
EXTENDED ESSAYS:Option BEE or BOE in Part B of the Final
Honour School of Mathematics
DISSERTATIONS:Option CCD or COD in Part C of the Final
Honour School of Mathematics
SOME IDEAS FOR PROJECTS
April 17, 2015
Contents
1 Logic 41.1 Model Theory — Strongly minimal structures . . . .
. . . . . 41.2 Model Theory — The model theory of cyclic groups. .
. . . . 41.3 Model theory of the real numbers . . . . . . . . . . .
. . . . . 51.4 Ominimal structures . . . . . . . . . . . . . . . .
. . . . . . . 51.5 Local equivalents of the Axiom of Choice . . . .
. . . . . . . . 61.6 Theories of the real numbers . . . . . . . . .
. . . . . . . . . . 6
2 Algebra 62.1 Quantum groups and crystal basis . . . . . . . .
. . . . . . . . 62.2 Affine algebraic groups schemes . . . . . . .
. . . . . . . . . . 72.3 Homotophy type theory . . . . . . . . . .
. . . . . . . . . . . 72.4 Relative algebraic geometry . . . . . .
. . . . . . . . . . . . . 72.5 BeilinsonBernstein localisation for
sl2 . . . . . . . . . . . . . 8
1

2.6 Converses of Lagrange’s Theorem . . . . . . . . . . . . . .
. . 82.7 Alhazen’s Problem . . . . . . . . . . . . . . . . . . . .
. . . . 82.8 Reduction of quadratic forms . . . . . . . . . . . . .
. . . . . 82.9 Uniserial group actions . . . . . . . . . . . . . .
. . . . . . . . 9
3 Geometry and Number Theory 93.1 Simple singularities and the
McKay correspondence . . . . . . 93.2 Recognizing the unknot . . .
. . . . . . . . . . . . . . . . . . . 103.3 Geometrisation of
3manifolds . . . . . . . . . . . . . . . . . . 113.4
Crystallographic groups . . . . . . . . . . . . . . . . . . . . .
123.5 Gromov’s theorem and groups of polynomial growth . . . . . .
123.6 Approximate groups . . . . . . . . . . . . . . . . . . . . .
. . 133.7 Higherorder Fourier analysis . . . . . . . . . . . . . .
. . . . . 13
4 Analysis 144.1 Abelian locally compact groups . . . . . . . .
. . . . . . . . . 144.2 Kazhdan’s property (T) . . . . . . . . . .
. . . . . . . . . . . 144.3 Analysis in a rational world . . . . .
. . . . . . . . . . . . . . 154.4 Analysis of holomorphic functions
with special values . . . . . 154.5 Integration . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 154.6 Measure Theory . . . .
. . . . . . . . . . . . . . . . . . . . . . 154.7 Univalent
functions . . . . . . . . . . . . . . . . . . . . . . . . 164.8
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . .
164.9 Special functions . . . . . . . . . . . . . . . . . . . . . .
. . . 164.10 Riemann surfaces . . . . . . . . . . . . . . . . . . .
. . . . . . 164.11 The Schwarzian derivative . . . . . . . . . . .
. . . . . . . . . 164.12 Chaos in nonlinear ordinary differential
equations . . . . . . . 17
5 Mathematical Methods and Applications 175.1 Mathematics and
the environment . . . . . . . . . . . . . . . . 175.2 Mathematical
biology and physiology . . . . . . . . . . . . . . 185.3 Fluid
dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 195.4
Mechanics of Solids . . . . . . . . . . . . . . . . . . . . . . . .
195.5 Multilayer Networks . . . . . . . . . . . . . . . . . . . . .
. . 19
6 Numerical Analysis 206.1 Stiff ordinary differential equations
. . . . . . . . . . . . . . . 206.2 Numerical approximation of
singular integrals . . . . . . . . . 216.3 Newton’s method for
nonlinear systems . . . . . . . . . . . . . 216.4 Finite element
methods for singularly perturbed problems . . 226.5 A posteriori
error analysis of finite element methods . . . . . . 22
2

6.6 Fast iterative methods for systems of linear equations . . .
. . 236.7 Superconvergence of the Collocation Method with Splines .
. . 23
7 Mathematical Physics 247.1 Extremum principles in theoretical
physics . . . . . . . . . . . 247.2 Hamilton–Jacobi theory . . . .
. . . . . . . . . . . . . . . . . 247.3 Concepts of quantum
mechanics . . . . . . . . . . . . . . . . . 257.4 Symmetries in
quantum mechanics . . . . . . . . . . . . . . . 257.5 Quantum
mechanics of scattering . . . . . . . . . . . . . . . . 257.6
Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . .
257.7 The microwave background . . . . . . . . . . . . . . . . . .
. 257.8 Tests of general relativity . . . . . . . . . . . . . . . .
. . . . . 257.9 Hot big bang versus steady state in cosmology . . .
. . . . . . 267.10 Appearance of moving objects in special
relativity . . . . . . . 267.11 Study of a special metric . . . . .
. . . . . . . . . . . . . . . . 267.12 SO(3), SU(2), Euler angles
and angular momentum . . . . . . 267.13 Momentum space in quantum
mechanics . . . . . . . . . . . . 267.14 Translation of some well
known theorem in euclidean space to
Minkowski space . . . . . . . . . . . . . . . . . . . . . . . .
. 26
8 Stochastics, Discrete Mathematics and Information 278.1
Combinatorics — Graphs of large chromatic number . . . . . 278.2
Mathematical models in finance . . . . . . . . . . . . . . . . .
278.3 Mathematical models in evolution . . . . . . . . . . . . . .
. . 278.4 Duality and Random Walks . . . . . . . . . . . . . . . .
. . . 288.5 The Coupling Method . . . . . . . . . . . . . . . . . .
. . . . 288.6 Applied Probability . . . . . . . . . . . . . . . . .
. . . . . . . 288.7 Operational Research . . . . . . . . . . . . .
. . . . . . . . . . 28
9 History of Mathematics 29
10 General 30
11 Titles of Previous Projects 3111.1 BE Extended Essays . . . .
. . . . . . . . . . . . . . . . . . . 3111.2 CD Dissertations . . .
. . . . . . . . . . . . . . . . . . . . . . 31
3

