EXTENDED ESSAYS: Option BEE or BOE in Part B of the Final … · 2.2 A ne algebraic groups schemes Part C dissertation A ne algebraic groups schemes: A ne algebraic groups schemes

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 O-minimal 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 Beilinson-Bernstein localisation for sl2 . . . . . . . . . . . . . 8

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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 3-manifolds . . . . . . . . . . . . . . . . . . 113.4 Crystallographic groups . . . . . . . . . . . . . . . . . . . . . 123.5 Gromov’s theorem and groups of polynomial growth . . . . . . 123.6 Approximate groups . . . . . . . . . . . . . . . . . . . . . . . 133.7 Higher-order 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

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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

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1 Logic

1.1 Model Theory — Strongly minimal structures

Part C dissertation.

Strongly minimal structures and their combinatorial geometries are im- por-tant notions of modern model theory. These notion are also central for un-derstanding principles of a model-theoretic classification of classical math-ematical 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 non-classical 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 model-theoretic limit of a se-quence 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 right-to-left 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

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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, Springer-Verlag, 2002.

P.J. Cohen, Decision procedures for real and p-adic fields, Comm. Pure Appl.Math. 22 (1969), 131–151.

1.4 O-minimal structures

Part C dissertation.

An o-minimal structure is a model-theoretic 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 o-minimal structures, LMS LectureNote Series 248, CUP, 1998.

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1.5 Local equivalents of the Axiom of Choice

The famous equivalence of the Axiom of Choice and the Well-Order 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 clas-sical 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)

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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 lan-guage 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 topolo-gies, 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)

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2.5 Beilinson-Bernstein localisation for sl2

Part C dissertation

Beilinson-Bernstein localisation for sl2: The Beilinson-Bernstein localisationtheorem is one of the most important tools in the representation theory ofsemi-simple Lie algebras. This project will cover the basics of category theory,the category of representations of sl2, the Weyl algebra, the categories of O-modules and D-modules on the projective line and the Beilinson-Bernsteinlocalisation 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 al-Haytham’, 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 ruler-and-compass 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 simi-lar 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

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non-singular 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 G-space Ω areequivalent:

(A) that the set of G-invariant equivalence relations on Ω (known as G-congruences 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.

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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.

Ito-Nakamura: 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, Arc-presentations of links: monotonic simplification. Fund.Math. 190 (2006), 29-76.

2. W. Haken, Theorie der Normalflchen. Acta Math. 105 (1961) 245-375.

3. J. Hass, J. Lagarias, The number of Reidemeister moves needed for un-knotting. J. Amer. Math. Soc. 14 (2001), no. 2, 399-428

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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, 185-211.

5. S. Matveev, Algorithmic topology and classification of 3-manifolds. Secondedition. Algorithms and Computation in Mathematics, 9. Springer, Berlin,2007. xiv+492 pp. ISBN: 978-3-540-45898-2

Contact: Prof. Marc Lackenby (lackenby@maths.ox.ac.uk)

3.3 Geometrisation of 3-manifolds

In the late 1970s, Bill Thurston revolutionised the study of 3-manifolds withthe introduction of his geometrisation conjecture [3]. Roughly speaking, thisproposed that any compact orientable 3-manifold has a “canonical decom-position into geometric pieces”. This conjecture was a far-reaching general-isation of the Poincare conjecture, which asserted that any 3-manifold ho-motopy equivalent to the 3-sphere is homeomorphic to the 3-sphere. 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 Math-ematics Monographs, 3. American Mathematical Society, Providence, RI;Clay Mathematics Institute, Cambridge, MA, 2007. xlii+521 pp. ISBN:978-0-8218-4328-4

2. W. Neumann, G. Swarup, Canonical decompositions of 3-manifolds. Geom.Topol. 1 (1997), 21-40

3. W. Thurston, Three-dimensional manifolds, Kleinian groups and hyper-bolic geometry. Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357-381.

4. W. Thurston, Three-dimensional geometry and topology. Vol. 1. Editedby Silvio Levy. Princeton Mathematical Series, 35. Princeton UniversityPress, Princeton, NJ, 1997. x+311 pp. ISBN: 0-691-08304-5

Contact: Prof. Marc Lackenby (lackenby@maths.ox.ac.uk)

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3.4 Crystallographic groups

A crystallographic group is a discrete, cocompact group of Euclidean isome-tries. 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 non-Euclidean geometries, such as spherical orhyperbolic geometry.

1. L. S. Charlap, Bieberbach groups and flat manifolds, Universitext. Springer-Verlag, 1986.

2. J. H. Conway, D. H. Huson, The Orbifold Notation for Two-DimensionalGroups, Structural Chemistry 2002, Vol. 13, Nos. 3-4, pp. 247-25

3. J. H. Conway; H. Burgiel, C. Goodman-Strauss (2008), The Symmetriesof Things. Worcester MA: A.K. Peters. ISBN 1-56881-220-5.

