pre-approved part c dissertations 2017-18 · algebraic geometry using category theory. this allows...

Pre-approved Part C Dissertations 2017-18

June 1, 2017

Contents

1 Introduction 2

2 Algebra 3

3 Analysis 10

4 Geometry, Number Theory and Topology 20

5 Logic 33

6 Mathematical Methods and Applications 39

7 Mathematical Physics 59

8 Stochastics, Discrete Mathematics and Information 63

9 History of Mathematics 71

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1 Introduction

This document contains the list of pre-approved Part C dissertations for2017-18. The dissertation topics are arranged into sections according to thesubject panel who proposed the topic. Some topics are relevant to more thanone subject panel and so are listed twice.

If you would like to undertake one of these dissertations you should makecontact with the named supervisor, to give you and the supervisor an oppor-tunity to think further about what the project might involve, and whetheryou are the right student to do it. You are advised to do this before the endof Trinity term.

Once you have chosen a topic and obtained the agreement of the supervisor,you should notify the Projects Committee that you are intending to offer adissertation by completing the form available at https://www.maths.ox.ac.uk/members/students/undergraduate-courses/teaching-and-learning/

projects/essays-and-dissertations. The deadline for submitting theform is 12noon on Friday of week 0, Michaelmas term 2017.

You are not limited to the list of dissertation topics in this booklet and maypropose your own topic if you wish. Further information on how to do this canbe found in the Project Guidance Notes available at https://www.maths.

ox.ac.uk/members/students/undergraduate-courses/teaching-and-learning/

projects/essays-and-dissertations.

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https://www.maths.ox.ac.uk/members/students/undergraduate-courses/teaching-and-learning/projects/essays-and-dissertations






2 Algebra

Suggested title of dissertation: Quantum groups and crystal basis

Dissertation supervisor: Kobi Kremnitzer ([email protected])

Maximum number of students: 2

Description of proposal: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.

Possible avenues of investigation:Knot invariants and the Jones polynomial. Different combinatorial descrip-tions of canonical basis. Modular representation theory and quantum groupsat roots of unity.

Pre-requisite courses (listed as essential, recommended, useful)Algebra II (essential), Introduction to representation theory (recommended).

Useful pre-reading (summer vacation)Kassel: Quantum Groups.

Further referencesJantzen: Lectures on quantum groups. Kassel: Quantum Groups. Kashi-wara: On Crystal bases.

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Suggested title of dissertation:Affine algebraic groups schemes



Description of proposal: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.

Possible avenues of investigation:Modular representation theory. Borel-Weil-Bott theorem. Cohomology ofalgebraic groups and Lie algebras.

Pre-requisite courses (listed as essential, recommended, useful)Algebra II (essential), Introduction to representation theory (recommended),Galois Theory (recommended).

Useful pre-reading (summer vacation)Milne: Algebraic groups.

Further referencesWaterhouse: Introduction to affine group schemes.

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Suggested title of dissertation: Homotophy type theory



Description of proposal: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.

Possible avenues of investigation:Formalising mathematics in HoTT.

Pre-requisite courses (listed as essential, recommended, useful)Logic (useful), Set Theory (useful), Topology and groups (useful).

Useful pre-reading (summer vacation)Robert Constable: The Triumph of Types: Creating a Logic of Computa-tional Reality. Bengt Nordstrm, Kent Petersson Jan M. Smith: Programmingin Martin-Lfs type theory.

Further referencesThe Univalent Foundations Program, Institute for Advanced Study: Homo-topy Type Theory: Univalent Foundations of Mathematics.

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Suggested title of dissertation: Relative algebraic geometry



Description of proposal: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.

Possible avenues of investigation:

Pre-requisite courses (listed as essential, recommended, useful)Algebra II (essential).

Useful pre-reading (summer vacation)F. William Lawvere and Stephen Schanuel: Conceptual Mathematics: a firstintroduction to categories. Ravi Vakil: Foundations of algebraic geometry.

Further referencesToen and Vaquie: Under Spec Z.

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Suggested title of dissertation: Beilinson-Bernstein localisation for sl2



Description of proposal: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.

Possible avenues of investigation:Category O and the Langlands classification. Modular representation theory.

Pre-requisite courses (listed as essential, recommended, useful)Algebra II (essential), Introduction to representation theory (recommended),Algebraic curves (recommended).

Useful pre-reading (summer vacation)F. William Lawvere and Stephen Schanuel: Conceptual Mathematics: a firstintroduction to categories. Ravi Vakil: Foundations of algebraic geometry.

Further referencesHotta, Ryoshi, Takeuchi, Kiyoshi, Tanisaki, Toshiyuki: D-Modules, PerverseSheaves, and Representation Theory. Dennis Gaitsgory: Geometric repre-sentation theory.

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Suggested title of dissertation: Representations of finite Hecke algebras

Dissertation supervisor: Dan Ciubotaru ([email protected])


Description of proposal:Finite Hecke algebras are certain deformations of finite real reflection groups(such as the symmetric group, the hyperoctahedral group etc.) that appearnaturally in many situations in representation theory, algebraic geometry,or number theory. The aim of this dissertation is to explore several resultsrelated to the structure and the representation theory of these algebras.

Possible avenues of investigation:Finite Coxeter groups, structural results, classification, conjugacy classes.

The explicit combinatorial realization of the simple Hecke algebra modulesover a field of characteristic 0; e.g., Specht modules for the Hecke algebraof Sn, Hoefsmit’s combinatorial construction of the simple modules for theHecke algebra of type Bn.

Modular representation theory for the Hecke algebra of Sn, e.g., the resultsof Dipper and James and the more recent results in the area (the expositorypaper of Kleshchev in Bull.A.M.S. is a good point of reference).

The representation theory of cyclotomic Hecke algebras for complex reflectiongroups, as in the work of S. Ariki.

Rationality question for the characters finite Hecke algebras.

Lusztig’s construction of a canonical line in a simple module of the Heckealgebra of Sn or more generally, for modules corresponding to special repre-sentations of finite Coxeter groups.

Pre-requisite courses (listed as essential, recommended, useful)Essential: the Part A courses in algebra. Recommended: Part B representa-tion theory.

Useful pre-reading (summer vacation)Coxeter groups: the first 2 chapters in J. Humphreys: “Reflection groupsand Coxeter groups”.

Finite Hecke algebras: Chapter 8 in “Characters of finite Coxeter groups andIwahori-Hecke algebras” by M. Geck and G. Pfeiffer.

A. Kleshchev’s expository paper “Representations of symmetric groups andrelated Hecke algebras”, arXiv:0909.4844.

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Further referencesDepending on the direction of investigation:

- Chapters 3 and 5 or Chapters 9 and 10 in the book of Geck and Pfeiffer;

- Mathas’ book “Iwahori-Hecke algebras and Schur algebras of the symmet-ric group”, University Lecture Series 15, American Mathematical Society,Providence, RI, 1999.

- S. Ariki: “On the decomposition numbers of the Hecke algebra of G(m,1,n)”,J. Math. Kyoto Univ. 36 (1996).

- R. Dipper and G. D. James, Representations of Hecke algebras of generallinear groups, Proc. London Math. Soc. 52 (1986)

- Lusztig’s paper arXiv:1507.02263.

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3 Analysis

Suggested title of dissertation: The method of alternating projections

Dissertation supervisor: Dr David Seifert ([email protected])


Description of proposal:Let X be a Hilbert space and suppose that M1, . . . ,MN are closed subspacesof X. Let Pk denote the orthogonal projection onto Mk, 1 ≤ k ≤ N , and letPM denote the orthogonal projection onto M = M1∩ . . .∩MN . Consider theoperator T : X → X given by T = PN . . . P2P1. Then according to a classicaltheorem of Halperin we have

limn→∞

‖T nx− PMx‖ = 0

for all x ∈ X. The result can be interpreted as saying that if we repeatedlyproject an arbitrary vector x ∈ X onto the subspaces M1, . . . ,MN in cyclicorder, then in the limit we approach the point in M which is closest to x.This so-called method of alternating projections has many applications, forinstance to the numerical solution of elliptic PDEs on composite domainsand of sparse systems of linear equations, but it is also interesting from atheoretical point of view. Halperin’s theorem moreover has an interestinghistory, having originally been discovered by von Neumann (in the case N =2) and then rediscovered in the context of various different applications.

The starting point of the project would be a complete proof of Halperin’stheorem, or some variant of it, perhaps giving a brief outline of the historyof the result. The main part of the dissertation could then focus eitheron certain applications of the method of alternating projections or on theabstract theory underlying it, for instance by investigating one or several ofthe following possible topics.


• extension to closed convex sets (with counterexample to Halperin’s the-orem)

• extension to the Banach space setting

• failure of Halperin’s theorem for non-cyclically repeated projections

• weak convergence for infinite products of projections

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• alternative proofs of Halperin’s theorem, for instance using the spectraltheorem (when N = 2) or the Katznelson-Tzafriri theorem

• rates of convergence in Halperin’s theorem

• application to the numerical solution of elliptic PDEs on compositedomains (the Schwarz alternating method)

• application to the iterative solution of sparse linear systems (the Kacz-marz method)

Pre-requisite courses:Essential: B4.1 Banach space, B4.2 Hilbert spaces. Recommended: C4.1Functional Analysis. Courses on numerical solution of differential equations(such as B6.1 and B6.2) or on the rigorous theory of PDEs (such as C4.3)may be useful for students wishing to investigate particular applications ofthe method of alternating projections.

Useful pre-reading (summer vacation)

• A. Netyanun and D.C. Solmon. Iterated products of projections in Hilbertspace. Amer. Math. Monthly, 113(7):644–648, 2006

• F. Deutsch. Best approximation in inner product spaces. CMS Booksin Mathematics. Springer, New York, 2001

Further references

• C. Badea, S. Grivaux, and V. Muller. The rate of convergence in themethod of alternating projections. Algebra i Analiz (St. PetersburgMath. J.), 23(3):1–30, 2011

• I. Halperin. The product of projection operators. Acta Sci. Math. (Szeged),23:96–99, 1962

• E. Kopecka and V. Muller. A product of three projections. Studia Math.223(2):175–186, 2014

• P.-L. Lions. On the Schwarz alternating method I. In First internationalsymposium on domain decomposition methods for partial differentialequations, pp. 1–42, 1988

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Suggested title of dissertation: Maximum principles for elliptic equations

Dissertation supervisor: Prof. Luc Nguyen ([email protected])


Description of proposal:The maximum principle proves extremely useful in the analysis of ellipticPDE. This project surveys known results on the maximum principle in bothqualitative and quantitative form, and possibly wanders in to the realmof nonlinear and possibly degenerate elliptic equations, where much less isknown.

Possible avenues of investigation:Classical maximum principle and strong maximum principle

Nash-Moser-De Giorgi estimates elliptic equations in divergence form

Alexandrov-Bakelmann-Pucci estimates elliptic equations in non-divergenceform

The user guide on viscosity solutions by Crandall, Ishii and Lions.

Pre-requisite courses:A4 Integration is required. Familiarity with Lebesgue spaces is essential.

Useful pre-reading (summer vacation)Evans’ Partial Differential Equations, Chapter 2 (Section 2.2), Chapter 5 andAppendix C

Further referencesHan and Lin’s Elliptic Partial Differential Equations

Caffarelli and Cabre’s Fully Nonlinear Elliptic Equations

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Suggested title of dissertation: Curve shortening flow

Dissertation supervisor: Prof. Melanie Rupflin ([email protected])


Description of proposal:Curve shortening flow is a geometric partial differential equation that changesclosed curves γ in the plane by moving in direction of the negative gradient ofthe length L and thus reduces the length of the curves. An interesting featureof the flow is that for convex curves the isoperimetric ratio L2

Areadecreases in

time and converges to the ratio of the circle, thus providing an alternativeproof of the isoperimetric inequality 4π · Area ≤ L2.