1 Logic
1.1 Model Theory — Strongly minimal structures
Part C dissertation.
Strongly minimal structures and their combinatorial geometries
are im portant notions of modern model theory. These notion are
also central for understanding principles of a modeltheoretic
classification of classical mathematical structures. There is an
extensive discussion of these in graduatetextbooks.The aims of the
project are:
1. to present and prove basic facts about strongly minimal
structures;
2. to present classical examples of strongly minimal structures
and todiscuss the existence of nonclassical once;
3. to discuss generalisations of strongly minimal
structures.
1.2 Model Theory — The model theory of cyclic groups.
Part C dissertation.
The aim of this project is to understand the modeltheoretic
limit of a sequence of finite cyclic groups and relate it to the
theory of the infinite cyclicgroup.The main intended result would
be the following.
Theorem Given a positive sentence ϕ in the standard language of
Abeliangroups, Z/nZ � ϕ for all n ∈ N, if and only if Z � ϕ.
The proof of the righttoleft implication is relatively easy
and is based onone of the basic facts of model theory, the
preservation of positive formulasunder homomorphisms.
The converse requires much more work and reading. Textbook
material onthe theory of abelian groups and the paper The
elementary theory of abelian
4

groups Ann. Math. Logic 4 (1972), 115–171. by P.Eklof and
E.Fisher isrecommended.
1.3 Model theory of the real numbers
Part C single or double unit dissertation.
According to Gödel’s theorem and its developments, the first
order theoryof the natural numbers is undecidable: there is no
decision procedure fordeciding whether statements are true or not.
However, some other naturalmathematical structures are decidable,
including the theory of the complexnumbers as an algebraically
closed field of characteristic zero, and the theoryof the real
numbers as an ordered field. How are such results established
andcan they be extended to more complicated structures?
D. Marker, Model theory: an introduction, Graduate Texts in
Mathematics217, SpringerVerlag, 2002.
P.J. Cohen, Decision procedures for real and padic fields,
Comm. Pure Appl.Math. 22 (1969), 131–151.
1.4 Ominimal structures
Part C dissertation.
An ominimal structure is a modeltheoretic structure M whose
underlyingdomain M possesses a dense linear order such that the
definable subsets ofM are “as simple as possible” : they are just
the finite unions of points andopen intervals. This simple
requirement has very strong consequences. Forexample, definable
functions in such a structure are continuous (in the ordertopology)
except at finitely many points of their domain. Further, if M isa
field, then definable functions are differentiable except at
finitely manypoints in their domain. A good deal of real (and
complex) analysis can bedeveloped in this setting, where the
underlying field may be very differentfrom the real or complex
numbers.
L. van den Dries, Tame topology and ominimal structures, LMS
LectureNote Series 248, CUP, 1998.
5

1.5 Local equivalents of the Axiom of Choice
The famous equivalence of the Axiom of Choice and the WellOrder
Principlecan be proved ‘locally’: a set X has a choice function if
and only if X is wellorderable. When we come to examine the
equivalence of the Axiom of Choicewith other assertions of Set
Theory we often find that mismatches appearin the local versions.
In Zorn’s Lemma, for example, if X has a choicefunction then in
every inductive partial ordering on X there are maximalelements; on
the other hand, the usual argument requires that there shouldbe
maximal elements in every inductive partial ordering of the power
set ofX2 to yield that there is a choice function on X. Investigate
such ‘gaps’ inlocal equivalents of the Axiom of choice.
1.6 Theories of the real numbers
The system of real numbers may be defined as a complete linearly
orderedfield. What this is may be defined in many ways. In
particular, manydifferent versions of the completeness axiom have
been proposed and used.Collect, compare and contrast these various
theories.
2 Algebra
2.1 Quantum groups and crystal basis
Part C dissertation
Quantum groups and crystal basis: Quantum groups are
deformations of classical groups. Using them it is possible to get
bases with very good propertiesfor representations of reductive
algebraic groups. This project will cover Hopfalgebras, comodules,
quantum groups, crystal basis and canonical basis. Theemphasis will
be on the sl2 case.
Contact: Prof. Kobi Kremnitzer
(kobi.kremnitzer@oriel.ox.ac.uk)
6

2.2 Affine algebraic groups schemes
Part C dissertation
Affine algebraic groups schemes: Affine algebraic groups schemes
are centralobjects in algebraic geometry and in representation
theory. This projectaim at introducing Hopf algebras, their
categories of comodules, differentexamples of commutative Hopf
algebras (affine algebraic group schemes),their Lie algebras and
descent theory.
Contact: Prof. Kobi Kremnitzer
(kobi.kremnitzer@oriel.ox.ac.uk)
2.3 Homotophy type theory
Part C dissertation
Homotopy type theory: Homotopy type theory is a new foundational
language for mathematics. In it basic notion from homotopy theory
are takenas primitive notions. This allows for very elegant and
simple presentation ofhomotopy theory and the theory of homotopy
types. The aim of this projectis to introduce the homotopy
category, introduce the language of homotopytype theory, develop
homotopy theory in this language and compute somehomotopy
types.
Contact: Prof. Kobi Kremnitzer
(kobi.kremnitzer@oriel.ox.ac.uk)
2.4 Relative algebraic geometry
Part C dissertation
Relative algebraic geometry: Relative algebraic geometry is an
approach toalgebraic geometry using category theory. This allows to
generalise algebraicgeometry to many different settings. This
project will cover basic notionsfrom category theory, symmetric
monoidal categories, Grothendieck topologies, algebraic geometry
relative to a symmetric monoidal category and theexample of usual
algebraic geometry and monoid algebraic geometry whichis a version
of the field with one element.
Contact: Prof. Kobi Kremnitzer
(kobi.kremnitzer@oriel.ox.ac.uk)
7

2.5 BeilinsonBernstein localisation for sl2
Part C dissertation
BeilinsonBernstein localisation for sl2: The
BeilinsonBernstein localisationtheorem is one of the most
important tools in the representation theory ofsemisimple Lie
algebras. This project will cover the basics of category theory,the
category of representations of sl2, the Weyl algebra, the
categories of Omodules and Dmodules on the projective line and
the BeilinsonBernsteinlocalisation theorem for sl2.
Contact: Prof. Kobi Kremnitzer
(kobi.kremnitzer@oriel.ox.ac.uk)
2.6 Converses of Lagrange’s Theorem
Lagrange’s Theorem states that if G is a finite group and H is a
subgroupthen H divides G. In general, if m is a divisor of G
there need not bea subgroup of order m. Investigate those finite
groups G with the propertythat for every divisor m of their order
there exists at least one subgroup oforder m.
2.7 Alhazen’s Problem
Alhazen’s Problem asks for the point P on a given spherical
mirror at whicha ray of light is reflected from a source at A to an
observer at B. See John D.Smith ‘The remarkable Ibn alHaytham’,
Math. Gazette, 76 (1992), 189–198for a good account of the problem.
It has been proved [Peter M. Neumann,‘Reflections on reflection in
a spherical mirror’, Amer. Math. Monthly, 105(1998)] that there is
in general no rulerandcompass construction for P .Also, Michael
Drexler & Martin J. Gander, in their paper ‘Circular
Billiard’in SIAM Review 1999, investigate for which configurations
there are fourreflection points, for which only two. Several
further questions of a similar nature suggest themselves, and
should be of a suitable standard for anundergraduate project.
2.8 Reduction of quadratic forms
One of the very useful theorems of algebra states that any real
quadratic form∑ai,jxixj can be changed to the form y
21+· · ·+y2p−y2p+1−· · ·−y2p+q by suitable
8