4. W. Thurston, Three-dimensional geometry and topology. Vol. 1. Editedby Silvio Levy. Princeton Mathematical Series, 35. Princeton UniversityPress, Princeton, NJ, 1997. x+311 pp. ISBN: 0-691-08304-5

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 Shalom-Tao used this to strengthen Gromov’s theorem giving

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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), 53-73.2. B. Kleiner, A new proof of Gromov’s theorem on groups of polynomialgrowth, Jour,.of the AMS, 23 (2010), no. 3, 815-829. 3. Shalom, Y.; Tao, T.,A finitary version of Gromov’s polynomial growth theorem. Geom. Funct.Anal. 20 (2010), no. 6, 1502-1547.

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 approxi-mate groups, so an essay should focus on one aspect together with some of itsapplications. Possibilities include, but are not limited to: (i) abelian approx-imate groups and Freiman’s theorem or (ii) approximate matrix groups andexpander graphs. A very brief introduction may be found in “What is an ap-proximate group?” available at http://www.ams.org/notices/201205/rtx120500655p.pdf

Contact: Prof. Ben Green (ben.green@maths.ox.ac.uk)

3.7 Higher-order Fourier analysis

Fourier analysis has been very successful in additive number theory. Forexample, using the Hardy-Littlewood method Vinogradov proved that everylarge odd number is the sum of three primes. However, Fourier analysis failsfor more complicated problems such as locating 4-term arithmetic progres-sions in a set (for example the set of primes). This essay should introduce thebasic objects of higher-order 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 4-term 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.

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http://www.ams.org/notices/201205/rtx120500655p.pdf

[2] B.J. Green Generalizing the Hardy-Littlewood 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 square-integrable functions on the real-line 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 prop-erty (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 topologi-cal groups, and how classical representation theory can be used to show that

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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 val-ues

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 Riemann-Stieltjes integra-tion. [Caution: the material for such a project should cite but not repeatthat given in the second-year Integration lectures.]

4.6 Measure Theory

The development of abstract measure theory yields rich rewards with a va-riety 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 (MacMil-lan, 1963); Kenneth Falconer Fractal Geometry (Wiley 1990). [Caution: thematerial for such a project may cite but should not repeat that given in thethird-year lecture course ‘Martingales Through Measure Theory’.]

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4.7 Univalent functions

A univalent map is a one-to-one 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

4-Theorem (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 discon-tinuity, 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 func-tion, the Gamma function.

4.10 Riemann surfaces

Explore the idea of the Riemann surface associated with a multi-valued con-formal map.

4.11 The Schwarzian derivative

The Schwarzian derivative has many interesting properties. Find out aboutthem.

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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 (one-humped) 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 bio-chemistry of individual species or functional types, allowing them to interactaccording to their position in the food web, and embedding the ecological sys-tem 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 Michaelis-Menten kinematics. A system-atic 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.

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REFS:R. Cropp & J.Norbury : J. Plankton Research vol 31 939-963 (2009)

R. Cropp, I.M. Moroz & J. Norbury: J. Marine Systems vol. 139 483-495(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 math-ematical and computational techniques. The models used could consist of,for example, ordinary and partial differential equation, stochastic differentialequation and individual-based 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 pre-dictions. 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 self-organisation

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in cellular and developmental biology. Nonlinearity 21:R251-R290.

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 10:229-266.

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: 1563-1578.

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; Saint-Venanttorsion 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 rela-tionships, change in time, and include other types of complications. Suchsystems include multiple subsystems and layers of connectivity, and it is im-portant to take such “multilayer” features into account to try to improve ourunderstanding of complex systems. Consequently, it is necessary to general-ize “traditional” network theory by developing (and validating) a frameworkand associated tools to study multilayer systems in a comprehensive fashion.

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In this project, the student will study either a theory-driven or application-driven 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 follow-up to the Part ANumerical Analysis course or in parallel with the third-year 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 one-step and linear multi-step 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 equa-tions II: Stiff and differential-algebraic problems (second ed.), Berlin: Springer-Verlag, ISBN 978-3-540-60452-5.

Lambert, J. D. (1992), N umerical Methods for Ordinary Differential Sys-tems, New York: Wiley, ISBN 978-0-471-92990-1.

Shampine, L. F.; Gear, C. W. (1979), A user’s view of solving stiff ordinarydifferential equations, SIAM Review 21 (1): 1–17.

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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, 1902-1925.

Davis, Philip J. and Rabinowitz, Philip. (1984), M ethods of Numerical In-tegration. 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. Con-sider 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 New-ton’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 fixed-point iteration?