Possible avenues of investigation:Properties of the flow for convex curves and the resulting proof of the isoperi-metric inequality using the maximum principle.

Pre-requisite courses:No specific courses are required, but the student should have an interest inboth Analysis and Geometry.

Useful pre-reading (summer vacation)Basics of local and global theory of planar curves as found e.g. in the firstparts of the chapters 8 and 9 of Differential Geometry: Manifolds, curvesand surfaces by Berger and Gastiaux or in another

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Suggested title of dissertation: Aspects of continuous function spaces [Aspecific project should have a more precise title finally ]

Dissertation supervisor: Prof. Hilary Priestley ([email protected])


Description of proposal:The proposal may be seen as grounded in Analysis, from a pure mathematicalperspective. It goes in a direction different from that followed in Part Canalysis courses: currently none of these covers Banach algebras or C∗ -algebras. A project in this area should appeal to a Part C student who wouldenjoy exploring connections between functional analysis and other branchesof pure mathematics. There is scope for the project to have a topological,algebraic, or categorical flavour, for example, according to which avenue ispursued.

Consider the spaces CR(X) and CC(X) of, respectively, real-valued andcomplex-valued continuous functions on a compact Hausdorff space X. Suchspaces feature strongly in Part B Banach Spaces as examples. But they havemuch richer structure than that of normed spaces. The discussion of theStone–Weierstarss Theorem in B4.1 gives a glimpse of this: multiplicativestructure and (in the real case) order structure come into play. There aremany possible project topics which stem from this.


These include, but are not confined to, the following

1. In which ways does C(X) determine X?

2. The Gelfand–Naimark Theorem and its place in the mathematical land-scape.

3. Continuous function spaces from a categorical perspective: a studyof the various equivalences and dual equivalences which can be set upbetween the category of compact Hausdorff spaces and continuous mapsand other categories.

The focus could be on the real case, on the complex case, or could involveboth, depending on the avenue to be investigated.

Pre-requisite courses:Essential: Part B Banach Spaces

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Recommended: Part A Topology (first half); Part A Rings and Modules(rings part only)

Useful: (depending on the subtopic chosen) Part C Analytic Topology; PartC Category Theory; some familiarity with measure theory

Useful pre-reading (summer vacation)G.F. Simmons, Introduction to Topology and Modern Analysis, McGraw Hill(1965), reprinted by Robert J. Krieger (1983)Highly recommended. A readable book which should be available in collegelibraries. The earlier parts cover background material, at a Part B level.Part III is directly relevant to the project proposal.

Z. Semadeni, Spaces of continuous functions on compact sets, Advances inMath. 1 fasc. 3 (1965), 319-382(.pdf file downloadable)This is not well written, but conveys some of the flavour, in particular inrelation to 1. above).

Further referencesG. Bezhanishvili, P.J. Morandi, B. Olberding, Bounded Archimedean `- al-gebras and Gelfand–Neumark–Stone algebras, Theory and Applications ofCategories, Vol. 28, no. 16 (2013), 435-475(.pdf file downloadable)This recent article is most closely connected to 2., 3. above, in the real case,but its introduction provides a contextual summary of the whole field.

L. Gillman, M. Jerison, Rings of Continuous Functions, Van Nostrand (1960)

Z. Semadeni, Banach Spaces of Continuous Functions, PWN-Polish ScientificPublishers (1971)Comprehensive, for reference only.

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Suggested title of dissertation: Hausdorff dimension, fractals and multi-fractals

Dissertation supervisor: Prof. Dmitry Belyaev ([email protected])

Maximum number of students: 1 or 2

Description of proposal:We start with a notion of Hausdorff measure and dimension which allows todetermine “size” of a set. With this notion we could define fractals as setswith non-integer dimension. A particularly important class of fractals areself-similar fractals which usually appear as limits of iterative constructions.A canonical example is the von Koch snowflake. The aim of the project is tostudy different notions of dimension and fractals as well as multifractals: setsthat have different behaviour at different scales, so they look like a mixtureof different self-similar fractals.


• General theory of dimensions of sets and measures. Connections be-tween them, Frostman’s lemma.

• Multifractal analysis.

• What is the dimension of the set of numbers such that all digits inits decimal expression have given frequencies? This gives a connectionbetween the notions of dimension, entropy, and dynamics.

• Many fractals of interest are of probabilistic nature. One of the mainexamples is the trajectory of Brownian motion, but there are manyother examples such as random snowflakes, random Cantor sets etc.This leads to a possibility of a more probability-oriented approach. Inparticular, even the basic question: How one defines dimension of arandom set or a measure, becomes non-trivial.

Pre-requisite courses (listed as essential, recommended, useful)

Essential: Part A Integration


Standard references about the dimensions of sets and measures are two booksby Falconer.

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• K. Falconer, Fractal geometry: mathematical foundations and applica-tions.

• K. Falconer, Techniques in fractal geometry.

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Suggested title of dissertation: Coefficient problems for univalent maps

Dissertation supervisor: Prof. Dmitry Belyaev ([email protected])


Description of proposal:The aim of the project is to study the coefficient problems for univalent func-tions in the disc. Let f =

∑∞0 anz

n be an analytic function in the unit disc.We assume that it is univalent, i.e. f(z1) = f(z2) if and only if z1 = z2. Wealso assume that the function is normalized by f(0) = 0 and f ′(0) = 1. A fa-mous Bieberbach conjecture states that |an| 6 n and the inequality is sharp.This conjecture has been proved by de Branges in 1985. If we additionallyassume that f is bounded, then the sharp upper bound is not known. Theaim of the project is to explore the existing results and their connectionswith other areas of analysis.

Possible avenues of investigation:Study sharp upper bounds for a first few coefficients.

Study estimates of the growth rate for the coefficients.

Pre-requisite courses (listed as essential, recommended, useful)It is strongly recommended to take Part A Integration and Part C ComplexAnalysis: Conformal Maps and Geometry

Useful pre-reading (summer vacation)Relevant chapters of

• Ch. Pommerenke, Univalent functions.

• Ch. Pommerenke, Boundary behaviour of conformal maps.

• P. Duren, Univalent functions.

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Suggested title of dissertation: Unique Continuation for Elliptic PDE

Dissertation supervisor: Prof. Yves Capdeboscq ([email protected])


Description of proposal:The so called Identity Theorem in Part A Complex Analysis shows thata holomorphic function can only have isolated zeros if it is not identicallynought.

This shows in turn that the same (interior) property holds true for solutionsof Laplace’s equation in dimension 2. The appropriate generalization of thisfact beyond the particular case of analytic functions is called the UniqueContinuation Property. This project focuses on the various proofs and openproblems associated with unique continuation.

Possible avenues of investigation:Beyond the harmonic case in dimension two, a variety of different directionthat can be taken. For example : regular problems is larger dimensions,or less regular problems in dimension 2 (heterogeneous coefficients or exoticgeometries), or quantitative vs. non quantitative results.

Technically, there are also very different sets of techniques, including

• The 3-spheres inequality are generalizations,

• Quasi-conformal maps,

• Quasi-analytic functions and Carleman inequalities.

Pre-requisite courses:There are no pre-requisites, however possibly useful courses are A Manifolds,B4.1, B4.2, and to a lesser extent A6, B3.2 and B5.2.

Useful pre-reading (summer vacation)A textbook on conformal maps or on elliptic PDEs, see for example thereading list of C4.1, C4.3 or C4.8.

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4 Geometry, Number Theory and Topology

Suggested title of dissertation: Graphical small cancellation theory

Dissertation supervisor: David Hume ([email protected])


Description of proposal:Graphical small cancellation theory provides a way of ensuring that a certaingraph appears as a subgraph of the Cayley graph of a group. It is one ofthe most powerful techniques for producing finitely generated groups withexceptional properties (for instance that contain an isometrically embeddedexpander). The goal of the project would be to explore this technique andunderstand some of its key applications.

Possible avenues of investigation:Strebel’s classification of bigons and triangles, and the geometry of van Kam-pen diagrams.

Rips’ technique for constructing small cancellation groups which admit sub-groups with unusual properties.

The work of Arzhantseva-Cashen-Gruber-Hume studying negative curvaturein small cancellation groups.

Osajda’s proof that certain expanders can be embedded into a group.


Topology and Groups would be useful, but not essential.

Useful pre-reading (summer vacation)Chapter V of Lyndon and Schupp’s book “Combinatorial Group Theory”(Springer, Classics in Mathematics) is an excellent introduction to clas-sical small cancellation theory. A more graphical approach is presentedin http://homepage.univie.ac.at/dominik.gruber/papers/graphical.pdf. Theambitious may want to delve a little into Gromov’s fantastically influencialpaper “Random walk in random groups” where graphical small cancellationtheory is first introduced.

Further referencesThere is an abundance of good references on small cancellation theory. Theintroduction of https://arxiv.org/abs/1602.03767 by Arzhantseva–Cashen–Gruber–Hume is a good and up-to-date source of suitable references.

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Suggested title of dissertation: Separation profiles of graphs and groups

Dissertation supervisor: David Hume ([email protected])


Description of proposal:Given a graph X, how many vertices are needed to break every subgraphwith at most n vertices in half? The answer, called the separation profile ofX, is a recent but very useful tool in geometry and the study of groups. It hasbeen shown to be closely related to expanders, volume growth, and certainnotions of finite dimensionality. It can also be used as an obstruction to onegroup being a subgroup of another. The aim of the project is to explore someof the existing calculations of this profile for nice graphs and survey some ofthe key open questions in the area.

Possible avenues of investigation:Benjamini–Schramm–Timar’s calculations of separation profiles for Euclideanand hyperbolic graphs and their bounds on the profiles of products.

Calculations of separation profiles of nilpotent, solvable and hyperbolic groups.

More ambitiously, the project may study closely related profiles with a moreanalytic flavour.


Nothing essential, a student who has taken Topology and Groups will havean advantage but only in the early stages.

Useful pre-reading (summer vacation)(At least) chapter I of the Topology and Groups notes. The paper(https://arxiv.org/abs/1004.0921) of Benjamini, Schramm and Timar shouldbe the main source. To better understand the examples mentioned may re-quire a survey on geometric group theory: for instancehttps://www.math.ucdavis.edu/ kapovich/280-2009/bhb-ggtcourse.pdf.

Further references“A continuum of expanders” by D. Hume (https://arxiv.org/abs/1410.0246)gives an equivalent definition of separation which is more closely linked toexpanders.

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Suggested title of dissertation: The classification of complex algebraicsurfaces

Dissertation supervisor: Prof. Balazs Szendroi ([email protected])


Description of proposal:Smooth algebraic surfaces over the complex numbers can be classified usingthe properties of their canonical divisor class. This classification is similar tothe classification of smooth algebraic curves (covered in the course AlgebraicCurves) into curves of genus 0 (rational curves, with negative degree canonicalclass), 1 (elliptic curves, canonical class trivial) and at least two (curves ofhigher genus, with positive degree canonical class). The case of surfaces ismore complicated, but follows a similar pattern. One new feature in the two-dimensional case is the existence of blow-ups and blow-downs. The aim of thedissertation is to understand the fundamental ingredients of the classification,and to study some of the possible cases that can arise.

Possible avenues of investigation:After stating the ingredients of the classification theorem, the dissertationcan explore some of the cases, such as rational surfaces, K3 surfaces, surfacesof general type, etc.