nonsingular linear change of variables, and that moreover the
numbers p, qare independent of the particular method used
(‘Sylvester’s Law of Inertia’).This theorem depends heavily on the
fact that the coefficients come from R.How far can analogous normal
form theorems be proved for quadratic formsover C, over Q, or over
other fields F , such as finite fields?
2.9 Uniserial group actions
Part B extended essay or Part C dissertation
For a (finite) group G, the following conditions on a transitive
Gspace Ω areequivalent:
(A) that the set of Ginvariant equivalence relations on Ω
(known as Gcongruences or just congruences on Ω) is linearly
ordered (by containment);
(B) that the collection of blocks of imprimitivity containing a
given point ofΩ is linearly ordered by containment;
(C) that the set of subgroups of G containing the stabiliser of
a given pointis linearly ordered by containment.
Alejandra Garrido recently came across uniserial group actions
in her DPhilwork. Examples suggest that it is unlikely that one can
say much about themin general. Nevertheless, it might be an
interesting project to investigate: canone find all uniserial group
actions of degree up to 20 or 30? Can one findall uniserial group
actions of degree p2 or degree pq where p, q are primenumbers? Can
one find all uniserial group actions for groups of order up tosome
bound (such as 60)? And several other such questions.
Contact: Dr Peter Neumann (peter.neumann@queens.ox.ac.uk) or
AlejandraGarrido (Alejandra.GarridoAngulo@maths.ox.ac.uk)
3 Geometry and Number Theory
3.1 Simple singularities and the McKay correspondence
Part B extended essay or Part C dissertation
Finite subgroups G of the group SU(2) can be classified into
types governedby one of the most ubiquitous patterns in
mathematics: the ADE pattern.
9

Geometrically, one can consider the quotient of the complex
plane C2 by thegroup G, obtaining what is known as a simple
singularity. Algebraically, astudy of the representation theory of
the finite group G leads to an orientedgraph called the McKay
quiver. There are fascinating relationships betweenthe geometry of
the quotient, as well as its resolution of singularities, on theone
hand, and the McKay quiver on the other. A deeper study also
bringsin another object classified by ADE patterns, the
corresponding simple Liealgebra; one can also look at the situation
in metric geometry, leading toa highly symmetric metric on these
geometries called a hyperkahler metric.There are many avenues this
project can take, looking at the big picture orconcentrating on one
particular corner, and on some of a large variety ofpossible
approaches into this rich field of ideas.
ItoNakamura: Hilbert schemes and simple singularities,
downloadable
fromhttp://homepages.warwick.ac.uk/staff/Miles.Reid/McKay/
Miles Reid, La correspondance de McKay,
http://xxx.lanl.gov/abs/math/9911165
Contact: Prof. Balazs Szendroi (szendroi@maths.ox.ac.uk)
3.2 Recognizing the unknot
Given a diagram of a knot, how do we decide whether it is the
unknot? Thisis a surprisingly difficult question to answer,
although there are now severalmethods for doing so. The first to
solve the problem was Wolfgang Haken[2], using his theory of normal
surfaces. In your dissertation, you mightexplain this theory [5],
and then perhaps go on to explore its consequencesand extensions,
for example the upper bound on Reidemeister moves by JoelHass and
Jeffrey Lagarias [3], or the existence of an NP algorithm to
solvethe problem [4]. Another more recent solution to the problem
was given byIvan Dynnikov [1], who used arc presentations of knots.
You might chooseto focus on this method instead.
References:
1. I. Dynnikov, Arcpresentations of links: monotonic
simplification. Fund.Math. 190 (2006), 2976.
2. W. Haken, Theorie der Normalflchen. Acta Math. 105 (1961)
245375.
3. J. Hass, J. Lagarias, The number of Reidemeister moves needed
for unknotting. J. Amer. Math. Soc. 14 (2001), no. 2, 399428
10
http://xxx.lanl.gov/abs/math/9911165

4. J. Hass, J. Lagarias, N. Pippenger, The computational
complexity of knotand link problems. J. ACM 46 (1999), no. 2,
185211.
5. S. Matveev, Algorithmic topology and classification of
3manifolds. Secondedition. Algorithms and Computation in
Mathematics, 9. Springer, Berlin,2007. xiv+492 pp. ISBN:
9783540458982
Contact: Prof. Marc Lackenby (lackenby@maths.ox.ac.uk)
3.3 Geometrisation of 3manifolds
In the late 1970s, Bill Thurston revolutionised the study of
3manifolds withthe introduction of his geometrisation conjecture
[3]. Roughly speaking, thisproposed that any compact orientable
3manifold has a “canonical decomposition into geometric pieces”.
This conjecture was a farreaching generalisation of the Poincare
conjecture, which asserted that any 3manifold homotopy equivalent
to the 3sphere is homeomorphic to the 3sphere. Thegeometrisation
conjecture, and hence the Poincare conjecture, were provedby
Perelman in 2003 [1]. Your dissertation should give a precise
explanationof the statement (but not the proof) of the
geometrisation conjecture. Thecanonical decomposition along spheres
and tori should be discussed [2], asshould the eight model
geometries, particularly hyperbolic geometry which isat the heart
of the conjecture. Your dissertation might then go on to
describesome applications of the conjecture.
1. J. Morgan, G. Tian, Ricci flow and the Poincar conjecture.
Clay Mathematics Monographs, 3. American Mathematical Society,
Providence, RI;Clay Mathematics Institute, Cambridge, MA, 2007.
xlii+521 pp. ISBN:9780821843284
2. W. Neumann, G. Swarup, Canonical decompositions of
3manifolds. Geom.Topol. 1 (1997), 2140
3. W. Thurston, Threedimensional manifolds, Kleinian groups and
hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.) 6 (1982), no.
3, 357381.
4. W. Thurston, Threedimensional geometry and topology. Vol. 1.
Editedby Silvio Levy. Princeton Mathematical Series, 35. Princeton
UniversityPress, Princeton, NJ, 1997. x+311 pp. ISBN:
0691083045
Contact: Prof. Marc Lackenby (lackenby@maths.ox.ac.uk)
11

