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Stem student stories from king's college london, how to apply for a mathematics phd at king’s: a step-by-step guide.

November 18, 2021 Lucy Emanuel Mathematics , Postgraduate 0

Starting Your Application

Your application will be submitted through King’s Apply . The first step is to create an account, though don’t be intimidated by the long list of titles, the standard ones (Ms, Mr, etc.) can be found by scrolling down slightly.

Course Finder

Once you have registered an account, you will be sent to the course finder, where you’ll need to select your programme. Typing in ‘maths’ does not work – only ‘mathematics’. Statistics must be searched for separately. You can find the applied programme by typing in your research group but be careful not to select an MSc or physics course. Several options will appear, but the correct courses are:

Applied Mathematics Research: Disordered Systems/Financial Mathematics/Probability/Theoretical Physics MPhil/PhD;Pure Mathematics Research MPhil/PhD;and Statistics Research MPhil/PhD.

Start Dates and English Language Requirements

After selecting your course, you will be asked to pick a start date. We will usually only consider applications for entry on October 1st. Other entry dates are exceptional and are only granted for deferrals or reasons relating to funding. We will not offer any internal or research council funding at these dates and may ask applicants to change this entry date or reject them outright.

The subsequent question, on English language requirements, is important for entry if you are an overseas applicant, you must prove that you meet the standard needed ( Band D ).

Residency, Nationality, Education, Employment, and References

These can be done in any order. The section on residency and nationality is necessary for visa reasons as well as determining your fee category. Based on your answers, you will be given a fee status assessment. You can ask for this to be reviewed after your application is submitted.You will also need to submit your education (usually just your BSc/MSc institution and classification) and employment history, as well as contact details for two referees.

Research Proposal

This may seem like the most daunting and important part of your application, so it is a good idea to discuss the level of detail expected here with the admissions tutor (and, for Pure Mathematics, a prospective supervisor). If you have already discussed your planned area of research with them, they may not expect you to write more than a few sentences.

We expect applicants to tick boxes 4 and 5 if they hope to be considered for funding from the department, including research council funding. Funding is allocated to applicants on merit, and admissions tutors will decide the source of funding that a student receives – this may not reflect what the applicant selected here. Funding offers will usually be sent out from March. There are also several centrally administered PhD funding schemes at King’s , shared across the university.

Attachment Summary and Submission

Make sure you have included everything that you wanted to at the attachment summary and check everything that you have written. Once you have done so, and are happy with your application, you are ready to submit, and can expect to hear from us shortly.

If you have any questions about any of this process, be sure to submit them to Gage Kumar Rull, PGR Programme Officer at [email protected].

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Spring 2024

The Seminar runs on Tuesdays at STRAND BLDG S4.29 . This term, some talks will run from 15:00 to 16:00 , and others will be in two parts – an introductory talk from 15:00 to 15:40 and a research talk from 15:45 to 16:30 .

23 April, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29

Speaker:  Dan Kaplan (University of Hasselt)

Title:  Quiver varieties for the working geometer

Abstract:  This talk is divided into two related, yet self-contained sections. The first section is an elementary introduction to (Nakajima) quiver varieties, beginning with representations of quivers and emphasizing small examples. The second section shifts gears to symplectic resolutions of singularities, including the minimal resolutions of du Val singularities and the Springer resolution of the nilpotent cone of a Lie algebra.

The sections unite as we construct symplectic resolutions for quiver varieties by varying a stability parameter. In joint work with Travis Schedler, we leverage these symplectic resolutions to build resolutions for spaces that are (analytically) locally quiver varieties. The key idea here is to choose local resolutions at the most singular points and then demonstrate that certain compatible, monodromy-free choices extend and glue to a global resolution.

30 April, 15:00-16:00, STRAND BLDG S4.29

Speaker:  Ulrike Tillmann (University of Oxford)

Title:  Homology stability for generalised Hurwitz spaces and asymptotic monopoles

Abstract:  Configuration spaces have played an important role in mathematics and its applications. In particular, the question of how their topology changes as the cardinality of the underlying configuration changes has been studied for some fifty years and has attracted renewed attention in the last decade.

While classically additional information is associated “locally” to the points of the configuration, there are interesting examples when this additional information is “non-local”. With Martin Palmer we have studied homology stability in some of these cases, including Hurwitz space and moduli spaces of asymptotic monopoles.

7 May, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29

Speaker:  Ilaria Di Dedda (King’s College London)

Title:  Type A symplectic Auslander correspondence

Abstract:  In this talk, we will study invariants of complex isolated hypersurface singularities. In the first half I will review the basics of Floer theory, and I will describe Fukaya-Seidel categories, a powerful and geometric derived invariant of singularities. In the second half, I will describe invariants of a special family of isolated singularities, whose Fukaya-Seidel categories play an important role in bordered Heegaard Floer theory. Motivated by representation theory, I will relate these singularities to abstract objects associated to algebras of type A (named after the quiver of Dynkin type A). I will introduce “type A symplectic Auslander correspondence”, a purely geometrical construction which realises a notable result in representation theory. Most of the talk will be example-based.