Pre-requisite courses (listed as essential, recommended, useful)Algebraic curves (essential) Commutative algebra (recommended)

Useful pre-reading (summer vacation)Reid: Undergraduate algebraic geometry, LMS Student Texts, CUP

Further referencesBeauville: Complex algebraic surfaces, LMS Student Texts, CUP Reid: Chap-ters on algebraic surfaces, http://arxiv.org/abs/alg-geom/9602006

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http://arxiv.org/abs/alg-geom/9602006

Suggested title of dissertation: Local Fields and the Hasse Principle

Dissertation supervisor: Victor Flynn ([email protected])


Description of proposal:The main theme of this Part C Dissertation will be to study local fieldsand their applications to number theory. This will include some of the maingeneral ideas and theorems: fields with non-Archimedean valuations, Hensel’slemma and some of its consequences, the p-adic numbers Qp, and extendingvaluations to finite field extensions, with examples to illustrate these ideas.There will then be an emphasis on violations of the Hasse principle (caseswhere there are solutions everywhere locally but not globally), looking atexamples where this occurs, applying techniques for showing that there aresolutions everywhere locally, and using ideas from Number Theory (such asquadratic reciprocity) to show that there are no solutions globally. The mainemphasis will be examples defined over Q (and where the completions areR and Qp), but there is also the possibility later of investigating examplesover number fields and using ideas from algebraic number theory. Thereare several texts (listed below) which are available as sources for the mainbackground ideas, and it is also intended to read some research articles fromthe last 50 years (two of these are listed below), to extract from these articlessome famous examples of violations of the Hasse principle. Since many of theinteresting examples arise from elliptic curves, there will also the scope forthe discussion of a few ideas about elliptic curves, such as descent methodsfor trying to find the rank of an elliptic curve.

Relevant courses:Essential: Part A Rings and Modules. Part A Number Theory. Part B GaloisTheory, Part B Algebraic Number Theory. (should also intend to take PartC Elliptic Curves, concurrently with the Dissertation).

Recommended: Part A Group Theory. Part B Algebraic Curves.

Useful. Part A Projective Geometry. Part B Commutative Algebra. (andPart C Algebraic Geometry, concurrently with the Dissertation).

Useful pre-reading (summer vacation)J.W.S. Cassels. Local Fields (1987). Cambridge University Press. ISBN:0-521-31525-5.

F.Q. Gouvea. p-adic Numbers: An Introduction. Springer, 2nd edition(2003). ISBN: 3-540-62911-4.

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J.W.S. Cassels. Lectures on Elliptic Curves. LMS Student Texts 24. Cam-bridge University Press, Cambridge, 1991. ISBN: 0-521-42530-1.

J.H. Silverman. The Arithmetic of Elliptic Curves. Graduate Texts in Math-ematics 106. Springer-Verlag, 1986. ISBN: 0-387-96203-4.

Further referencesB.J. Birch and H.P.F. Swinnerton-Dyer. The Hasse problem for rationalsurfaces., J. Reine Angew. Math., 274/275 (1975), 164-174.

E.V. Flynn and J. Redmond, Application of Covering Techniques to Familiesof Curves, J. Number Theory 101 (2003), 376-397.

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Suggested title of dissertation: Prime numbers in residue classes

Dissertation supervisor: Prof. Ben Green ([email protected])


Description of proposal:The prime number theorem states that π(x), the number of primes less thanor equal to x, is asymptotic to x/ log x. This is the main result of the courseC3.8 Analytic number theory. One may also ask how many primes less thanor equal to x are congruent to a modulo q, for q > 1. The aim of thisdissertation is to understand some aspects of this question. The essay shouldbe thought of as a second course in analytic number theory, following onfrom C3.8.

Possible avenues of investigation:The essay should begin by discussing Dirichlet characters and giving a proofof some version of the prime number theorem in a residue class. Further top-ics might include a discussion of Siegel zeros, the Bombieri–Vinogradov the-orem (which gives results as strong as the Generalised Riemann Hypothesis“on average”), a look at the work of Friedlander and Granville on limitationsto the well-distribution of primes in residue classes with large moduli q, andupper bounds via sieve methods.

Pre-requisite courses (listed as essential, recommended, useful)C 3.8 Analytic number theory (to be taken in conjunction with the essay)

Useful pre-reading (summer vacation)Course notes from C3.8 last year (available online)

H. Davenport, Multiplicative Number Theory, Springer Graduate Texts inMathematics 74.

Further references

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Suggested title of dissertation: Freiman’s theorem and related topics

Dissertation supervisor: Prof. Ben Green ([email protected])


Description of proposal:Let A be a finite set of integers. The sumset A + A is the set of all sumsa1+a2 with a1, a2 ∈ A. Freiman’s theorem is a partial answer to the followingquestion: what is the structure of A if |A + A| is not too much larger than|A|? The aim of this essay is to give Ruzsa’s proof of this theorem, developingthe necessary tools from additive combinatorics along the way, and then toexplore further topics.

Possible avenues of investigation:Further topics might include a discussion of Freiman’s theorem in othergroups, for example Fn

2 or non-abelian groups.

Pre-requisite courses (listed as essential, recommended, useful)None in particular.

Useful pre-reading/further referencesThere is a lot of material online about this topic. Potentially interestedstudents might browse around for a little while and then come and talk to me.My review of the book Additive Combinatorics by Tao and Vu, available athttp://people.maths.ox.ac.uk/greenbj/papers/book-review.pdf, gives a shortintroduction to this general area.

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Suggested title of dissertation: Unramified extensions of number fields

Dissertation supervisor: Prof. Minhyong Kim ([email protected])


Description of proposal:The purpose of this project is to study the fundamental group of rings ofintegers in algebraic number fields. In concrete terms, this amounts to con-structing unramified extensions of number fields. There are examples in theliterature where many examples have been constructed for solvable groupsand some simple non-abelian groups. We will try to construct more examplesusing a combination of numerical work, theory of Galois representations, andanalytic techniques.

Possible avenues of investigation:We will start with some previous papers on the subject including Unramifiedalternating extensions of quadratic fields by Kiran Kedlaya and the papersof my former Ph.D student Kwang-seob Kim (Google kwang seob kim un-ramified). I havent thought about how one might extend these techniques,but am willing to help students think about possibilities.

Pre-requisite courses (listed as essential, recommended, useful)Essential: A3, A5, ASO: Number Theory, B3.1, B3.3, B3.4

Recommended: B2.1, B3.5

Useful: B3.2

Useful pre-reading (summer vacation)James Milne: Lecture notes on algebraic number theory, Lecture notes onClass Field Theory

F. Diamond and J. Shurman: A first course on modular forms

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Suggested title of dissertation: Number of points on varieties over finitefields. Correlations between primes.

Dissertation supervisor: Prof. Minhyong Kim ([email protected])


Description of proposal:A rather deep phenomenon of number theory comes up in studying the num-ber of solutions to congruence equations like x3 + y3 + z3 = 0 mod p. Onecan look at two kinds of variations, a “vertical” variation over finite fieldswith prime power numbers of elements for a fixed prime, and a “horizontal”one, where p varies. We will try to investigate this phenomenon throughvarious examples, especially looking for correlations.

Possible avenues of investigation:Some famous results include the Weil conjectures, the Sato-Tate conjecturesand relation to modular forms. Even though the general machinery is ad-vanced, the student will likely end up trying to acquire some concrete feelfor the advanced literature in this direction.

Pre-requisite coursesEssential: A3, A5, ASO: Number Theory, B3.1, B3.3, B3.4

Recommended: B2.1 , B3.5

Useful: B3.2

Useful pre-reading (summer vacation)Serre: A course in arithmeticSilverman-Tate: Rational points on elliptic curves.James Mile: Lecture notes on Class Field Theory

Further references

28

Suggested title of dissertation: Abelian extensions of the rationals andof quadratic fields.

Dissertation supervisor: Alan Lauder ([email protected])


Description of proposal:Extensions of the rationals with an abelian Galois group are subfields ofcyclotomic fields, and as such can be explicitly described via values of theexponential function e2πiz at rational arguments (Kronecker-Weber Theo-rem). The “dream of youth” of Kronecker (1823 - 1891) extends this toimaginary quadratic fields by replacing the exponential function by a cer-tain modular function j(z) and rational arguments by quadratic irrationalsin the upper half of the complex plane. Hilbert’s 12th problem asks (rathervaguely) whether these ideas can be taken further.

The dissertation would explore this circle of ideas, probably focussing mainlyon the more classical complex analytic work from the late 19th and early 20thcentury, but could alternatively also look at the application of more recentp-adic methods.

Possible avenues of investigation:Kronecker-Weber Theorem (for the rational field)

Kronecker’s Jugendtraum (for imaginary quadratic fields)

Darmon-Dasgupta’s p-adic conjecture (for real quadratic fields).

Pre-requisite courses (listed as essential, recommended, useful)Part B Algebraic Number Theory, and thus also its prerequisite courses, areessential. A liking for both algebraic and analytic methods is desired, as theclassical constructions are very much complex analytic.

Useful pre-reading (summer vacation)D. Cox, Primes of the form x2 + ny2 (an excellent resource for the projectitself).

Further referencesH. Darmon and S. Dasgupta, Elliptic units for real quadratic fields.

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Suggested title of dissertation: Geometric Class Field Theory

Dissertation supervisor: Damian Rossler ([email protected])

Maximum number of students: None specified

Description of proposal:The main theorem of class field theory gives a description of abelian exten-sions of global fields (ie number fields or functions fields of curves over finitefields) in terms of the idele classgroup of the field. It is possible to provethe main theorem of class field theory using an axiomatic approach whichworks for any global field (although the verification of the axioms is harderfor number fields) but in the case of functions fields of curves over finitefields, the main theorem can be proven using geometric methods, involvingthe generalised Jacobians of curves. This method was initiated by Lang andRosenlicht in the fifties.

The aim of the dissertation is to describe the main results of the class fieldtheory of function fields of curves over finite fields and give an outline of theproofs. The main reference is the book [1] which describes and develops allthe results of Lang and Rosenlicht. A very concise summary of the theory isgiven in [2]. A very good reference for general class field theory is the bookIf the student prefers it, he/she could also treat geometric class field theoryas a special case of general class field theory but he/she would then have toverify the axioms in this special case.

Pre-requisite courses, useful pre-reading (summer vacation)B3.1 Galois Theory (essential)B3.3 Algebraic Curves (recommended)B3.4 Algebraic Number Theory (recommended)C3.4 Algebraic Geometry (essential)C2.6 Introduction to Schemes (recommended)C3.7 Elliptic Curves (recommended)

Further references

[1] Algebraic Groups and Class Fields by J.-P. Serre (Springer, 1975)

[2] Brian Conrad : Geometric Global Class Field Theory (not published butcan be found on his homepage at Stanford).

[3] Artin, Tate: Class Field Theory (AMS Chelsea Publishing).

30

Suggested title of dissertation: The Semistable Reduction Theorem forCurves over Function Fields.

Dissertation supervisor: Damian Rossler ([email protected])

Maximum number of students: None specified

Description of proposal:The aim of this dissertation is to present the details of the proof by Artin-Winters of the semistable reduction theorem for curves over function fields.This theorem states that up to a ramified base-change, a curve over a functionfield always has a semistable model over the smooth model of the functionfield. The details are given in the article [1].

In the book [2] all the necessary background material is described and theproof of Artin-Winters is also described.

The proof of this theorem is much easier in characteristic 0 and the disserta-tion could start with that case. For this, a good reference is [3] where a moresophisticated proof (which uses etale cohomology) of the general theorem isalso given.

Possible avenues of investigation, pre-requisite courses, useful pre-reading (summer vacation)B3.3 Algebraic Curves (essential)C3.4 Algebraic Geometry (recommended)C2.6 Introduction to Schemes (essential)B8.5 Graph Theory (recommended)

Please see Prof. Rossler.