3.4 Crystallographic groups
A crystallographic group is a discrete, cocompact group of
Euclidean isometries. These groups are of interest to chemists as
well as mathematicians,because the symmetry group of any
crystalline arrangement of atoms is acrystallographic group. In
dimension 2, crystallographic groups are knownas “wallpaper”
groups, and it is a famous theorem that there are 17 of them(up to
a suitable notion of equivalence). A fairly simple proof of this
wasgiven by John Conway and Bill Thurston, using the theory of
orbifolds. Yourdissertation might explain their proof [2, 3], and
then go on to examine moreadvanced material. This might include
Bieberbach’s theorem, which statesthat any crystallographic group
has a finite index subgroup that is an abeliangroup of translations
[1, 3]. You might explain why, in each dimension, thereare only
finitely many crystallographic groups, and you might investigatehow
the number of crystallographic groups grows as the dimension
increases[1]. You could also consider nonEuclidean geometries,
such as spherical orhyperbolic geometry.
1. L. S. Charlap, Bieberbach groups and flat manifolds,
Universitext. SpringerVerlag, 1986.
2. J. H. Conway, D. H. Huson, The Orbifold Notation for
TwoDimensionalGroups, Structural Chemistry 2002, Vol. 13, Nos.
34, pp. 24725
3. J. H. Conway; H. Burgiel, C. GoodmanStrauss (2008), The
Symmetriesof Things. Worcester MA: A.K. Peters. ISBN
1568812205.
4. W. Thurston, Threedimensional geometry and topology. Vol. 1.
Editedby Silvio Levy. Princeton Mathematical Series, 35. Princeton
UniversityPress, Princeton, NJ, 1997. x+311 pp. ISBN:
0691083045
Contact: Prof. Marc Lackenby (lackenby@maths.ox.ac.uk)
3.5 Gromov’s theorem and groups of polynomial growth
The growth function f(n) of a finitely generated group G is
defined to be thenumber of vertices in a ball of radius n of a
Cayley graph of G. Gromov in aseminal paper showed that if this
growth function is bounded by a polynomialthen G has a finite index
nilpotent subgroup. Gromov’s original proof relieson the solution
of Hilbert’s 5th problem.
Recently Kleiner gave a different proof of Gromov’s theorem
using analysison groups and ShalomTao used this to strengthen
Gromov’s theorem giving
12

an effective version whereby it is enough to have a bound for
f(n) for a singlesufficiently large n . The project should
introduce the subject of group growthand give an exposition of a
proof and remaining open questions related tothis.
1. M. Gromov, Inst. Hautes tudes Sci. Publ. Math. No. 53 (1981),
5373.2. B. Kleiner, A new proof of Gromov’s theorem on groups of
polynomialgrowth, Jour,.of the AMS, 23 (2010), no. 3, 815829. 3.
Shalom, Y.; Tao, T.,A finitary version of Gromov’s polynomial
growth theorem. Geom. Funct.Anal. 20 (2010), no. 6, 15021547.
Contact: Prof. Panos Papazoglou
(panagiotis.papazoglou@maths.ox.ac.uk)
3.6 Approximate groups
An approximate group is a finite subset A of a group G which is
“almost”closed under multiplication. A great deal has been written
about approximate groups, so an essay should focus on one aspect
together with some of itsapplications. Possibilities include, but
are not limited to: (i) abelian approximate groups and Freiman’s
theorem or (ii) approximate matrix groups andexpander graphs. A
very brief introduction may be found in “What is an approximate
group?” available at
http://www.ams.org/notices/201205/rtx120500655p.pdf
Contact: Prof. Ben Green (ben.green@maths.ox.ac.uk)
3.7 Higherorder Fourier analysis
Fourier analysis has been very successful in additive number
theory. Forexample, using the HardyLittlewood method Vinogradov
proved that everylarge odd number is the sum of three primes.
However, Fourier analysis failsfor more complicated problems such
as locating 4term arithmetic progressions in a set (for example
the set of primes). This essay should introduce thebasic objects of
higherorder Fourier analysis such as the Gowers norms, andthen go
on to give an application. A natural one would be to describe
TimGowers’s proof of Szemeredi’s theorem for 4term arithmetic
progressions [1],or some aspects of the work of Green and Tao on
asymptotics for progressionsof primes [2]. Both [1] and [2] are
freely available on the respective authors’webpages.
[1] W.T. Gowers, A new proof of Szemerédi’s theorem for
progressions oflength four, GAFA 8 (1998), no. 3, 529–551.
13
http://www.ams.org/notices/201205/rtx120500655p.pdf

[2] B.J. Green Generalizing the HardyLittlewood method for
primes, Proc.Intern. Cong. Math. (Madrid 2006), Vol. 2,
373–399.
Contact: Prof. Ben Green (ben.green@maths.ox.ac.uk)
4 Analysis
4.1 Abelian locally compact groups
Part C dissertation.
Abstract: The Fourier transform defined on the Hilbert space of
squareintegrable functions on the realline can be readily
extended to all locallycompact abelian groups by the introduction
of the dual group. An essaycould look at what locally compact
groups are, what the Haar measure is,how to define the Fourier
transform, and what Pontryagin duality is. Ideasfrom functional
analysis, group theory and measure (integration) theory areall
involved.
4.2 Kazhdan’s property (T)
Part C dissertation.
Kazhdan’s property (T) was introduced in a short paper, defined
in termsof the Fell topology on the space of representations of a
group. However,we can state the property, informally, in an easy
way: a group has property (T) if whenever, given a (unitary)
representation, there is a unit vectorwhich is only perturbed
slightly by the group action, then there is actually aunit vector
fixed by the group action. Such groups hence have very
”rigid”actions. Recently property (T) has found its way into a
diverse collectionof applications, from number theory, measure
theory, and operator algebras,to graph theory and computer science.
An essay could either explore someof these applications: for
example, the applications to graph theory andcomputer science can
be stated using a minimal amount of representationtheory.
Alternatively, one could look at the theory of representing
topological groups, and how classical representation theory can be
used to show that
14

some concrete groups have property (T). A longer essay could
touch uponboth.
4.3 Analysis in a rational world
The basic concepts of real analysis make perfectly good sense
for functionsf : Q → Q and in a world where only rational numbers
are contemplated.Thus, for example, continuity and
differentiability are defined in exactly thesame way as for
functions f : R→ R (but note that f ′(x) has to be rationalfor all
x ∈ Q). In this rational world we find that basic theorems such as
theIntermediate Value Theorem, Rolle’s Theorem, and the Mean Value
Theoremfail. How badly do they fail? What can be rescued? What can
one say aboutsolutions to differential equations such as f ′ = f ,
or f ′′ = f?
4.4 Analysis of holomorphic functions with special values
We denote by Q[i] the set of complex numbers whose real and
imaginaryparts are rational. Investigate holomorphic functions f :
C → C with theproperty that z ∈ Q[i]⇒ f(z) ∈ Q[i].
4.5 Integration
Compare and contrast Lebesgue integration and RiemannStieltjes
integration. [Caution: the material for such a project should cite
but not repeatthat given in the secondyear Integration
lectures.]
4.6 Measure Theory
The development of abstract measure theory yields rich rewards
with a variety of generalisations and applications, for example,
fractals, stochasticintegration. References include: Robert
Strichartz The Way of Analysis(Jones and Bartlett Publishers,
2000); H. L. Royden Real Analysis (MacMillan, 1963); Kenneth
Falconer Fractal Geometry (Wiley 1990). [Caution: thematerial for
such a project may cite but should not repeat that given in
thethirdyear lecture course ‘Martingales Through Measure
Theory’.]
15

