Supporting information from the supervisor about the project.

Please limit each section to 100 words.
 
1. Brief outline of the project *

McKay Quivers of Finite Matrix Groups

The project concerns finite subgroups G in SL(2,CC) or
SL(3,CC). These were studied from the 19th century. About 25
years ago, I initiated a new research direction called the
higher dimensional McKay correspondence, relating the
representation theory of G with the geometry of the resolution
of singularities of the orbit space X = CC^n/G (or orbifold).
The McKay quiver Q of G is a directed graph, and its
calculation is a first step in this study. In good cases, the
representation theory of Q provides the resolution of
singularities of X.

Describe how:
2. The student will have the opportunity to achieve a substantial outcome *

The proposal builds on ideas of the 3rd year algebra courses,
particularly groups and their representations. The research
project considers suitable classes of matrix groups, usually
2 x 2 or 3 x 3, especially those containing a large Abelian
normal subgroup. The aim is to work out the representation
theory of the bigger group, and in particular its McKay
quiver, in terms of that of the Abelian subgroup. Although the
study starts from basic undergraduate algebra, it leads on to
research topics in modern algebraic geometry, particularly
orbifolds and their resolution and the McKay correspondence.


3. The student will be able to develop a range of research skills *

Beyond a broad sweep of basic group theory and representation
theory, the project covers several bordering areas of
geometry, topology and algebraic geometry. The student will
need to absorb the material of research papers, including
several substantial modern papers, and the theses of several
of my PhD students. There has been quite a lot of recent
research involving moduli spaces of representaion of quivers,
which goes well beyond undergraduate algebra. The student will
need to learn to juggle with two of three of these theories at
the same time.


4. The student will be able to develop a range of transferable skills *

The study involves different aspects of computer algebra, such
as the symbolic computation program Magma, which was
originally developed for groups and representations, but which
is currently finding much wider applications in geometry and
topology. The skills involved in this are of key importance in
many branches of applicable math, notably the cryptography
used in the financial sector and international espionage.

Presenting the outcome as a written document (in LaTeX with
embedded graphics and computer routines) also involves
acquiring useful skills.


5. The project links with ongoing research being undertaken in
your department or centre *

The McKay correspondence is a large component of my research
output: 6 of my research papers since 1992 are related to the
McKay correspondence, as well as my EPSRC project EP/H023267/1
"Orbifolds and birational geometry" and a section of the
currently running Warwick EPSRC Symposium. In addition,
several of my Warwick PhD students wrote theses around this
topic. In addition, this kind of representation theory and
computer algebra is part of the standard repertory of several
of the Warwick algebraists. There is also an interesting link
with the geometric group theory studied by the Warwick
topologists.


6. You will support the student to acquire any necessary
discipline-specific skills and knowledge needed to complete
the project successfully. *

I have successfully supervised several previous project
students, notably Ben Wormleighton, who did a URSS project
with me in summer 2013, and is currently talking to Ed Pearce
about his proposed project, and the visiting Erasmus student
Javier Carrasco, who wrote a 4th year MMath research project
under my supervision in 2014, and who produce a dissertation
that is an extremely useful summary of the literature on
finite subgroups of SL(2,CC) and SL(3,CC).


Do you feel that there are any ethical considerations in this project? *
Yes No  NO

Please give the total amount the student is applying for from URSS (£) *

Please summarise the funding breakdown here to ensure you have
both agreed on the purposes of the funding. *

Please give the code for transfer of the bursary if necessary? *

What type of code is this? *

Cost code Internal order code Project code

Please provide the name and email of a finance contact for
your department to whom all financial queries may be directed:
 
Finance contact name *
Finance contact email *







 
Title of research project *
McKay quivers of finite matrix groups

This title MUST be exactly the same as that on the supervisor
application form. If it is not, the forms may not be linked
together and may not be scored.
 
Briefly outline your project (max. 100 words) *
What benefits could your project bring (e.g. to the community,
wider world, or knowledge base – max 100 words)? *
Project department/centre *
First name of lead academic supervisor *
Family name of lead academic supervisor *
Name(s) of Additional Supervisor(s) (one per line)
Proposed start date of project *
 
You must ensure that your project finishes far enough in
advance of the URSS deadline to allow you to complete and
submit the URSS required tasks.
 
Proposed end date of project *
Total duration of project (weeks) *
Will you be travelling outside of the UK as part of your
project? *
Yes No
If yes, are you going to one of the following:
If not one of the above, please give the location (leave blank
if not applicable):

As a pilot in 2015, certain projects have been invited as part
of an exchange with institutions in Brazil and Colombia. If
your project has not already been agreed for this initiative,
then it is not eligible this year.
 
Is your project part of this initiative? *
Yes No
Do you feel that there are any ethical considerations in your
project? *
Yes No
Supporting information from you, the student about the
benefits to you of completing a project:

Please limit each section to 100 words.
 