References

[1] Artin, M.; Winters, G. Degenerate fibres and stable reduction of curves.Topology 10 (1971), 373–383

[2] Liu, Qing Algebraic geometry and arithmetic curves. Oxford GraduateTexts in Mathematics, 6. Oxford Science Publications. Oxford UniversityPress, Oxford, 2002

[3] Abbes, Ahmed Reduction semi-stable des courbes d’apres Artin, Deligne,Grothendieck, Mumford, Saito, Winters, . Courbes semi-stables et groupefondamental en geometrie algebrique (Luminy, 1998), 59110, Progr. Math.,187, Birkhuser, Basel, 2000

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Suggested title of dissertation: The Zilber-Pink conjecture

Dissertation supervisor: Jonathan Pila ([email protected])


Description of proposal:The project will study the Zilber-Pink conjecture (ZP): its provenance, for-mulation, and implications, which include several classical conjectures includ-ing the Manin-Mumford and Mordell-Lang conjectures (solved), the Andre-Oort conjecture (many cases solved but still open in general), and muchmore.

Possible avenues of investigation:The project could be pursued in a number of directions, according to inter-ests:

Connections with Schanuel’s conjecture and model theory of ez:

Various classical (or not so classical) special cases of ZP;

Arithmetic of special points;

Variations.

Pre-requisite courses (listed as essential, recommended, useful)Essential: Some number theory, Galois theory.

Useful (depending on directions taken): Algebraic number theory, Logic,Model Theory, Modular Forms/Elliptic Curves

Useful pre-reading (summer vacation)J. Pila, O-minimality and diophantine geometry, Proc ICM 2014, Seoul.

U. Zannier, Some problems of unlikely intersections in geometry and arith-metic, Annals Math. Studies 181, Princeton UP, 2012.

Further referencesE. Bombieri, D. Masser, U. Zannier, Intersecting a curve with algebraic sub-groups of multiplicative groups, IMRN 20 (1999), 1119–1140.

E. Bombieri, D. Masser, U. Zannier, On unlikely intersections of complexvarieties with tori, Acta Arithm. 133 (2008), 309–323.

B. Zilber, Exponential sums equations and the Schanuel conjecture, J. Lon-don Math. Soc. 65 (2002), 27–44.

32

5 Logic

Suggested title of dissertation: Continuous model theory

Dissertation supervisor: Ehud Hrushovski ([email protected])


Description of proposal:An introduction to continuous logic. The survey referenced below gives agood idea of the scope. It should lead up to basic stability (including defin-ability of types), and quantifier elimination in some significant stable exam-ples, such as Hilbert spaces and probability algebras, and either L1-latticesor possibly a new variant. In these examples, what does stable independenceamount to and what classical notion is it related to?

Possible avenues of investigation:A good start would be the papers

Ita Ben Yaacov and Alexander Usvyatsov, Continuous first order logic and lo-cal stability, Transactions of the American Mathematical Society 362 (2010),no. 10, 5213-5259

and the survey

Ita Ben Yaacov, Alexander Berenstein, C. Ward Henson, and AlexanderUsvyatsov, Model theory for metric structures, Model theory with applica-tions to algebra and analysis. Vol. 2, London Math. Soc. Lecture Note Ser.,vol. 350, Cambridge Univ. Press, Cambridge, 2008, pp. 315-427.

Both are available in Ben Yaacov’s page http://math.univ-lyon1.fr/~begnac/pub.html


Model theory (can be taken concurrently). A familiarity with elementarysoft analysis (Stone-Weierstrass theorem; basics of metric and of measurespaces.)


Begin with the survey and references therein.

Further references

A classical text on the subject:

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http://math.univ-lyon1.fr/~begnac/pub.html

http://math.univ-lyon1.fr/~begnac/pub.html

Chang, Chen-chung; Keisler, H. Jerome Continuous model theory. Annals ofMathematics Studies, No. 58 Princeton Univ. Press, Princeton, N.J. 1966xii+166 pp.

34

Suggested title of dissertation: Topics in o-minimality



Description of proposal:The project will study the provenance and basics of o-minimality: Tarski’stheorem on decidability of the real field and Tarski’s question about the modeltheory of real exponentiation; the isolation of the notion of o-minimality andits basic properties by van den Dries and Pillay-Steinhorn; further topics.

Possible avenues of investigation:The project will study basic properties including the proof and some con-sequences of the Cell Decomposition Theorem. Further topics according tostudent interests:

The growth dichotomy of Miller;

Complex analysis in o-minimal structures (Peterzil-Starchenko);

Diophantine applications;

Pfaffian functions and “Fewnomials”.

Pre-requisite courses (listed as essential, recommended, useful)Essential: Part B Logic and Part C Model Theory

Useful: Basic complex variables, number theory, depending on topics.

Useful pre-reading (summer vacation)L. van den Dries, Tame topology and o-minimal structures , LMS LectureNote Series 248, CUP, 1998.

Further referencesvan den Dries, Remarks on Tarski’s problem concerning (R,+,×, exp), LogicColloquium ’82, Lolli, Longo, amd Mrcja, editors, North Holland, 1984.

A. Khovanskii, Fewnomials , AMS Translations 88, AMS, 1991.

C. Miller, Exponentiation is hard to avoid, Proc. AMS 122 (1994), 257–259.

K. Peterzil and S. Starchenko, Tame complex geometry and o-minimality,Proc ICM 2010, Hyderabad.

S. Starchenko, Notes on o-minimality , available on Fields Institute website.

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Suggested title of dissertation: The Zilber-Pink conjecture



Description of proposal:The project will study the Zilber-Pink conjecture (ZP): its provenance, for-mulation, and implications, which include several classical conjectures includ-ing the Manin-Mumford and Mordell-Lang conjectures (solved), the Andre-Oort conjecture (many cases solved but still open in general), and muchmore.

Possible avenues of investigation:The project could be pursued in a number of directions, according to inter-ests:

Connections with Schanuel’s conjecture and model theory of ez:

Various classical (or not so classical) special cases of ZP;

Arithmetic of special points;

Variations.

Pre-requisite courses (listed as essential, recommended, useful)Essential: Some number theory, Galois theory.

Useful (depending on directions taken): Algebraic number theory, Logic,Model Theory, Modular Forms/Elliptic Curves

Useful pre-reading (summer vacation)J. Pila, O-minimality and diophantine geometry, Proc ICM 2014, Seoul.

U. Zannier, Some problems of unlikely intersections in geometry and arith-metic, Annals Math. Studies 181, Princeton UP, 2012.

Further referencesE. Bombieri, D. Masser, U. Zannier, Intersecting a curve with algebraic sub-groups of multiplicative groups, IMRN 20 (1999), 1119–1140.

E. Bombieri, D. Masser, U. Zannier, On unlikely intersections of complexvarieties with tori, Acta Arithm. 133 (2008), 309–323.

B. Zilber, Exponential sums equations and the Schanuel conjecture, J. Lon-don Math. Soc. 65 (2002), 27–44.

36

Suggested title of dissertation: The role of choice principles in ordertheory

Dissertation supervisor: Hilary Priestley ([email protected])


Description of proposal:Zorn’s Lemma and the Axiom of Choice are studied in B1.2 Set Theory buthave a role in mathematics well beyond set theory. This proposal wouldinvolve the exploration of some of the ways in which these principles andweaker versions of them come into play in the theory of ordered structures(partially ordered sets, lattices, Boolean algebras).

The topic would provide an opportunity to work with a range of algebraicand relational structures which are not systematically studied in pure coursesin Parts B and C. The subject has a number of different facets, with differentflavours and connections with a variety of other areas, including theoreticalcomputer science (domain theory).

Possible avenues of investigation:According to choice:

• The many ways in which choice principles are exploited in one or morebranches of order theory and/or lattice theory. (A survey-style project.)

• Order theory in a constructive world: many results are easily provedwith the aid of AC, ZL, or an equivalent but how far can one getwith ZF rather than ZFC, provided suitable modifications are made toconcepts and proofs?

• An investigation of results in order theory which are equivalent to AC orto weaker versions thereof, rather than merely implied by these choiceprinciples.


Essential: B1.2 Set Theory,

Also of possible relevance, depending on the focus: Part C Axiomatic SetTheory (though advanced set theory would be unlikely to be needed), PartC Analytic Topology, Part C Category Theory.


B.A. Davey and H.A. Priestley, Introduction to Lattices and Order, 2ndedition (CUP 2002), in particular Ch. 10. At undergraduate textbook level.

37

P.J. Cameron, Sets, Logic and Categories (Springer, 1999). General back-ground.

Further references

Available on request. Would draw on research literature mostly from thepast 15 years.

38

6 Mathematical Methods and Applications

Members of the Mathematical Method Panel are happy to supervise projectsin their research area(s). The Mathematical Methods Panel runs the B5and C5 courses. Please see https://www.maths.ox.ac.uk/groups/ociam/

people and https://www.maths.ox.ac.uk/groups/mathematical-biology/

people for the list of faculty in this area.

Suggested title of dissertation: Ecological niche formation in spatialmodels of coexistence dynamics

Dissertation supervisor: Dr Robert Van Gorder ([email protected])


Description of proposal:Several recent papers have considered extensions to a reaction-advection-diffusion system originally proposed in [1] as a model for colony formationand aggregation for population dynamics. In two spatial dimensions, theautonomous non-dimensional model can be written as [2]

∂uk∂t

= δ1∆uk −∇ · (ukwk) + rkuk

(Ak(x, y, t)−

K∑j=1

akjuj

), (1)

− εk∆wk + wk = ∇

(Ak(x, y, t)−

K∑j=1

akjuj

), (2)

where uk(x, y, t) is the biomass or population density of the kth species(k = 1, 2, . . . , K), wk is the local direction in which the species will advectat given space and time coordinates, δk > 0, εk > 0, and rk > 0 are givendiffusion, advection, and growth-rate parameters, respectively, while the pa-rameters ajk model inter and intra-species interactions. This can be thoughtof as a system of reaction-advection-diffusion equations, where the advectionis motivated biologically by populations choosing deterministically to movetoward favorable conditions and away from unfavorable regions. This modelwas studied in 1D domains for K = 2 (two-species) in [1] and in 2D domainsfor K = 2 in [2]. When the Ak’s are constant in all variables, spatially uni-form steady states were found corresponding to extinction of one or bothspecies, or a coexistence state [2]. In particular, the paper showed that if theuniform coexistence state existed, then it was always locally stable. A chiefresult of the work was that the long time asymptotics were dominated bythe kinetics, and no spatial patterning was observed over long time scales.

39

https://www.maths.ox.ac.uk/groups/ociam/people

https://www.maths.ox.ac.uk/groups/ociam/people

https://www.maths.ox.ac.uk/groups/mathematical-biology/people

https://www.maths.ox.ac.uk/groups/mathematical-biology/people

Meanwhile, for spatially varying Ak’s (i.e., Ak = Ak(x, y)), we obtain pat-tern formation as the respective species will be drawn to certain areas of thespatial domain.

We shall consider this model in order to see if spatial patterning and segre-gation can result from Ak(x, y) which are compactly supported over disjointsubsets of R2. This means that the resource available to one species willbe separated in space from resources available to other species. For largeenough time, we will then hope to see niche formation, where species parti-tion spatially around their respective resources. Then, we shall consider whathappens when the resources begin to overlap, with the expectation that lessfit species (here modeled by sub-optimal parameters relative to other species)are crowded out by species which are more fit. These results will cast light onroutes to spatial partitioning and niche formation within species competingfor various resources.

Possible avenues of investigation:We shall be most interested in cases where resources available to each speciesare located over spatially distinct regions, which should permit partitioningof the populations and result in pattern formation. However, we can alsoconsider the case where resources overlap, resulting on regions of competi-tion. While space-varying but temporally constant resources were employedin other works, we may also consider the case where resources are depletedor “used-up” by populations, as this scarcity of resources may result in in-teresting dynamics. Mathematically, this would involve defining appropriatedifferential equations for the Ak functions, rather than simply prescribingtheir functional forms.

Pre-requisite courses (listed as essential, recommended, useful)Anything involving the analytical or numerical solution of PDEs would beuseful. No biology background is needed.