4.7 Univalent functions
A univalent map is a onetoone conformal map z = f(ζ) from the
unit discto a domain Ω in the complex plane. It is usual to
normalize f by assumingthat f(0) = 0 and f ′(0) = 1. Among many
interesting results in the field arethe Koebe 1
4Theorem (the distance from 0 to the boundary of Ω is never
less
than 14, with equality for f(ζ) = ζ/(1 − ζ)2) and the celebrated
Bieberbach
conjecture (proved in 1985 by De Branges) that if f is univalent
and has theTaylor series f(ζ) = ζ +
∑∞2 anζ
n, then an ≤ n, with equality for the samemap. Proofs of the
conjecture for small values of n are not hard.
4.8 Distributions
Investigate the theory of distributions and their applications.
Probably themost immediate starting point is the delta function (cf
point masses, sources,or charges) and its relation to the
derivative of a function with a jump discontinuity, but a proper
theoretical development is not hard to set out. Furthertopics
include the relation with Fourier transforms.
4.9 Special functions
There are many possible project topics here, for example: Bessel
functionsand their applications, hypergeometric functions (and the
connection withcomplex differential equations and conformal maps),
the Riemann zeta function, the Gamma function.
4.10 Riemann surfaces
Explore the idea of the Riemann surface associated with a
multivalued conformal map.
4.11 The Schwarzian derivative
The Schwarzian derivative has many interesting properties. Find
out aboutthem.
16

4.12 Chaos in nonlinear ordinary differential equations
Explore the connection between the various kinds of homoclinic
bifurcationsand the onset of chaos in ordinary differential
equations.
Investigate the occurrence of stochasticity in Hamiltonian
systems, as anintegrable system is perturbed more and more
strongly.
The period doubling sequence for unimodal (onehumped) maps is
well known.What happens for other maps, e.g. cubics?
5 Mathematical Methods and Applications
5.1 Mathematics and the environment
The analysis of low dimensional Plankton models
Part B extended essay or Part C dissertation
The increasing exploitation of marine resources has driven a
demand forcomplex biogeochemical models of the oceans and the life
they contain. Thecurrent models are constructed from the ‘bottom
up’, considering the biochemistry of individual species or
functional types, allowing them to interactaccording to their
position in the food web, and embedding the ecological system in a
physical model of ocean dynamics. The resulting ecology
simulationmodels typically have no conservation laws and the
ecology often produces‘emergent properties’, that is, surprising
behaviours for which there is noobvious explanation. Because
realistic models have too many experimentallypoorly defined
parameters (often in excess of 100), there is a need to
analysesimpler models. A recent approach by Cropp and Norbury
(2007) involvesthe construction of complex ecosystem models by
imposing conservation ofmass with explicit resource limitation at
all trophic levels (i.e. positionsoccupied in a food chain). The
project aims to analyse models containingtwo ‘predators’ and two
‘prey’ with MichaelisMenten kinematics. A systematic approach to
elicit the bifurcation structure and routes to chaos usingparameter
values, appropriate to different ocean areas would be adopted.In
particular the influence of nonlinearity in the functional (life)
forms onthe stability properties of the system and the bifurcation
properties of themodel will be comprehensively numerically
enumerated and mathematicallyanalysed.
17

REFS:R. Cropp & J.Norbury : J. Plankton Research vol 31
939963 (2009)
R. Cropp, I.M. Moroz & J. Norbury: J. Marine Systems vol.
139 483495(2014).
Contact: Prof. Irene Moroz (moroz@maths.ox.ac.uk)
Part C dissertations
Surging glaciers.Observations and theories concerning glacier
surges.
Waves on riversFormation of waves on rivers (roll waves, tidal
bores).
Formation of aeolian and fluvial bedforms such as dunes.
5.2 Mathematical biology and physiology
There are many topics that could be explored in relation to the
Part B andPart C courses in the field mathematical biology and
ecology. In each case,the aim will be to explore a particular
biological phenomenon using mathematical and computational
techniques. The models used could consist of,for example, ordinary
and partial differential equation, stochastic differentialequation
and individualbased approaches. Exploration of the models willbe
carried out using a suitable combination of analytical and
computationalmethods, with the aim being the generation of
experimentally testable predictions. Some possible areas for
investigation include (but are not limitedto):
• physiological/biological fluid mechanics;
• tissue engineering;
• the mechanics of growth processes;
• pattern formation;
• cell movement, signalling and interaction;
• multiscale modelling.
ReferencesR. E. Baker, E. A. Gaffney and P. K. Maini (2008).
PDEs for selforganisation
18

in cellular and developmental biology. Nonlinearity
21:R251R290.
R. D. O’Dea, H. M. Byrne and S. L. Waters (2013). Continuum
modellingof in vitro tissue engineering: a review. Studies in
Mechanobiology, TissueEngineering and Biomaterials 10:229266.
H. M. Byrne, T. Alarcon, M. R. Owen, S. D. Webb and P. K. Maini
(2006).Modelling aspects of cancer dynamics: a review.
Philosophical Transactionsof the Royal Society A 364:
15631578.
5.3 Fluid dynamics
There are many topics that could be done in parallel with the
course forPaper B6; consult the lecturer for possibilities.
Describe some of the phenomena which occur in rotating flows,
and theirapplication in meteorology.
Linear and nonlinear stability theory of Rayleigh–Bénard
convection.
Transition to turbulence in shear flows.
5.4 Mechanics of Solids
Write on the linear theory of elastic solids, including for
instance some ofthe following topics: the stress–strain relation in
an anisotropic crystallinematerial and its relation to the
symmetries of the material; SaintVenanttorsion of a prismatic beam
of isotropic material; wave motion in an isotropicelastic material,
including waves in thin rods and thin plates.
5.5 Multilayer Networks
In most natural and engineered systems, a set of entities
interact with eachother in complicated patterns that can encompass
multiple types of relationships, change in time, and include other
types of complications. Suchsystems include multiple subsystems and
layers of connectivity, and it is important to take such
“multilayer” features into account to try to improve
ourunderstanding of complex systems. Consequently, it is necessary
to generalize “traditional” network theory by developing (and
validating) a frameworkand associated tools to study multilayer
systems in a comprehensive fashion.
19

