
minute  exam  (B1)  and  continuous  assessment  (A2).  
 
Pool  A:  
M4S18A1:  Multivariate  Analysis  [90  minute  exam]  
Key  topics  of  linear  algebra.  Standard  multivariate  notations.  The  covariance  matrix.  The  multivariate  normal  
distribution.  Principle  Component  Analysis.  Multivariate  Discriminant  Analysis.  The  central  and  non-­central  
chi-­square  distributions.  The  Kronecker  product.  The  Wishart  distribution.  The  F-­distribution  and  Hotelling’s  
T2  statistic.  Selected  likelihood  ratio  tests.  Ordinary,  multiple  and  partial  correlation  coefficients.  
 
M4S18A2:  Machine  Learning  [Continuous  assessment  through  coursework]  
Introduction  to  statistical  pattern  recognition  and  machine  learning.  Methods  for  feature  extraction,  
dimensionality  reduction,  data  clustering  and  pattern  classification.  State-­of-­art  approaches  such  as  support  
vector  machines  and  ensemble  learning  methods.  Real-­world  applications  to  real  data  sets.  

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Pool  B:  
M4S18B1:  Graphical  Models  [90  minute  exam]  
  Graphical  modelling  for  both  (a)  a  vector  of  random  variables,  and  (b)  vector-­valued  time  series.    Conditional  
independence.  Dependence  structure  and  graphical  representation.  Markov  properties.  Conditional  
independence  graphs.  Decomposable  models.  Graphical  Gaussian  models.  Model  selection.  Directed  acyclic  
graphs  (DAGs),  Bayesian  networks.  Graphical  modelling  of  time  series  (model  selection,  Kullback-­Leibler  
approach).  (Some  prior  knowledge  of  time  series  analysis  would  be  helpful  for  part  (b),  the  last  section.)  
 
M4S18B2:  Bayesian  Methods  [90  minute  exam  +  Coursework]  
       Probability  (definitions  and  interpretations);;  Bayes’s  theorem  and  the    
         law  of  total  probability;;  parameter  estimation;;  multi-­level  models;;    
         model  comparison;;  hypothesis  testing;;  information  theory;;    
         the  maximum  entropy  principle;;  and  experimental  design.  
 
OTHER  “NON-­MATHEMATICAL”  MATHEMATICS  MODULES  
 
 
M3B  THE  MATHEMATICS  OF  BUSINESS  AND  ECONOMICS  
 
Term  1    
 
This  module  aims  to:    
Give  a  broad  mathematical  introduction  to  both  microeconomics  and  macroeconomics  with  a  particular  
emphasis  on  the  former.  
Consider  the  motivations  and  optimal  behaviours  of  both  firms  and  consumers  in  the  marketplace,  and  show  
how  this  leads  to  the  widely  observed  laws  of  supply  and  demand.  
Look  at  the  interaction  of  firms  and  consumers  in  markets  of  varying  levels  of  competition.  
Discuss  the  interplay  of  firms,  households  and  the  government  on  a  microeconomic  scale.  
   
Syllabus:  
Theory  of  the  firm  
Profit  maximisation  for  a  competitive  firm.  Cost  minimisation.  Geometry  of  costs.  Profit  maximisation  for  a  non-­
competitive  firm.  
Theory  of  the  consumer  
Consumer  preferences  and  utility  maximisation.  The  Slutsky  equation.  
Levels  of  competition  in  a  market  
Consumers’  and  Producers’  surplus.  Deadweight  loss.  
Macroeconomic  theory  
Circular  flow  of  income.  Cross  Domestic  Product.  Social  welfare  and  allocation  of  income.  
 
Mathematical  Methods:  
(Constraint)  Optimisation.  Quasi-­concavity.  Preferences  relations  and  orders.  
 
 
M3C  INTRODUCTION  TO  HIGH  PERFORMANCE  SCIENTIFIC  COMPUTING  
 
Term  1  
 
High-­performance  computing  centres  on  the  solution  of  large-­scale  problems  that  require  substantial  
computational  power.  This  will  be  a  practical  module  that  introduces  a  range  of  powerful  tools  that  can  be  used  
to  efficiently  solve  such  problems.  By  the  end  of  the  module,  which  will  be  examined  by  projects,  students  will  be  
prepared  to  tackle  research  problems  using  the  tools  of  modern  high-­performance  scientific  computing  in  an  
informed,  effective,  and  efficient  manner.  
           
Contents:            
Getting  started:  working  with  UNIX  at  the  command  line          
Software  version  control  with  git  and  Bitbucket  

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Programming  and  scientific  computing  with  Python  
Modular  programming  with  modern  Fortran,  using  scientific  libraries,  interfacing  Python  and  Fortran  
OpenMP  (with  Fortran)  for  parallel  programming  of  shared-­memory  computers  
MPI  (with  Fortran)  for  programming  on  distributed-­memory  machines  such  as  clusters  
Cloud  computing  
Good  programming  practice:  planning,  unit  testing,  debugging,  validation  (to  be  integrated  with  the  above  topics  
and  the  programming  assignments.)    
 