Useful pre-reading (summer vacation)Reference [1] gives the original model on 1D space domains, while in [2] weextend that model to 2D domains, including heterogeneous domains givennon-constant Ak. We can discuss additional readings on pattern formationand Turing instability.

Further references1. P. Grindrod, Models of individual aggregation or clustering in single andmulti-species communities, Journal of Mathematical Biology 26 (1988) 651-660.2. L. Kurowski, A. L. Krause, H. Mizuguchi, P. Grindrod, and R. A. Van

40

Gorder, Two-species migration and clustering in two-dimensional domains,Bulletin of Mathematical Biology, preprint (2017).

41

Suggested title of dissertation: Gierer-Meinhardt systems with advection



Description of proposal:The Gierer-Meinhardt system was introduced in [1] to model head formationof hydra, an animal of a few millimeters in length, made up of approximately100,000 cells of about 15 different types. It consists of a “head” regionlocated at one end along its length. Typical experiments with hydra involveremoving part of the “head” region and transplanting it to other parts ofthe body column. Then, a new “head” will form if the transplanted area issufficiently far from the (old) head. These observations led to the assumptionof the existence of two chemical substances-a slowly diffusing activator u anda rapidly diffusing inhibitor v. The ratio of their diffusion rates, denotedby ε, is assumed to be small. This results in a system of reaction-diffusionequations, the Gierer-Meinhardt system.

While the Gierer-Meinhardt system has been studied in a number of cases, weshall be interested in the role advection plays on the dynamics of the chemicalsystem mentioned above. Including advection, one obtains the PDE system

∂u

∂t= ε2∇2u+ (µ · ∇)u− u+

up

vq, (3)

∂v

∂t= D∇2v + (ν · ∇)v − v +

ur

vs. (4)

Here µ = (µ1, µ2) and ν = (ν1, ν2) are the direction vectors for the advectionalong the domain, ε << 1 and D are diffusion parameters, and p > 1, q > 0,r > 1, s > 0, where we require qr−(p−1)(s+1) > 0. To avoid singular cases,we require u(x, t) > 0 and v(x, t) > 0 for all x ∈ S2 and all t > 0. Physically,u and v represent the concentrations of biochemicals called activator andinhibitor, respectively.

It is this advection form of the Gierer-Meinhardt system that we shall bemost interested in studying.

Possible avenues of investigation:While advection has previously been considered in Gierer-Meinhardt sys-tems, and some very special solutions obtained, we shall be interested in amore comprehensive study of the dynamics of this system under advection.We can consider these dynamics on compact subsets of the plane or on thesphere. This shall involve numerical simulation, while analytical tools may

42

also be useful in certain reductions of the model. We may also consider moresophisticated nonlinear advection terms, time permitting.

Pre-requisite courses (listed as essential, recommended, useful)Anything involving the analytical or numerical solution of PDEs would beuseful. No biology background is needed.

Useful pre-reading (summer vacation)Steady state solutions on a ball in Rn were studied in [2]. An asymptoticstudy of the equations on Rn was considered in [3] (with the focus being onsolution in R2). Physical motivation for including advection in such mod-els (on the plane) was discussed in [4,5], where some solutions were alsodiscussed. Basic notes on pattern formation and Turing instability can befound online.

Further references1. A. Gierer and H. Meinhardt, A theory of biological pattern formation,Kybernetik (Berlin) 12 (1972) 30-39.2. W.-M. Ni, and J. Wei, On positive solutions concentrating on spheres forthe GiererMeinhardt system, Journal of Differential Equations 221 (2006)158-189.3. J. Wei and M. Winter, On the Two-Dimensional Gierer–Meinhardt Systemwith Strong Coupling, SIAM Journal on Mathematical Analysis 30 (1999)1241-1263.4. R. A. Satnoianu and M. Menzinger, A general mechanism for “inexact”phase differences in reaction-diffusion-advection systems, Physics Letters A304 (2002) 149-156.5. G. P. Bernasconi and J. Boissonade, Phyllotactic order induced by sym-metry breaking in advected Turing patterns, Physics Letters A 232 (1997)224-230.

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Suggested title of dissertation: Nonlinear waves in the Landau-Lifshitz-Gilbert equation



Description of proposal: The Landau-Lifshitz-Gilbert (LLG) equationdescribes the dynamics for the spin in ferromagnetic materials. For a review,derivation, and more comprehensive physical motivation, see [1]. A varietyof solutions exist [1,2], and often these solutions are found after convertingthe LLG equation (a nonlinear vector PDE) into a complex scalar PDE bymeans of a transformation of dependent variable. For example, the LLGequation for a 1D isotropic chain is given by the vector PDE

St = S× Sxx + λ {Sxx − (S · Sxx)S} .

In [3], this equation was shown to be equivalent to the complex scalar PDE

iqt + qxx + 2|q|2q = iλ

{qxx − 2q

∫ x

−∞(qq∗y − q∗qy)dy

}.

Another example, the LLG equation corresponding to a SO(3) invariantdeformed Heisenberg spin equation

St = S× Sxx + λSx|Sx|2

can be put into the form of a complex scalar PDE [4]

iqt + qxx +1

2|q|2q = iλ

(|q|2q

)x.

A list of other examples is given in the review paper [1].

Note that for |λ| << 1, both complex scalar PDEs are perturbations of acubic nonlinear Schrodinger (NLS) equation

iqt + qxx + γ|q|2q = 0 .

Such equations are know to have soliton solutions. A soliton is a self-reinforcing solitary wave packet which keeps its form while propagating ata constant velocity. One famous example is the soliton solution to the KdVequation. It is natural to wonder what happens to a soliton or solitary wavesolution of the cubic NLS equation when λ > 0 in any of the LLG reductions,and this will be the focus of the present project.

44

Possible avenues of investigation:Since the equations we study are nonlinear even as we take the perturba-tion parameter to zero, we shall need a way of dealing with nonlinear basestates. Therefore, we will employ fairly general soliton perturbation theory[5,6]. Base state solutions to NLS can take the form of the bright soliton(sech), dark soliton (tanh), or periodic solutions (involving Jacobi ellipticfunctions); only the first of these has so far attracted considerable attentionin literature on the LLG equation. After mastering the method, the nextstep is to apply the approach to the study of various LLG equations, in orderto better understand the solutions to each. Finally, we will consider howproperties of these various solutions can tell us something about the physicsemergent from the LLG model.

Pre-requisite courses (listed as essential, recommended, useful)Anything involving the analytical or numerical solution of PDEs would beuseful. No physics background is needed.

Useful pre-reading (summer vacation)See the review article [1] and then the relevant papers it cites, for derivationsand physical motivation. Then, after learning about the various reductionsof the LLG equation, one can go through the soliton perturbation paper [5]in order to learn the relevant analytical method (see [6] for another exampleof the method).

Further references1. M. Lakshmanan, The fascinating world of the LandauLifshitzGilbert equa-tion: an overview, Philosophical Transactions of the Royal Society of LondonA: Mathematical, Physical and Engineering Sciences 369 (2011) 1280-1300.2. S. Gutierrez and A. de Laire, Self-similar solutions of the one-dimensionalLandau-Lifshitz-Gilbert equation, Nonlinearity 28 (2015) 1307.3. M. Daniel and M. Lakshmanan, Perturbation of solitons in the classicalcontinuum isotropic Heisenberg spin system, Physica A 120 (1983) 125-152.4. K. Porsezian, M. Daniel, and M. Lakshmanan, On the integrability aspectsof the one-dimensional classical continuum isotropic biquadratic Heisenbergspin chain, Journal of Mathematical Physics 33 (1992) 1807-1816.5. R. L. Herman, A direct approach to studying soliton perturbations, Jour-nal of Physics A: Mathematical and General 23 (1990) 2327.6. R. A. Van Gorder, First-order soliton perturbation theory for a general-ized KdV model with stochastic forcing and damping, Journal of Physics A:Mathematical and Theoretical 44 (2010) 015201.

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Suggested title of dissertation: Dynamics and networks

Dissertation supervisor:Dr Robert Van Gorder ([email protected])

Maximum number of students: n ≥ 1

Description of proposal:A network is a set of objects (called nodes) that are connected together.The connections between the nodes are called edges or links. One commonexample you’ve likely considered would be a lattice in one or more dimensions.However, many other network structures arise in applications.

A number of interesting problems in applied mathematics can be phrasedin terms of dynamical systems on networks. Here, to each node there is as-signed a differential equation (or system of equations), and the edges betweennodes dictate which systems are connected to each other. For instance, thedifferential equation could model information packets or a flu virus, whichis then spread to other individuals which have some connection to that in-dividual. For such applications, one interesting feature is that the structureof the network (namely, the way in which the nodes of the network are con-nected to other nodes) can impact the collective dynamics of the system as awhole. For instance, a flu may spread more rapidly through a well-connectedpopulation than a more sparsely connected one.

It is also possible to consider dynamic networks, in which the actual networkstructure changes over time. Here, the edges can be taken to depend on time,so that two nodes may be connected for some time interval, and disconnectedfor some other time interval. This will generally modify the dynamics of adynamical system defined on a network. One example would be transitionbetween locations when one thoroughfare is under construction or blockedby an accident and inaccessible. Another example is if mobile phone towersmalfunction or are shut off.

For a dissertation in this area, one would consider a particular problem whichinvolves dynamical systems defined on the nodes of some network (or familyof networks) such that the network structure itself plays a role in the dynam-ics, or a project also involving dynamic networks. I anticipate a number ofprojects being available, and individual applications can be tailored to theinterests of the student. Recent student projects have focused on applica-tions in physics, mechanics, biology, ecology, game theory. Some projectscould also be mode more theoretical, rather than application driven, if thatis of interest. As there are many routes such projects can take, and manyapplications to be considered, I would be open to multiple projects in thisarea.

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Disclaimer: Note that the focus here will be on dynamical systems defined onnetworks or dynamic networks, rather than on static network science or datascience along the lines of what might be covered in the Networks course. Whilethere are a few lectures on dynamics and networks in the Networks course,these will be rather low-level, compared to the research-level topics you wouldtake if doing this dissertation. As such, topics will also be disjoint from whatwould be appropriate for the Networks course mini-projects. Furthermore,you need not take the Networks course in order to complete a project in thisarea.

We shall now list some specific examples of topics which may be undertaken.We give examples from pattern formation, neuroscience, physics, and PDEtheory. These are not exhaustive.

(i) Control of Turing Instabilities in Reaction-Diffusion Systems onNetworks: The theory of pattern formation for reaction-diffusion systemshas a rich history going back to Turing [1]. Recently, a method of controllingTuring instabilities within reaction-diffusion systems, with the goal of drivingsolutions of such systems toward particular patterns, was considered [2]. Theauthors used tools from dynamical systems and the spectral theory of graphsto pose an approximate control problem to drive the state of a system to aspecific pattern that is admissible. Biologically, such results suggest a wayof understanding guided self-organization and pattern formation. Recentexperimental models have been proposed in order to understand the controlof pattern formation emergent from chemical and biological systems withcomplex spatiotemporal dynamics [3]. An interesting example of a temporalcontrol in bacteria was shown in [4].

In this project we would consider reaction-diffusion systems on networks,with the aim of developing a control protocol similar to that presented in [2].The generalization to this case is not obvious, but it appears that an exactcontrol strategy might be feasible by exploiting the finite-dimensionality ofreaction-diffusion systems on networks. We believe that a more thoroughcharacterization of the space of admissible patterns might also be feasible forparticular network topologies. A specific example of a control strategy forTuring pattern formation on networks can be found in [5], although this isquite different from the control strategy of [2]. Still, this example demon-strates that such problems on networks are tractable.