In this project, the student will study either a theorydriven
or applicationdriven project in multilayer networks.
Main reference:M. Kivela et al, 2014;
http://arxiv.org/abs/1309.7233
Contact: Prof. Mason Porter (porterm@maths.ox.ac.uk)
6 Numerical Analysis
There are many topics that could be addressed as a followup to
the Part ANumerical Analysis course or in parallel with the
thirdyear courses in thisarea. Here is a small sample of possible
projects; the lecturer may be able tosuggest some others.
6.1 Stiff ordinary differential equations
Part B extended essay
Consider an initial value problem for the differential equation
�y′ = f(x, y), ora system of differential equations of the form �y′
= f(x,y), where 0 < �� 1 isa small parameter. An interesting and
practically relevant question concernsthe construction and analysis
of numerical methods for the accurate solutionof such problems. How
do standard onestep and linear multistep methodsbehave when � is
very small? How would you improve the performance ofthese methods
by adapting the computational mesh? Is it possible to designspecial
methods which provide accurate approximations for such stiff
initialvalue problems?
Gear, C. W. (1981), N umerical solution of ordinary differential
equations: Isthere anything left to do?, SIAM Review 23 (1):
10–24.
Hairer, Ernst; Wanner, Gerhard (1996), Solving ordinary
differential equations II: Stiff and differentialalgebraic
problems (second ed.), Berlin: SpringerVerlag, ISBN
9783540604525.
Lambert, J. D. (1992), N umerical Methods for Ordinary
Differential Systems, New York: Wiley, ISBN 9780471929901.
Shampine, L. F.; Gear, C. W. (1979), A user’s view of solving
stiff ordinarydifferential equations, SIAM Review 21 (1): 1–17.
20
http://arxiv.org/abs/1309.7233

6.2 Numerical approximation of singular integrals
Part B extended essay
You are probably familiar with simple numerical integration
rules such as thetrapezium rule or Simpson’s rule; but how would
you evaluate numerically
the integral∫ 10
ex2
x1/2(1−x)1/3 dx or∫∞0
sinx5
x2(1+x)55? There are special techniques
for the numerical approximation of such integrals. What are they
and whatcan be said about their accuracy?
Beale, J.T. and Lai, M.C. (2001), A method for computing nearly
singularintegrals. SIAM Journal on Numerical Analysis, 38,
19021925.
Davis, Philip J. and Rabinowitz, Philip. (1984), M ethods of
Numerical Integration. 2nd ed. Academic Press, San Diego.
Linz, P. (1985), On the approximate computation of certain
strongly singularintegrals, Computing, 35, 345–353.
6.3 Newton’s method for nonlinear systems
Part B extended essay
Newton’s method is a standard technique for solving a nonlinear
equation ofthe form f(x) = 0 where f is a continuously
differentiable function. Consider the generalisation of Newton’s
method for solving the nonlinear systemf(x, y) = 0, g(x, y) = 0.
What can be said about the convergence of Newton’s method? What is
a good choice of starting value and how to locate it?How does the
speed of convergence of Newton’s method compare with thatof a
simple fixedpoint iteration?
Kendall E. Atkinson (1989), An Introduction to Numerical
Analysis, JohnWiley & Sons, Inc, ISBN 0471624896.
Peter Deuflhard, (2004), N ewton Methods for Nonlinear Problems.
Affine Invariance and Adaptive Algorithms. Springer Series in
Computational Mathematics, Vol. 35. Springer, Berlin. ISBN
3540210997.
J. M. Ortega, W. C. Rheinboldt (2000), I terative Solution of
Nonlinear Equations in Several Variables. Classics in Applied
Mathematics, SIAM. ISBN0898714613.
Endre Süli and David Mayers (2006), An Introduction to
Numerical Analysis,2nd. printing, Cambridge University Press, 2003.
ISBN 0521007941.
21

Tjalling J. Ypma (1995), H istorical development of the
NewtonRaphsonmethod, SIAM Review 37 (4), 531–551.
doi:10.1137/1037125
6.4 Finite element methods for singularly perturbedproblems
Part C dissertation
Conventional finite element methods are known to provide poor
approximations when applied to singularly perturbed twopoint
boundary value problems of the form −�u′′ + u′ = 0, u(0) = 0, u(1)
= 1: while the analyticalsolution is a smooth function, its
numerical approximation exhibits unacceptable oscillations. Why
does this happen? Can you improve matters by usinga suitable
nonuniform mesh? How should such a mesh be designed?
Wouldgeneralising the concept of Galerkin finite element method by
allowing a trialspace that is different from the test space cure
the problem?
HansGöerg Roos, Martin Stynes and Lutz Tobiska (2008), Robust
Numerical Methods for Singularly Perturbed Differential Equations:
ConvectionDiffusionReaction and Flow Problems Springer Series in
ComputationalMathematics, Vol. 24 2nd ed.
Claes Johnson (2009), N umerical Solution of Partial
Differential Equationsby the Finite Element Method (Dover Books on
Mathematics), Dover. ISBN10: 048646900X — ISBN13:
9780486469003
Endre Süli (2012), F inite Element Methods for Partial
Differential Equations,
http://people.maths.ox.ac.uk/suli/fem.pdf
6.5 A posteriori error analysis of finite element methods
Part C dissertation
Conventional a priori error estimates for finite element
approximations ofboundary value problems are of limited practical
use since they bound thecomputational error in terms of powers of
the mesh size and norms of derivatives of the unknown analytical
solution. How would you derive an a posteriori bound on the error
by exploiting a computed solution? Now suppose thatyou have
estimated the size of the error by means of an a posteriori
bound;
22
http://people.maths.ox.ac.uk/suli/fem.pdf