Kendall E. Atkinson (1989), An Introduction to Numerical Analysis, JohnWiley & Sons, Inc, ISBN 0-471-62489-6.

Peter Deuflhard, (2004), N ewton Methods for Nonlinear Problems. Affine In-variance and Adaptive Algorithms. Springer Series in Computational Math-ematics, Vol. 35. Springer, Berlin. ISBN 3-540-21099-7.

J. M. Ortega, W. C. Rheinboldt (2000), I terative Solution of Nonlinear Equa-tions in Several Variables. Classics in Applied Mathematics, SIAM. ISBN0-89871-461-3.

Endre Süli and David Mayers (2006), An Introduction to Numerical Analysis,2nd. printing, Cambridge University Press, 2003. ISBN 0-521-00794-1.

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Tjalling J. Ypma (1995), H istorical development of the Newton-Raphsonmethod, 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 approxima-tions when applied to singularly perturbed two-point boundary value prob-lems of the form −�u′′ + u′ = 0, u(0) = 0, u(1) = 1: while the analyticalsolution is a smooth function, its numerical approximation exhibits unaccept-able oscillations. Why does this happen? Can you improve matters by usinga suitable non-uniform 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?

Hans-Göerg Roos, Martin Stynes and Lutz Tobiska (2008), Robust Numer-ical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction 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. ISBN-10: 048646900X — ISBN-13: 978-0486469003

Endre Süli (2012), F inite Element Methods for Partial Differential Equa-tions, http://people.maths.ox.ac.uk/suli/fem.pdf

6.5 A posteriori error analysis of finite element meth-ods

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 deriva-tives of the unknown analytical solution. How would you derive an a posteri-ori 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;

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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, Springer-Verlag.

R. Verfürth (2013), A Posteriori Error Estimation Techniques for Finite El-ement Methods, Oxford University Press.

6.6 Fast iterative methods for systems of linear equa-tions

Part C dissertation

Gaussian elimination is a standard technique for solving a system of linearequations of the form Ax = b where A is a non-singular 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 perfor-mance of various iterative methods (such as the Conjugate Gradient Methodand its relatives) for the solution of sparse systems. How would you accel-erate the convergence of an iterative method by preconditioning A, i.e. bypre-multiplying the system by a non-singular 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 978-0-521-55569-2.

Howard C. Elman, David J. Silvester, and Andrew J. Wathen (2005): F initeElements and Fast Iterative Solvers with Applications in IncompressibleFluid Dynamics, ISBN 978-0-19-852868-5.

van der Vorst, H. A. (2003), I terative Krylov Methods for Large LinearSystems, Cambridge University Press, Cambridge. ISBN 0-521-81828-1.

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

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particular with two-point 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 B-splines, 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 978-0-387-95366-3.

C. de Boor and B. Swatz (1973), C ollocation at Gaussian Points, SIAM J.Numer. Anal, 10, 582-606.

Powell, M.J.D. (1981), Approximation Theory and Methods, Cambridge Uni-versity Press. ISBN 978-0-521-29514-9.

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) varia-tional 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

Action-angle variables and the early development of quantum theory.

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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, particle-fieldsystems can be studied in a simple manner.

7.5 Quantum mechanics of scattering

There are many calculational examples which are within reach of the ad-vanced undergraduate. Examples: the Born approximation; simple atomsand molecules (in electric and/or magnetic fields); one- or two-electron sys-tems.

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).

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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 relativ-ity

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 2-to-1 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 eu-clidean space to Minkowski space

Many interesting problems relating to Euclidean and hyperbolic geometrycan be tackled.

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8 Stochastics, Discrete Mathematics and In-

formation

8.1 Combinatorics — Graphs of large chromatic num-ber

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 χ-bounded classes and theGyárfás-Sumner Conjecture.

8.2 Mathematical models in finance

Possible topics might include a rigorous discussion of Itô’s lemma and its rela-tion 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 random-ness 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.

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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 dis-tributions 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 satu-rate 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.

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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 athird-year 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 Church-Turing 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.

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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.

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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 Prob-lem

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Mean-Variance 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 Time-dependent Co-authorship 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 Radon-Nikodym Theorem

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LogicModel Theory — Strongly minimal structuresModel Theory — The model theory of cyclic groups.Model theory of the real numbersO-minimal structuresLocal equivalents of the Axiom of ChoiceTheories of the real numbers

AlgebraQuantum groups and crystal basisAffine algebraic groups schemesHomotophy type theoryRelative algebraic geometryBeilinson-Bernstein 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 3-manifoldsCrystallographic groupsGromov's theorem and groups of polynomial growthApproximate groupsHigher-order 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