47

(ii) Dynamics of the Kuramoto and Haken-Kelso-Bunz models onnetworks: Two similar models of phase coupling dynamics have been ex-plored in neuroscience: the Kuramoto model

θn = ωn +K

N

N∑m=1

sin(θm − θn) (5)

and the Haken-Kelso-Bunz model

φ = ω − a sinφ− 2b sin 2φ . (6)

Generally, the Haken-Kelso-Bunz equation models the system of two oscil-lators, quantified by the phase difference, whereas the Kuramoto equationmodels n oscillators, quantified by the mean phase. In particular, the HKBmodel captures a subcritical pitchfork bifurcation based on empirical obser-vations [6].

Of great interest are special cases that occur between strong and weak cou-pling: for fixed parameters, the order parameters can show both coupledand decoupled tendencies. In the Kuramoto and HKB models respectively,these states are referred to as “chimeras” [7] and “metastability” [8]. Theseregimes are of particular interest to neuroscience, because they are believedto enable flexible changes in the nervous system [9].

This project will involve a systematic comparison of the Kuramoto and HKBmodels, a generalization for n oscillators that link both models, and an exam-ination of different network topologies for their coupling patterns. Recently,networks of HKB oscillators have been investigated [10]. In particular, em-pirical studies have tested human participants with different connectivities[11, 12, 13], and one can visualize these various connectivities mathematicallyin terms of networks. We shall aim to provide a more detailed and rigorousof the dynamics possible under various network configurations.

(iii) Granular Material on a Lattice or Other Network StructuresA granular material is a conglomeration of discrete solid, macroscopic parti-cles characterized by a loss of energy whenever the particles interact. Propa-gation of macroscopic wave packets (due to a disturbance or force applied in-ternally or at a boundary) within granular media or granular crystals (tightlypacked granular media with sufficient structure) finds applications in ar-eas such as medical physics, Bose-Einstein condensates, nonlinear optics,atomic physics, and elastic collisions, with specific applications in molecularchains, second-harmonic generation, arrays of repelling magnets, nonlinear

48

resonances, shock mitigation, energy localization, stress wave control, non-linear acoustic lenses (see [14] and references therein).

Many models of wave propagation through an array of beads on 1D finite orinfinite lattices take the form [15]

d2undt2

= V ′(un − un−1)− V ′(un+1 − un) +W (un) .

Here un is the displacement of then nth bead from it’s initial position, Vand W are functions which govern the particular physics of the problem.Regarding the domain, we have either n ∈ {0, 1, . . . , N} or n ∈ Z. In theformer case, boundary conditions would be needed, while in the latter case,solution would be expected to satisfy relevant decay criteria as n → ±∞.To study 2D granular crystals, one would consider similar systems on twodimensional finite or infinite lattices. It is not necessary for such domainsto be rectangular lattices; other lattices are both possible and useful. Itwould also be possible to consider other network structures than lattices.For instance, Y-shaped configurations have appeared in some experiments.Of course, any configurations should be selected in such a way that theymight be realized in real-world experiments.

There have been a variety of solutions to systems of this type, with solitarywaves [16, 17], dark breathers [18], plane solitary waves [19], periodic travel-ing waves and compactons [20] among solutions being found. There have beenstudies on interactions with linear media (such as a wall placed at a bound-ary) [21], and the scattering of waves at surfaces or interfaces has also beenstudied [22]. While many results assume homogeneous media, there havebeen solitary waves observed in heterogeneous yet ordered (periodic) media[23, 24]. Furthermore, although one-dimensional or quasi-one-dimensionalchains are most common, some work has been done on fundamentally 2Dproblems (see, for example, [25]).

One interesting area to consider would be the propagation of waves in mediaconsisting of two different materials, one permitting fast propagation, theother permitting slow propagation. With an external potential, we may evenbe able to slow or even stop a wave, before releasing it. In such cases, it isnatural to ask if we can we preserve the wave structure. Or, will part of thewave always be reflected at a boundary between different media? If we thentransition back to the ‘fast’ media, are we able to recover the previous ‘fast’wave speed?

Consider various 2D lattice domains or other more exotic network configu-rations in 2D or 3D. It is natural to ask how the shape of the domain will

49

influence the propagation and reflection of waves. Once this is better under-stood, then one can consider how such waves would propagate in the presenceof heterogeneous media.

(iv) When do Continuum and Discrete Models Agree? In areas suchas fluids and physics, you have a vague idea that there are a very large numberof particles which make up some bit of matter, and if the number of particlesis “large enough” then one may represent the matter as a continuum ratherthan trying to keep track of the discrete particles. However, if the numberof particles is too small, then a continuum assumption can fail. In thisproject, we shall consider continuum versus discrete approaches to modelingphenomenon in intermediate regimes.

Importantly, such intermediate regimes emerge when one is attempting tonumerically solve a PDE by, say, a finite difference method. In a way, thisis the reverse of the physics problem described above, as we are trying toapproximate a continuum model with a discrete or network model. Supposeone wishes to solve a PDE involving space and time on the plane. One mayinstead convert the PDE into a system of ODEs on a 2D lattice approximatingthe plane, and solve the resulting system. Suppose one wishes to solve asimilar PDE on a 1D interval with spatially periodic conditions? Then,perhaps one will represent this a system of ODEs on a cycle graph, whichcan then be solved. When will these approximations be valid?

What if one should wish to consider the other direction? Suppose one has anatomized problem, with interacting particles represented by a network; hereinteractions between particles are represented by edges. If particles interactwith neighbors in a standard way (i.e. as a lattice), then perhaps a standardPDE can be obtained in a continuum limit. However, what if you had somesort of asymmetric network, rather than a lattice? How would this modifythe continuum PDE you derive? What if the underlying network is that of acomplete graph, for which every particle interacts with every other particle?Would a non-local model (involving integrals) be better than a local model?

An application such as this would explore the more mathematical propertiesof going between network and continuum spatial models.

Pre-requisite courses (listed as essential, recommended, useful)Anything involving the analytical or numerical solution of ODE systems (orPDE if a project such as the latter suggestion is undertaken) would be useful,as you will be studying the dynamics of systems of differential equations un-

50

der various network structures. Prior knowledge of networks or graph theoryis optional, and can be learned along the way. No background on a spe-cific application area is needed. The course in Networks is optional and notneeded for anything you would do here.

Useful pre-reading (summer vacation)The review article [26] is a good place to start. For examples of papers origi-nating from student projects related to dynamics on networks, see below forapplications in ecology [27], game theory [28], and physics [29].

References

[1] A. M. Turing, The chemical basis of morphogenesis, Philosophical Trans-actions of the Royal Society of London B: Biological Sciences 237 (1952)37-72.

[2] K. Kashima, T. Ogawa, and T. Sakurai, Selective pattern formationcontrol: Spatial spectrum consensus and Turing instability approach,Automatica 56 (2015) 25-35.

[3] A. S. Mikhailov and K. Showalter, Control of waves, patterns and tur-bulence in chemical systems, Physics Reports 425 (2006) 79-194.

[4] S. Payne, B. Li, Y. Cao, D. Schaeffer, M. D. Ryser, and L. You, Temporalcontrol of self-organized pattern formation without morphogen gradientsin bacteria, Molecular Systems Biology 9 (2013) 697.

[5] S. Hata, H. Nakao, and A. S. Mikhailov, Global feedback control ofTuring patterns in network-organized activator-inhibitor systems, EPL(Europhysics Letters) 98 (2012) 64004.

[6] H. Haken, J. A. S. Kelso, and H. Bunz, A theoretical model of phasetransitions in human hand movements, Biological Cybernetics 51(5)(1985) 347–356.

[7] D. M. Abrams and S. H. Strogatz, Chimera states for coupled oscillators,Physical Review Letters. 93(17) (2004) 174102.

[8] J. A. S. Kelso, Multistability and metastability: understanding dynamiccoordination in the brain, Philosophical Transactions of the Royal Soci-ety of London B: Biological Sciences 367(1591) (2012) 906–918.

51

[9] E. Tognoli and J. A. S. Kelso, The metastable brain, Neuron 81(1)(2014) 35–48.

[10] F. Alderisio, B. G. Bardy, and M. Bernardo, Entertainment and synchro-nization in networks of rayleigh–van der pol oscillators with diffusive andhaken–kelso–bunz couplings, Biological Cybernetics, pages 1–19, 2016.

[11] F. Alderisio, G. Fiore, R. N. Salesse, B. G. Bardy, and M. di Bernardo,Interaction patterns and individual dynamics shape the way we move insynchrony. arXiv preprint arXiv:1607.02175, 2016.

[12] F. Alderisio, M. Lombardi, G. Fiore, and M. di Bernardo, Study ofmovement coordination in human ensembles via a novel computer-basedset-up, arXiv preprint arXiv:1608.04652, 2016.

[13] M. Richardson, R. Garcia, T. Frank, M. Gregor, and K. Marsh, Measur-ing group synchrony: a cluster-phase method for analyzing multivariatemovement time-series, Frontiers in Physiology 3 (2012) 405.

[14] M. Moleron, A. Leonard, and C. Daraio, Solitary waves in a chain ofrepelling magnets, Journal of Applied Physics 115 (2014) 184901.

[15] S. Sen, J. Hong, Jonghun Bang, E. Avalos, and R. Doney, Solitary wavesin the granular chain, Physics Reports 462 (2008) 21-66.

[16] C. Coste, E. Falcon, and S. Fauve, Solitary waves in a chain of beadsunder Hertz contact, Physical Review E 56 (1997) 6104.

[17] R. S. MacKay, Solitary waves in a chain of beads under Hertz contact,Physics Letters A 251 (1999) 191-192.

[18] C. Chong, P. G. Kevrekidis, G. Theocharis, and C. Daraio,Darkbreathers in granular crystals, Physical Review E 87 (2013) 042202.

[19] M. Manjunath, A. P. Awasthi, and P. H. Geubelle, Family of planesolitary waves in dimer granular crystals, Physical Review E 90 (2014)032209.

[20] G. James, Periodic travelling waves and compactons in granular chains,Journal of Nonlinear Science 22 (2012) 813-848.

[21] J. Yang, C. Silvestro, D. Khatri, L. De Nardo, and C. Daraio, Interactionof highly nonlinear solitary waves with linear elastic media, PhysicalReview E 83 (2011) 046606.

[22] L. Vergara, Scattering of solitary waves from interfaces in granular me-dia, Physical Review Letters 95 (2005) 108002.

52

[23] M. A. Porter, C. Daraio, I. Szelengowicz, E. B. Herbold, and P. G.Kevrekidis, Highly nonlinear solitary waves in heterogeneous periodicgranular media, Physica D 238 (2009) 666-676.

[24] K. R. Jayaprakash, Y. Starosvetsky, and A. F. Vakakis, New family ofsolitary waves in granular dimer chains with no precompression, PhysicalReview E 83 (2011) 036606.

[25] A. Leonard, C. Chong, P. G. Kevrekidis, and C. Daraio, Traveling wavesin 2D hexagonal granular crystal lattices, Granular Matter 16 (2014)531-542.

[26] M. A. Porter and J. P. Gleeson, Dynamical systems on networks: Atutorial. arXiv preprint arXiv:1403.7663 (2014).

[27] L. Shen and R. A. Van Gorder, Predator-Prey-Subsidy Population Dy-namics on Stepping-Stone Domains, Journal of Theoretical Biology 420(2017) 241-258.

[28] X. Meng, R. A. Van Gorder and M. A. Porter, Opinion formation anddistribution in a bounded confidence model on various networks, arXivpreprint arXiv:1701.02070

[29] A. Kekic and R. A. Van Gorder, Wave propagation across interfacesbetween two Hertz-like granular crystals with different interaction ex-ponents, arXiv preprint arXiv:1612.07617

53

Suggested title of dissertation: Size-structured plant populations



Description of proposal:The principal feature that distinguishes plant competition from animal com-petition is that most plants require exactly the same resources. Plant speciestherefore cannot avoid competition with others by partitioning resources asmany animal competitors do. Yet most terrestrial communities support mul-tiple coexisting plant species. Such observations raise the compelling questionof how such plant species manage to avoid competitive exclusion. Althoughseveral mechanisms have been proposed, most involve environmental varia-tion through horizontal space (or time) so that each species enjoys competi-tive superiority somewhere (or sometime).