how would you adapt the computational mesh to ensure that the
error in agiven norm does not exceed a given tolerance?
S. Brenner & R. Scott (1996), T he Mathematical Theory of
Finite ElementMethods, SpringerVerlag.
R. Verfürth (2013), A Posteriori Error Estimation Techniques
for Finite Element Methods, Oxford University Press.
6.6 Fast iterative methods for systems of linear equations
Part C dissertation
Gaussian elimination is a standard technique for solving a
system of linearequations of the form Ax = b where A is a
nonsingular matrix. Supposethat A is a sparse matrix (i.e. it
contains a very large number of zero entries).Is it a good idea to
solve such a system by Gaussian elimination, or would itperhaps be
better to use an iterative method instead? Consider the
performance of various iterative methods (such as the Conjugate
Gradient Methodand its relatives) for the solution of sparse
systems. How would you accelerate the convergence of an iterative
method by preconditioning A, i.e. bypremultiplying the system by a
nonsingular matrix P such that P ≈ A−1,and solving PAx = Pb
instead? How would you choose P?
Axelsson, Owe (1996), I terative Solution Methods, Cambridge
UniversityPress. p. 6722. ISBN 9780521555692.
Howard C. Elman, David J. Silvester, and Andrew J. Wathen
(2005): F initeElements and Fast Iterative Solvers with
Applications in IncompressibleFluid Dynamics, ISBN
9780198528685.
van der Vorst, H. A. (2003), I terative Krylov Methods for Large
LinearSystems, Cambridge University Press, Cambridge. ISBN
0521818281.
6.7 Superconvergence of the Collocation Method withSplines
Part C dissertation
There is a wide variety of methods for finding numerical
solutions to secondorder ordinary differential equations (ODEs).
This project is concerned in
23

particular with twopoint boundary value problems (BVPs), where
there areboundary conditions given at two particular values of x,
say x = a and x = b.One method of solution is the collocation
method, where the solution y(x) isrequired to satisfy the boundary
conditions at x = a and x = b, and also tosatisfy the ODE at a
finite number of points, called collocation sites. In thisproject
you will investigate how this, combined with the use of Bsplines,
canproduce surprisingly efficient and accurate solutions to certain
kinds of ODEs.There is plenty of scope for experimentation with
MATLAB: for example youcould compare what happens with different
choices of collocation sites, or youcould find solutions to
different kinds of ODEs (both linear and nonlinearBVPs).
de Boor, Carl (2001), A Practical Guide to Splines, Springer.
ISBN 9780387953663.
C. de Boor and B. Swatz (1973), C ollocation at Gaussian Points,
SIAM J.Numer. Anal, 10, 582606.
Powell, M.J.D. (1981), Approximation Theory and Methods,
Cambridge University Press. ISBN 9780521295149.
Contact: Dr Cath Wilkins (wilkins@maths.ox.ac.uk)
7 Mathematical Physics
7.1 Extremum principles in theoretical physics
For example: (a) Fermat’s principle; (b) Hamilton’s principle; .
. .; (f) variational methods in quantum mechanics; . . .; (r)
geodesics in general relativityfor a (given) Schwarzschild metric;
. . .; (z) path integral formulations ofquantum mechanics [rather
advanced].
7.2 Hamilton–Jacobi theory
Actionangle variables and the early development of quantum
theory.
24

7.3 Concepts of quantum mechanics
For example: Schrödinger’s cat; Bell’s inequality. Another
possibility is acritical summary of the Bohr–Einstein
dialogues.
7.4 Symmetries in quantum mechanics
Various groups which occur naturally in quantum mechanics,
particlefieldsystems can be studied in a simple manner.
7.5 Quantum mechanics of scattering
There are many calculational examples which are within reach of
the advanced undergraduate. Examples: the Born approximation;
simple atomsand molecules (in electric and/or magnetic fields);
one or twoelectron systems.
7.6 Waveguides
The undergraduate can start with the excellent exposition on
waveguides inthe Feynman Lectures, and go a bit further from
there.
7.7 The microwave background
A systematic list of the consequences of the microwave
background wouldmake a worthwhile and approachable project.
7.8 Tests of general relativity
For example, simple calculations involved in gravitational wave
experiments;gravitational lensing effect (quite a lot of
interesting geometry there).
25

7.9 Hot big bang versus steady state in cosmology
This is a topic that can be expanded in various interesting
ways, expositoryand historical.
7.10 Appearance of moving objects in special relativity
In spite of Lorentz contraction, a spherical object does look
spherical to anobserver. Why?
7.11 Study of a special metric
Even apart from the Schwarzschild metric, general relativity is
full of specialmetrics (“exact solutions”) which repay even
simplistic studies. In particular,the NUT metrics have intriguing
topological properties.
7.12 SO(3), SU(2), Euler angles and angular momentum
Work out the exact 2to1 map of SU(2) to SO(3); generalize
Euler angles tohigher dimensions; angular momentum and spin.
7.13 Momentum space in quantum mechanics
States as functions of momentum. Fourier transform. Plancherel
theorem.Time evolution.
7.14 Translation of some well known theorem in euclidean space
to Minkowski space
Many interesting problems relating to Euclidean and hyperbolic
geometrycan be tackled.
26

8 Stochastics, Discrete Mathematics and In
formation
8.1 Combinatorics — Graphs of large chromatic number
BE Extended Essay.
It is easy to see that the chromatic number of a graph is at
least as large asits clique number (the number of vertices in a
largest complete subgraph).A graph is perfect if its chromatic
number equals its clique number, andthe same holds for all its
induced subgraphs. An essay on this topic couldinvestigate the
theory of perfect graphs, including the theorem of Lovászthat a
graph is perfect if and only if its complement is perfect. An
essayshould certainly discuss the Strong Perfect Graph Theorem of
Chudnovsky,Robertson, Seymour and Thomas, which gives a structural
characterizationof perfect graphs, although there would not be
space to include a proof.
Alternatively, the essay might look at what subgraphs must
appear in graphswith large chromatic number. Relevant topics would
include graphs withlarge girth and large chromatic number, and
χbounded classes and theGyárfásSumner Conjecture.
8.2 Mathematical models in finance
Possible topics might include a rigorous discussion of Itô’s
lemma and its relation to random walks; the rôle of martingales
in models of markets; stochasticvolatility; time series methods.
The lecturer may be able to suggest others.
8.3 Mathematical models in evolution
Modern methods for understanding genetic data, and using this to
learnabout the processes of evolution, rely heavily on mathematical
models. Thesemodels usually involve probability, to capture the
various sources of randomness in genetics processes. Study these
models and their uses to untangleevolutionary questions, such as
how different species are related, or what wecan learn about early
human evolution from genetics data.
27