1. Why do you wish to participate? *
2. What would you like to learn? *
3. Which transferable and research skills would you like to
develop? *
4. What relevant experience do you have that would help you
with your project? *

5. How do you see this experience supporting your current
studies, choice of further study and/or future career? *

Have you undertaken a URSS project previously? *
Yes No
If yes, please state the year, your reasons for wanting to do
another project and what additional skills and experience you
will gain from this project.

 
How much are you applying for from URSS (£ – max. £1000) *
Please include a concise breakdown of your anticipated costs
that your bursary would support. This is a field included in
the form your prospective supervisor will have completed, so
please discuss it with him/her. Your breakdowns should be the
same *







Dear Ed,

I don't think I can really improve this all that much.

You could put just a bit more emphasis on the substance
of the project -- representation theory, calculations in
more substantial depth than u/g exercise sheets, and an
appreciation of how undergraduate algebra meshes in to
more advance topics in algebra, geometry and topology.
After all, the question "what do you want to learn?"
might include some substance as well as platitudes
about general interest.

That said, what you've done so far is pretty much par for the
course. Plug it in when you are ready, and I will plug in
my sections.

Miles




Title of research project
McKay Quivers of Finite Matrix Groups

Briefly outline your project

The project is concerned with groups, which can be viewed as
the symmetries of geometrical objects such as the set of
reflections and rotations of a cube, a sphere, or a crystal
lattice. We try to study the structure of groups associated
with more complex shapes such as the symmetries of a
dodecahedron or a twisted prism. We aim to better understand
the representations of these groups by building up from
smaller subgroups. In particular, we will investigate methods
of computing the McKay quiver for the larger group in terms of
a given suitable subgroup.

What benefits could your project bring (e.g. to the community,
wider world, or knowledge base)?

The study of symmetry groups has applications in chemistry and
crystallography, in particular in classifying crystal
structures, regular solids, and the symmetries of molecules,
which in turn can lead to the development of novel medicines
through a greater understanding of their physical properties.

In addition to the practical applications, groups have many
theoretical applications in the fields of geometry, topology
and algebraic geometry. In particular, our project involves
representation theory and is related to recent research areas
in the resolution of singularities of orbifolds and the McKay
correspondence in modern algebraic geometry.

Proposed start date of project
Saturday 20th June 2015

Proposed end date of project
Saturday 15th August 2015

Total duration of project

8 Weeks (56 days)

Supporting information from you, the student, about the benefits to you of completing a project:

1. Why do you wish to participate?

I wish to gain an understanding of what it is to do
mathematical research as I am considering whether I’d like to
do a PhD and pursue an academic career in mathematics.

I believe that the project will give me an invaluable
opportunity to explore the concepts and ideas introduced in my
lecture courses further, whilst also getting a taste of what
it’s like to be at the forefront of academic research.

I’ve also been inspired by older peers who have spoken about
their own URSS research projects through Warwick Maths
Society’s academic talks series.

2. What would you like to learn?

The main thing I would like to learn from the project is what
the world of research is like. Furthermore, I would like to
learn about the representation theory of groups in more depth
and how it relates to algebraic geometry in the context of my
project. I would also like to learn how to effectively present
findings to other people in a clear and precise manner. Also,
I would like to learn some programming skills (which I will do
when doing the project) which are specifically useful for
mathematical research, and explore the links between maths and
other subjects.

3. Which transferable and research skills would you like to develop?

I hope to gain an improvement of some skills, such as
presenting work to others in an understandable and precise
manner, how to research a problem with no explicit answer, and
communicating ideas and concepts effectively, all of which
would be beneficial in any line of work beyond university.

Further, I would like to develop research specific skills,
such as searching for and reviewing existing literature,
preparing a mathematics paper for a technical audience,
working with a computer algebra system, and working
collaboratively on a project over a sustained period of time.

4. What relevant experience do you have that would help you
with your project?

The project will be a rather new experience to me. In college
and university I have done projects which have involved a
large element of independent study and learning new topics.
This has given me experience in working on my own, time
management, and resolving problems as they arise.

Currently following the Groups and Representations lecture
course has given me exposure to necessary background knowledge
for the project and a sense for the difficulty which lies
ahead.

5. How do you see this experience supporting your current studies, Choice of further study and/or future career?

By undertaking this project, I hope to develop my problem
solving skills; hence improving how I approach my studies
during my degree. Studying representation theory in such depth
will also impact my future module choices as I will likely opt
for related modules such as Algebraic Geometry, and Geometric
Group Theory in my fourth year.

Furthermore, I hope to go into research after my degree, so
this would give me the perfect insight and preparation for my
future. If I do not go into research, I will still have many
transferable skills which can be applied in any situation.

How much are you applying for from URSS?
£1000
Please include a concise breakdown of your anticipated costs that your bursary would support.
Off-campus accommodation (utilities included) at £98 per week
Subsistence (Self-catering) at £25 per week
£123 per week over 8 weeks = £984
Contingency for bicycle maintenance and repairs £16
Total £1000