Previously, population growth for a single species was assumed to obey adynamical system of the form [1]

du

dt= u

(∫ ∞0

ϕ(S(z)u)s(z) dz − C). (7)

Here u(t) denotes the leaf area index (i.e., leaf area overlying one unit ofground area) of a plant population at time t ≥ 0. The probability densityfunction, s(z), denotes the population’s fixed vertical leaf profile and S(z) =∫∞zs(ξ) dξ is the proportion of its leaves which lie above height z ≥ 0. Then

the leaf area at height z is s(z)u dz and the overlying leaf area is S(z)u. As aconsequence of self-shading, the gain function ϕ(x) (the rate at which leavesperform photosynthesis) was assumed to be a continuous and decreasingfunction of overlying leaf area x ≥ 0 with ϕ(x) → 0 as x → ∞. Thepositive constant C represents the tissue maintenance rate associated withone unit of leaf area. It was shown that a population can persist if and only ifcanopy photosynthesis in an infinitesimal population exceeds its maintenancerequirement: ϕ(0) > C. When the population does persist, it approaches aglobally asymptotically stable equilibrium leaf area u∗ determined by theimplicit equation

∫ 1

0ϕ(αu∗) dα = C.

In the case of two or more species, (7) can be generalized to account for thecompetition between species, and one obtains a Kolmogorov-type competi-tion model for the two species [2,3].

The prototype model (7) for one species assumes that the population has avertical leaf profile s(z) which does not change in time, and hence plants do

54

not grow in height. In order to relax this assumption to allow height growth,in this project we will consider a size-structured version in which a populationis made up of many stems of different heights. For simplicity, we assume thatall stems of height h ≥ 0 have the same vertical leaf profile sh(z). It followsthat all stems of height h have the same proportion Sh(z) =

∫∞zsh(ξ) dξ of

their leaves which lie above height z ≥ 0. Let uh(t) be the leaf area indexfor stems of height h and let v(h, t) be the population density of stems ofheight h. Then the leaf area at height z which belongs to stems of height his sh(z)uh dz and the combined overlying leaf area (belonging to all stems) isT (u, z) =

∫∞zSξ(z)uξ dξ. We propose a model for the dynamics of leaf area

and stem height to be given by a McKendrick-von Foerster equation coupledto the dynamics (7):

∂v

∂t+

{A+

uhuh

}∂v

∂h+µv = 0 and

duhdt

= uh

{∫ h

0

ϕh (T (u, z)) sh(z) dz−Ch}.

(8)The partial differential equation for v is a McKendrick-von Foerster equa-tion, while the equation for u is a Kolomogrov-type equation which has beengeneralized from (7) to account for the height dependence. The equations arecoupled through the expression A + uh

uh(with A constant), which represents

the rate at which stems increase in height and is positively related to changesin leaf area [4,5].

Possible avenues of investigation:The system (8) involves a PDE of v coupled to a kind of integral equation inu. It may be possible to obtain asymptotic results in some idealized cases,however we should expect most results to be numerical in nature when wetry to solve a sufficiently general form of the model.

Pre-requisite courses (listed as essential, recommended, useful)Anything involving the analytical or numerical solution of ODEs/PDEs wouldbe useful. No biology background is needed.

Useful pre-reading (summer vacation)References [1-3] will provide a good overview and motivation for the problem.Additional reading material will be provided.

Further references1. R. R. Vance and A. L. Nevai, Plant population growth and competitionin a light gradient: a mathematical model of canopy partitioning, J. Theor.Biol. 245 (2007) 210–219.2. A. L. Nevai and R. R. Vance, Plant interspecies competition for sunlight:a mathematical model of canopy partitioning, J. Math. Biol. 55 (2007) 105–

55

145.3. W. Just and A. L. Nevai, A Kolmogorov-type competition model withmultiple coexistence states and its applications to plant competition for sun-light, J. Math. Anal. Appl. 348 (2008) 620–636.4. J. M. Cushing, An Introduction to Structured Population Dynamics. Con-ference Series in Applied Mathematics. Vol. 71. SIAM, Philadelphia, 1998.5. A. Calsina and J. Saldana, A model of physiologically structured popu-lation dynamics with a nonlinear individual growth rate, J. Math. Biol. 33(1995) 335–364.

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Suggested title of dissertation: Equitable graph partitions and the de-formed graph Laplacian

Dissertation supervisor: Dr Neave O’Clery ([email protected])


Description of proposal:Previous work (see [1] and [5] below) has unearthed the intrinsic relationshipbetween the external equitable partition a graph partition that groups nodeswith constant out-degree within each cell and is closely related to symmetriesin the graph- and the spectral properties of the graph Laplacian (a matrixclosely related to the graph adjacency matrix). In particular, if a non-trivialexternal equitable partition exists, their exists a smaller weighted quotientgraph whose Laplacian exhibits identical spectral properties. These proper-ties result in dynamical invariance for simple dynamical systems on the fullgraph given an appropriate initial condition (e.g. consensus dynamics ubiqui-tous in the engineering control literature). This project will seek to examinethe recently proposed deformed graph Laplacian a new form of Laplacianrelated to non-back-tracking random walks. The idea is to understand thedifferent features of the two Laplacians in terms of their spectral properties,and look for an analogous graph partition (similar to the EEP) that resultsin dynamical invariance for dynamics governed by the deformed Laplacian.

Possible avenues of investigation:The existence of an external equitable partition is associated with invariancein the spectral properties of the Laplacian. E.g., there exists, in most cases,repeated entries in the associated eigenvectors. This project could take asimilar approach, and look for conditions (in terms of the graph structure)under which invariance would emerge in the spectral properties of the de-formed Laplacian.

Pre-requisite courses (listed as essential, recommended, useful)Recommended: ASO: Graph Theory and/or B8.5 Graph Theory and/or C8.3Combinatorics and/or C5.4 Networks


[1] O’Clery, Neave, et al. “Observability and coarse graining of consensusdynamics through the external equitable partition.” Physical Review E 88.4(2013): 042805.

[2] Grindrod, Peter, Desmond J. Higham, and Vanni Noferini. “The deformedgraph Laplacian and its applications to network centrality analysis.” SIAM

57

Journal on Matrix Analysis and Applications (2017).

[3] Saade, Alaa, Florent Krzakala, and Lenka Zdeborova. “Spectral clus-tering of graphs with the bethe hessian.” Advances in Neural InformationProcessing Systems. 2014.

Further references

[4] Godsil, Chris, and Gordon F. Royle. Algebraic graph theory. Vol. 207.Springer Science & Business Media, 2013.

[5] Schaub, Michael T., et al. “Graph partitions and cluster synchronizationin networks of oscillators.” Chaos: An Interdisciplinary Journal of NonlinearScience 26.9 (2016): 094821.

58

7 Mathematical Physics

Suggested title of dissertation: Scattering theory

Dissertation supervisor: Prof. Lionel Mason


Description of proposal:This project concerns the scattering of quantum mechanical particles off aprescribed potential. A potential is is assumed to be given that falls off atlarge distances. Solutions to the Schrodinger equation with this potentialare constructed that correspond to particles coming in from infinity, inter-acting with the potential and then radiating back out to infinity. This can beanalyzed via integral equations that give rise, either to a perturbative frame-work for computing the scattering or to a means of proving existence anduniqueness of solutions. The scattering is encoded in certain reflection andtransmission coefficients that describe the probabilities of scattering at dif-ferent angles. These can be approximated and computed for different choicesof potential.


1. The traditional theoretical physics formulation of this theory can befound in many quantum mechanics textbooks. It follows by setting upGreens functions and performing perturbative expansions and expan-sions in spherical harmonics and partial waves.

2. A quite different direction is to study the inverse scattering problem,that is, the reconstruction of the scattering potential from the scat-tering data. In the one-dimensional case, this has rich and deep con-nections with integrable systems, nonlinear differential equations forwhich the inverse scattering problem provides a nonlinear analogue ofa Fourier transform. These often have particle-like solutions called soli-tons.

3. Someone with an analysis of PDE background might like to follow theLax-Philips approach which is based on the wave equation rather thanthe Schrodinger equation. A geometric analyst might like to followMelrose’s Geometric Scattering theory (but this is quite advanced).

Pre-requisite courses (listed as essential, recommended, useful)Essential: Part A quantum mechanics.Recommended: Part B Further quantum mechanics.

59

Useful pre-reading (summer vacation)Standard theoretical physics approaches can be found in College libraries andcan be taken from any one of:Schiff, Quantum mechanics, Chapter 5 and 9 in 3rd edition.Mandl, Quantum mechanics, Chapter 8.

For inverse scattering and the relationship with integrable systems, seeAblowitz and Clarkson, Solitons, Nonlinear Evolution Equations and InverseScattering, LMS lecture note series 149, CUP, 1991.Drazin & Johnson, Solitons, an introduction, CUP 1989.

For the more analytic and geometrical approaches, the basic reference is:Lax and Philips, Scattering theory, Academic press 1989.

Further referencesMore modern but more physics based treatments can be found in

Binney and Skinner, the Physics of Quantum mechanics, Oxford, 2014, hap-ter 13.

Weinberg, Lectures on quantum mechanics, Cambridge 2013, chapters 7 and8.

An interesting reference for the geometric analysis approach is:

Melrose, Geometric scattering theory, Cambridge 1995.

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Suggested title of dissertation: Gross-Pitaevskii equation on a Rieman-nian manifold

Dissertation supervisor:Dr Robert Van Gorder ([email protected])


Description of proposal: After standard scalings, the non-dimensionalform of the Gross-Pitaevskii (GP) equation reads

iΨt +4Ψ =(εV (x) + κ|Ψ|2

)Ψ , (9)

where x ∈ Rn, Ψ = Ψ(x, t) defined as a map Ψ : Rn × R → C giving acomplex wave function, ε ∈ R is a constant, V (x) defined as V : Rn → C acomplex potential function (although most often it is taken to be real, viz.,V : Rn → R), and i2 = −1. By 4 we denote the Laplacian operator onRn, while | · | denotes the complex modulus. When V ≡ 0 we have the cubicnonlinear Schrodinger (NLS) equation, for which case the constant κ is eitherκ = +1 (the defocusing case) or κ = −1 (the focusing case), or κ = 0 (fromwhich ones recovers the linear Schrodinger equation). When V 6= 0, it hasthe interpretation as a confining potential, and this is useful in the study of,for instance, Bose-Einstein condensation.

We shall be concerned with the generalization of the GP equation to the casewhere the underlying spatial structure is that of a more general manifold thanRn. To this effect, let us consider an n-dimensional manifold M. By 4Mwe denote the Laplace-Beltrami operator on M. Consider a wave functionΨ : M × R → C such that Ψ ∈ C2(M) in space and Ψ ∈ C1(R) in time.Then, the non-dimensional form of the GP equation defined over the spaceM is given by

iΨt +4MΨ =(εV (x) + κ|Ψ|2

)Ψ . (10)

Here, the potential function V is defined over the manifold, as appropriatefor the application at hand. It is assumed to be time-independent for ourinterests, although time-dependent potentials could also be considered. If weassume thatM is a differentiable manifold with Riemannian metric g onM,then we refer to (M, g) as a Riemannian manifold.

Possible avenues of investigation: Equation (10) and related equationshave been studied when V ≡ 0; for some theoretical results see [1,2,3,4].However, the potential V is useful and sometimes essential when attemptingto study certain phenomenon, such as Bose-Einstein condensates (BECs)[5,6]. Therefore, we shall be interesting in studying (10) under non-zero V ,for various choices of (M, g). There are a number of directions one can take,

61

and the precise focus of the project will be agreed on after consultation withthe student. Stability of BECs is a physically relevant topic [7,8,9], andone might consider how the curvature of the underlying space will modifystability properties of BECs.