8.4 Duality and Random Walks
Duality, the relation between a set of paths and the reversed
set is a classicaltool in the study of random walks. One project
would be to review someproblems in which this technique has been
successfully applied, notably inthe queuing literature.
For the very able student there is the possibility of novel work
on the studyand classification of dual times for walks in Rn or in
the understanding ofduality relations that have arisen from recent
work on stopping rules forMarkov chains.
8.5 The Coupling Method
Loosely, coupling refers to the study of one or two marginal
probability distributions by way of the construction of an
appropriate joint distribution.The success of this method lies in
its ability to convert difficult analyticproblems into ones of
probabilistic construction. There are many areas inapplied
probability in which to study how this most elegant method is
used,including Poisson Approximation, Random Walks and Markov
Chains.
8.6 Applied Probability
Queues are often used to model communication and manufacturing
systems:a topic with manufacturing applications would be to discuss
the stability andbehaviour of networks of queues when the arrivals
at a network almost saturate its service capacity; a topic with
applications in the telecommunicationscontext would be to discuss
rare events in large systems.
8.7 Operational Research
Gather appropriate data from a filling station or a supermarket,
and use it,with the help of an appropriate simulation software
package, to investigatethe characteristics of the queues which
would form under different possibleservice regimes.
28

9 History of Mathematics
It is difficult to offer specific projects in the history of
mathematics becausethe possibilities are so varied and the choice
will depend very much on eachstudent’s personal inclination and
skills. Those who have taken O1 as athirdyear option will already
have a good grounding in the history behindthe present day
mathematics curriculum and may choose to go more deeplyinto a
particular topic, person, or debate. Others may wish to work on
amore general theme. There is also plenty of untranslated source
materialand those with some Latin, French, or German might like to
undertake atranslation and commentary; there is no better way to
enter into the mind ofa first rank mathematician, and Euler,
Lagrange, and Cauchy, for example,all offer material that is both
accessible and engaging.
To give some idea of the range and style of projects that are
possible, hereare examples of some topics that have been the
subject of some recent OEessays or OD dissertations:
Mathematics and World War II
Mathematics in the Early Years of the St Petersburg Academy of
Sciences
Robert Recorde’s presentation of Euclidean geometry
Hilbert’s seventh and eighth problems
The lives and work of Emilie du Chatelet and Sophie Germain
The life and work of Edmund Halley
Arthur Cayley’s contribution to group theory
A translation (from Latin) on summation of series by Euler
A comparison of the contributions to analysis of Cauchy and
Bolzano
A translation (from French) of Galois’ work on finite fields
A Historical Study of the ChurchTuring Thesis
The Historical Development of the Theory of Matrices
Anyone interested in working on a historical subject is
encouraged to comeand discuss ideas with Dr Chris Hollings
(christopher.hollings@maths.ox.ac.uk>)before the end of Trinity
Term.
29

10 General
A host of tractable and interesting problems are to be found in
journals suchas
• The American Mathematical Monthly ,
• The Mathematical Gazette,
• The Mathematical Intelligencer,
• SIAM Review.
Many of these have a past, and ramifications, the tracing of
which wouldprovide a project—including, perhaps, the solution of
the problem, thoughthis would not be essential.
30

11 Titles of Previous Projects
Listed below are the titles of some projects undertaken by
students in recentyears.
11.1 BE Extended Essays
Alhazen’s Problem and Galois Groups
A Study on Waring’s Problem
Investigating the Community Structure of Federal Election
Commission Dataon Congressional Donations
The Applications of Quantum Phase Estimation
Graphs of Large Chromatic Number
Mathematical Models of Dermal Wound Healing
Probabilistic Models Behind Spread Betting in Football
A Study of Distribution Theory in One Dimension
11.2 CD Dissertations
Extended Series Solutions to Flow Through Curved Pipes (single
unit)
Thin Viscous Flow over a Flexible Beam (single unit)
Frobenius Algebras, Topological Quantum Field Theories and the
CobordismHypothesis (single unit)
Of Galois Groups in Polynomial Time (single unit)
A Comparison of Intuitionistic and Classical Mathematics and
Logic
The Poincare Conjecture for Dimensions 5 and Above
Numerical Solution of Elliptic PDEs with Stochastic
Coefficients
Trees and Excursions
Hilbert’s Irreducibility Theorem and Applications to the Inverse
Galois Problem
31

MeanVariance Portfolio Selection Under Constraints
Contact Line Dynamics of an Evaporating Droplet
Irreducible Representations of Some Finite Groups Over Finite
Fields
The Hole in the Wall: a look at the singular barrier found in a
model of cellinvasion
Stochastic Models of Chemical Processes
An Analysis of Changes in Mathematical Subfields via
Timedependent Coauthorship Network
Baroclinic Instability in The Earth’s Atmosphere
Homotopy Types
Catalysis in Quantum State Transformations
Transcendental Numbers and Algebraic Independence
Measure Theory, Riez Spaces and the RadonNikodym Theorem
32
LogicModel Theory — Strongly minimal structuresModel Theory —
The model theory of cyclic groups.Model theory of the real
numbersOminimal structuresLocal equivalents of the Axiom of
ChoiceTheories of the real numbers
AlgebraQuantum groups and crystal basisAffine algebraic groups
schemesHomotophy type theoryRelative algebraic
geometryBeilinsonBernstein localisation for sl2Converses of
Lagrange's TheoremAlhazen's ProblemReduction of quadratic
formsUniserial group actions
Geometry and Number TheorySimple singularities and the McKay
correspondenceRecognizing the unknotGeometrisation of
3manifoldsCrystallographic groupsGromov's theorem and groups of
polynomial growthApproximate groupsHigherorder Fourier
analysis
AnalysisAbelian locally compact groupsKazhdan's property
(T)Analysis in a rational worldAnalysis of holomorphic functions
with special valuesIntegrationMeasure TheoryUnivalent
functionsDistributionsSpecial functionsRiemann surfacesThe
Schwarzian derivativeChaos in nonlinear ordinary differential
equations
Mathematical Methods and ApplicationsMathematics and the
environmentMathematical biology and physiologyFluid
dynamicsMechanics of SolidsMultilayer Networks
Numerical AnalysisStiff ordinary differential equationsNumerical
approximation of singular integralsNewton's method for nonlinear
systemsFinite element methods for singularly perturbed problemsA
posteriori error analysis of finite element methodsFast iterative
methods for systems of linear equationsSuperconvergence of the
Collocation Method with Splines
Mathematical PhysicsExtremum principles in theoretical
physicsHamilton–Jacobi theoryConcepts of quantum
mechanicsSymmetries in quantum mechanicsQuantum mechanics of
scatteringWaveguidesThe microwave backgroundTests of general
relativityHot big bang versus steady state in cosmologyAppearance
of moving objects in special relativityStudy of a special
metricSO(3), SU(2), Euler angles and angular momentumMomentum space
in quantum mechanicsTranslation of some well known theorem in
euclidean space to Minkowski space
Stochastics, Discrete Mathematics and InformationCombinatorics —
Graphs of large chromatic numberMathematical models in
financeMathematical models in evolutionDuality and Random WalksThe
Coupling MethodApplied ProbabilityOperational Research
History of MathematicsGeneralTitles of Previous ProjectsBE
Extended EssaysCD Dissertations