 
 
M3T   COMMUNICATING  MATHEMATICS  
 
 (Terms  2  &  3)  
 
(Note:  only  G104  students  who  have  already  registered  for  it  may  take  this  module  in  their  4th  year)  
 
This  module  will  give  students  the  opportunity  to  observe  and  assist  with  teaching  of  Mathematics  in  local  
schools.    Entry  to  the  module  is  by  interview  in  the  preceding  June  and  numbers  will  be  limited.  It  is  required  for  
anyone  on  the  Mathematics  with  Education  degree  coding.  
 
For  those  selected  there  will  follow  a  one  day  training  course  in  presentation  skills  and  other  aspects  of  teaching.    
Students  will  be  assigned  to  a  school  where  they  will  spend  ten  half  days  in  Term  2,  under  the  supervision  of  a  
teacher.    Assessment  will  be  based  on  a  portfolio  of  activities  in  the  school,  a  special  project,  evaluation  by  the  
school  teacher  and  an  oral  presentation.  
 
CENTRE  FOR  LANGUAGE,  CULTURE  AND  COMMUNICATION/BUSINESS  
SCHOOL  
 
Students  may  consider  broadening  their  study  programme  by  taking  advantage  of  the  CLCC/Business  School  
provision.      
 
Note  that  Centre  for  Co-­Curricular  Studies  modules  extend  throughout  Terms  1  and  2  and  some  
modules  may  be  examined  in  January.    Taking  the  HSCS3006  Humanities  Project  normally  also  requires  
explicit  permission  from  the  Centre  for  Co-­Curricular  Studies.  
 
Module  
Module  Titles   Terms   ECTS  Values  
Codes  
HGC31     Lessons  from  History   1  +  2   6  
HGC33   Creative  Futures   1  +  2   6  
HSCS3001   Advanced  Creative  Writing   1  +  2   6  
HSCS3002   History  of  Science,  Technology  and  Industry   1  +  2   6  
HSCS3003   Philosophy  of  Mind   1  +  2   6  
HSCS3004   Contemporary  Philosophy   1  +  2   6  
HSCS3006   Humanities  Project   1  +  2   6  
HSCS3007   Conflict,  Crime  and  Justice   1  +  2   6  
HSCS3008   Visual  Culture,  Knowledge  and  Power   1  +  2   6  
HSCS3011   Psychology  of  Music   1  +  2   6  
HSCS3012   How  do  you  Know?   1  +  2   6  
HSCS2007   Music  Technology   1  +  2   6  
BS0808   Finance  and  Financial  Management   2   6  
BS0820   Managing  Innovation   1   6  
 
 
Syllabus  and  timetabling  information  can  be  viewed  online  at:    
CLCC:  http://www.imperial.ac.uk/horizons  
Business  School:  http://wwwf.imperial.ac.uk/business-­school/programmes/undergraduate-­study/bpes-­
programme/  

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Note  that  places  in  CLCC  and  Business  School  modules  are  normally  limited  and  registration  should  be  done  
separately  via  the  Centre  for  Co-­Curricular  Studies  and  Business  School  websites.  
 
Note  that  a  change  in  degree  code  registration  can  lead  to  your  registration  for  a  CLCC/  BPES  module  being  
revoked,  so  you  must  contact  the  CLCC/  BPES  programme  if  you  are  planning  to  make  such  a  change.  Save  a  
screenshot  of  your  registration  to  help  in  any  dispute.  
 
 
IMPERIAL  HORIZONS  
 
The  College  has  created  the  ‘Imperial  Horizons’  programme  to  broaden  students’  education  and  enhance  their  
career  prospects.    This  programme  is  open  to  all  undergraduate  students.  
 
The  Department  of  Mathematics  always  endeavours  to  avoid  timetabling  Mathematics  modules  during  the  times  
allocated  for  Horizons  modules.  
 
Note  that  modules  on  this  programme  (except  for  the  ones  listed  separately  above  as  approved  
modules  for  4th  year  students)  do  not  contribute  to  degree  Honours  marks  but  they  do  have  an  ECTS  
value  of  6.  
 
Further  information  about  the  ‘Horizons’  programme  can  be  found  at:  http://www.imperial.ac.uk/horizons  
 
 
CHANGE  OF  PROGRAMME  –  Tier  4  visa  holders.  
 
Holders  of  Tier  4  visas  who  are  considering  changing,  or  who  are  required  to  change  between  BSc  and  MSci  
programmes  should  consult  the  information  available  at:  
 
http://www.imperial.ac.uk/study/international-­students/visas-­and-­immigration/changes-­to-­course-­of-­study/  
 
and  contact  the  International  Student  Support  Team  if  necessary.  
 
 

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