Pre-requisite courses (listed as essential, recommended, useful)Experience with PDEs and nonlinear ODEs: analytic methods and basicfamiliarity with Matlab or the equivalent. Knowledge of differential geometryor topology, or an interest in learning a few results in these areas, could proveuseful. Additional mathematical tools or physical knowledge of the area arenot essential at the outset, and can be picked up over the course of theproject.

Useful pre-reading (summer vacation)I can provide useful notes and readings once the particular direction of theproject is agreed upon.

Further references1. N. Burq, P. Gerard, and N. Tzvetkov, American Journal of Mathematics126(3) (2004) 569.2. Z. Hani, Communications in Partial Differential Equations 37(7) (2012)1186.3. A. D. Ionescu and B. Pausader, Communications in Mathematical Physics312(3) (2012) 781.4. Z. Hani and B. Pausader, Communications on Pure and Applied Mathe-matics 67(9) (2014) 1466.5. K. Mallory and R. A. Van Gorder, Physical Review E 92 (2015) 013201.6. G. Karali and C. Sourdis, Archive for Rational Mechanics and Analysis217 (2015) 439.7. J. Zhang, Journal of Statistical Physics 101(3-4) (2000) 731.8. E. A. Donley et al., Nature 412(6844) (2001) 295.9. J. C. Bronski et al., Physical Review E 63 (2001) 036612.

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8 Stochastics, Discrete Mathematics and In-

formation

Suggested title of dissertation: Equitable graph partitions and the de-formed graph Laplacian

Dissertation supervisor: Dr Neave O’Clery ([email protected])


Description of proposal:Previous work (see [1] and [5] below) has unearthed the intrinsic relationshipbetween the external equitable partition a graph partition that groups nodeswith constant out-degree within each cell and is closely related to symmetriesin the graph- and the spectral properties of the graph Laplacian (a matrixclosely related to the graph adjacency matrix). In particular, if a non-trivialexternal equitable partition exists, their exists a smaller weighted quotientgraph whose Laplacian exhibits identical spectral properties. These proper-ties result in dynamical invariance for simple dynamical systems on the fullgraph given an appropriate initial condition (e.g. consensus dynamics ubiqui-tous in the engineering control literature). This project will seek to examinethe recently proposed deformed graph Laplacian a new form of Laplacianrelated to non-back-tracking random walks. The idea is to understand thedifferent features of the two Laplacians in terms of their spectral properties,and look for an analogous graph partition (similar to the EEP) that resultsin dynamical invariance for dynamics governed by the deformed Laplacian.

Possible avenues of investigation:The existence of an external equitable partition is associated with invariancein the spectral properties of the Laplacian. E.g., there exists, in most cases,repeated entries in the associated eigenvectors. This project could take asimilar approach, and look for conditions (in terms of the graph structure)under which invariance would emerge in the spectral properties of the de-formed Laplacian.

Pre-requisite courses (listed as essential, recommended, useful)Recommended: ASO: Graph Theory and/or B8.5 Graph Theory and/or C8.3Combinatorics and/or C5.4 Networks


[1] O’Clery, Neave, et al. “Observability and coarse graining of consensus

63

dynamics through the external equitable partition.” Physical Review E 88.4(2013): 042805.

[2] Grindrod, Peter, Desmond J. Higham, and Vanni Noferini. “The deformedgraph Laplacian and its applications to network centrality analysis.” SIAMJournal on Matrix Analysis and Applications (2017).

[3] Saade, Alaa, Florent Krzakala, and Lenka Zdeborova. “Spectral clus-tering of graphs with the bethe hessian.” Advances in Neural InformationProcessing Systems. 2014.

Further references

[4] Godsil, Chris, and Gordon F. Royle. Algebraic graph theory. Vol. 207.Springer Science & Business Media, 2013.

[5] Schaub, Michael T., et al. “Graph partitions and cluster synchronizationin networks of oscillators.” Chaos: An Interdisciplinary Journal of NonlinearScience 26.9 (2016): 094821.

64

Suggested title of dissertation: Stochastic partial differential equationsin credit risk

Dissertation supervisor: Prof. Ben Hambly ([email protected])


Description of proposal:Structural models for credit risk assume that the value of a company evolvesas a diffusion process and default occurs when this process reaches a lowerboundary. When considering credit products based on large markets it isnatural to take large portfolio limits of these models and, in the case wherethere are global factors influencing the market, it is natural for stochasticpartial differential equations (SPDEs) to appear to describe the evolution ofthe limit empirical measure of the firms. The loss process from the portfolio,the process which is need for the pricing of credit derivatives, is a function ofthe solution to the SPDE. This project would look at large portfolio modelsfor credit risk and then consider some variations of the current research.

Possible avenues of investigation:1. What happens when we combine this structural approach with the reducedform modelling approach in which credit events happen ‘out of the blue’.In this case there is a Cox process indicating a credit event which occursat random according to some random hazard rate process rather than forstructural reasons. Combining these two ideas will lead to a modification ofthe SPDE and introduce new parameters which could improve the model fit.

2. In the initial models all firms have the same parameters and it is justtheir idiosyncratic noise and starting position which lead to the differencesin the evolution of prices. If we allow different parameters for the differentdiffusions we should get a model where the limit empirical measure evolves asa mixture of solutions of SPDEs. Is it possible to find good approximationsto this mixture and determine the effect upon the prices of credit derivatives?

Pre-requisite courses (listed as essential, recommended, useful)‘Martingales through Measure Theory’ and ‘Continuous Martingales andStochastic Calculus’ are highly recommended.

Useful pre-reading (summer vacation)It would be useful to do some background reading on credit risk. For examplethe book by Schonbucher. For the more model specific background there arepapers on my website; for example1. N. Bush, B.M. Hambly, H. Haworth, L. Jin and C. Reisinger, Stochasticevolution equations in portfolio credit modelling. SIAM J. Fin. Math. 2,

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627–664, 2011.2. B.M. Hambly and S. Ledger, A stochastic McKean-Vlasov equation forabsorbing diffusions on the half-line.3. F. Ahmad, B.M. Hambly and S. Ledger, A stochastic partial differentialequation model for mortgage backed securities.

Further references

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Suggested title of dissertation: The Erdos-Hajnal Conjecture

Dissertation supervisor: Alex Scott ([email protected])

Maximum number of students: Not specified

Description of proposal:Ramsey’s Theorem implies that every graph on n vertices contains a completesubgraph or an independent set of size at least c log n. In general, we can’texpect more than this: random graphs show that there are graphs with nocomplete subgraph or independent set of size more than O(log n). But whatif we consider graphs that do not contain some fixed graph as an inducedsubgraph? The Erdos-Hajnal Conjecture says that in this case there is amuch larger independent set. More precisely, the conjecture asserts that forevery fixed graph H there is a constant εH > 0 such that every graph on nvertices without an induced copy of H contains either a complete subgraphor an independent set of size at least cnεH .

The conjecture is still open, even for some very small graphs such as C5. Theaim of the dissertation is to investigate what partial results are known, andlook at current approaches to the problem.

Possible avenues of investigation:Topics to look at could include: graphs H for which the conjecture is knownto hold, and the substitution property (see [1] and possibly [3]); equivalentformulations in terms of tournaments (see [1]); forbidding a graph and itscomplement (see [2]); bipartite subgraphs of H-free graphs (see [5]).

Pre-requisite courses:Part B Graph Theory is an essential prerequisite.

Useful pre-reading (summer vacation):Good starting points are the papers of Erdos and Hajnal [4] and Alon, Pachand Solymosi [1].

References

[1] Noga Alon, Janos Pach, and Jozsef Solymosi, Ramsey-type theorems withforbidden subgraphs, Combinatorica 21 (2001), 155–170.

[2] Nicola Bousquet, Aurelie Lagoutte and Stephan Thomasse,The Erdos-Hajnal conjecture for paths and antipaths, seehttps://arxiv.org/abs/1303.5205

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[3] Maria Chudnovsky and Shmuel Safra, The Erdos-Hajnal conjecture forbull-free graphs, Journal of Combinatorial Theory, Series B, 98 (2008),1301–1310.

[4] Paul Erdos and Andras Hajnal, Ramsey-type theorems, Discrete AppliedMathematics 25 (1989), 37–52.

[5] Jacob Fox and Benny Sudakov, Density theorems for bipartite graphsand related Ramsey-type results, Combinatorica 29 (2009), 153–196.

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Suggested title of dissertation: Diffusion maps for machine learningproblems

Dissertation supervisor: Harald Oberhauser ([email protected])


Description of proposal:A common situation in machine learning/statistics is that data lies in a high-dimensional, linear space but concentrates in this space on a low dimensionalmanifold (nonlinear space). Therefore methods need to be developed thatcan learn the geometry of this smaller (nonlinear) subspace. A related goalis to find meaningful structures in the data. A very successful approach thataddresses (and often solves) both questions are so-called diffusion maps inwhich a Markov process is used to define a metric on the data. The goalof the project is to overview and benchmark state of the art constructions,apply them to machine learning problems of dynamical systems, networks,etc.

Possible avenues of investigation:Efficient algorithms, convergence properties and statistical learning guaran-tees, kernelization, extensions to continuous time.

Pre-requisite courses (listed as essential, recommended, useful)B8.1, B8.2, B4.2 essentialB8.5 recommendedBasic programming (Python recommended but this can be picked up duringthe project)


1. Ronald R. Coifman, Stphane Lafon, Diffusion maps, Applied and Com-putational Harmonic Analysis, Volume 21, Issue 1, 2006, Pages 5-30,

2. Smale, Steve, and Ding-Xuan Zhou. ”Geometry on probability spaces.”Constructive Approximation 30.3 (2009): 311-323.

Further references

1. Murphy, Kevin P. Machine learning: a probabilistic perspective. MITpress, 2012.

2. Friedman, Jerome, Trevor Hastie, and Robert Tibshirani. The elementsof statistical learning. Vol. 1. Springer, Berlin: Springer series instatistics, 2001.

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Suggested title of dissertation: Large deviation principle for Gaussianmeasures or Brownian motion

Dissertation supervisor: Zhongmin Qian ([email protected])


Description of proposal:Large deviation principles (LDP) are mathematical statements about par-ticular convergence rates of a family of laws, where the laws are defined interms of distributions of stochastic processes. LDPs thus have important ap-plications in analysis, statistics and mathematical physics which deal largecomplex and random systems.. For the dissertation, the candidate will studythe general theory of large deviations, and carry out a research on the largedeviation principle for stochastic processes related to Brownian motion: fora diffusion or for a non-Markovian type Gaussian process.

Possible avenues of investigation:The candidate will learn how to formulate a large deviation principle in termsof a rate function, various continuity or contraction principles, the functionalintegral form of LDPs and etc. The candidate then carry out a careful studyof the large deviation principle for a diffusion process or a Gaussian processessuch as Gaussian variables or a Gaussian measure defined by a Gaussianprocess e.g. fractional Brownian motion.

Pre-requisite courses (listed as essential, recommended, useful):Essential: Part A Probability, Part A Integration and Part B Martingales.

Useful pre-reading (summer vacation):Part C Lecture notes on Stochastic Differential Equations

Further references:1) Jean-Dominique Deuschel and Daniel W. Stroock: Large Deviations. AMSChelsea Pub. Chapter 1 and Chapter 2.

2) M.D.Donsker and S.R.S. Varadhan: “Asymptotic evaluation of certainWiener integrals for large time,” pp. 15–33 in Functional integration andits applications (London, April 1974). Edited by A.M.Arthurs. ClarendonPress (Oxford), 1975.

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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 ([email protected])before the end of Trinity Term.

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pre-approved part c dissertations 2017-18 · algebraic geometry using category theory. this allows...

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