14 May, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29

Speaker:  Eloise Hamilton (University of Cambridge)

Title:  Geometric Invariant Theory and moduli of representations of quivers [Part 1] Non-Reductive Geometric Invariant Theory and moduli of representations of quivers with multiplicities [Part 2]

Abstract:  Geometric Invariant Theory (GIT) is a powerful theory for constructing quotients in algebraic geometry. An important classical application of GIT is to the construction of moduli spaces of representations of quivers. I will explain the basics of GIT and this particular application in the first part of the talk. In the second part of the talk I will turn to a natural generalisation of representations of quivers, called representations of quivers with multiplicities. These are representations of quivers in the category of modules over a truncated polynomial ring, instead of in the category of vector spaces. Unfortunately, moduli spaces of such objects cannot be constructed using GIT because the group involved is not reductive, yet reductivity plays a key role in GIT. I will explain recent work, joint with Victoria Hoskins and Joshua Jackson, in which we extend existing work on a non-reductive version of GIT to enable the construction of moduli spaces for representations of quivers with multiplicities.

21 May, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29

Speaker:  Anna Felikson (Durham University)

Title:  Polytopal realizations of non-crystallographic associahedra

Abstract:  An associahedron is a polytope arising from combinatorics of Catalan-type objects (for example, from a collection of all triangulations of a given polygon). Fomin and Zelevinsky found a way to construct the same combinatorial structure from considering the Coxeter group of type A_n. This allowed them to define a generalized associahedron for every finite reflection group. For generalized associahedra arising from crystallographic reflection groups, it was also shown that they can be realized as polytopes. We use the folding technique to construct polytopal realisations of generalized associahedra for all non-simply-laced root systems, including non-crystallographic ones. This is a joint work with Pavel Tumarkin and Emine Yildrim.

In the first half of the talk, I will sketch the history of the associahedron and introduce generalised associahedra, then in the second half we will discuss how to produce the associahedra in the non-crystallographic case. The talk will not require any special background.

28 May, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29

Speaker:  Nick Lindsay (University of Cologne)

11 June, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29

Speaker:  Filippo Baroni (University of Oxford)

Mathematics

Geometry and Topology

Our research interests range from low-dimensional topology and geometric group theory, through algebraic geometry to symplectic geometry, gauge theory, differential geometry and geometric analysis.

The UCL Geometry and Topology Group is part of the UCL Mathematics Department. We have eight faculty members, three postdocs and 14 PhD students. Our research interests include differential geometry and geometric analysis, symplectic geometry, gauge theory, low-dimensional topology and geometric group theory. We are involved with the London School of Geometry and Number Theory (LSGNT) , a graduate programme spanning University College London, King's College London and Imperial College London.

Costante Bellettini works in geometric analysis, with special emphasis on regularity questions arising in the calculus of variations and in calibrated geometry, often using methods from geometric measure theory and partial differential equations. He is particularly interested in the impact of such regularity results on questions arising in differential geometry.

Dario Beraldo works in geometric representation theory, especially on the geometric Langlands program. His other interests include derived algebraic geometry, higher category theory, connections to number theory and to mathematical physics.

Aleksander Doan works in differential geometry and symplectic topology. He is particularly interested in the partial differential equations of gauge theory and their relation to enumerative invariants of manifolds with special holonomy, such as Calabi-Yau manifolds. 

Lorenzo Foscolo works in differential geometry. His current research interests focus on manifolds with special and exceptional holonomy, in particular G2 and hyperkähler metrics, and gauge theory, and are often directly inspired by theoretical physics.

Jeffrey Galkowski works in microlocal analysis with an emphasis on high frequency spectral and scattering theory problems motivated by mathematical physics. Typically, these problems are posed on non-trivial manifolds where the underlying geometry plays a crucial role.

Selim Ghazouani ’s work is about deformation spaces of geometric structures on manifolds. More specifically, I have worked on moduli spaces of structures on surfaces, complex hyperbolic geometry and Lorentzian geometry. I also try to apply geometric techniques to rigidity questions in dynamical systems.

Francis Johnson : Topology of manifolds; low-dimensional topology; the D(2) problem, more generally problems involving the fundamental group; Lie groups and their discrete subgroups; homological algebra; geometric invariant theory.

Mikhail Karpukhin works in geometric analysis. He is primarily interested in harmonic maps, minimal surfaces and their connection to optimisation problems for eigenvalues of elliptic operators.

Lars Louder works in geometric group theory and low dimensional topology, primarily free groups and limit groups.

Yusra Naqvi works in geometric group theory and is especially interested in Coxeter groups, Chevalley groups and buildings. She also works in algebraic combinatorics.

Ruth Reynolds   works in noncommutative ring theory, algebraic geometry, and their interactions. Recently her interests include the behaviour of idealizers, a subring of a noncommutative ring, which often exhibit interesting and pathological behaviour. 

Ed Segal works in algebraic geometry and homological algebra, with a lot of inspiration from theoretical physics. His interests include derived categories, matrix factorizations, non-commutative resolutions, and topological field theories (mainly the B-model).

Michael Singer has various research interests in differential geometry.  He is currently working on projects in Kaehler geometry, hyperKaehler and ant-self-dual Einstein metrics in four dimensions, and moduli spaces arising in mathematical physics (in particular euclidean monopoles).

Postdoctoral Research Associates

Stephane Geudens is an IMSS fellow working in Poisson geometry and related areas. His current research deals with deformation problems and rigidity phenomena in this context.

Marcello Malagutti works in Harmonic analysis, geometrical analysis of PDEs, microlocal analysis, spectral analysis and scattering theory as well as quantum field theory and quantum mechanics.

Federico Trinca works in differential geometry and geometric analysis including minimal submanifolds, calibrated geometry and manifolds of special holonomy. 

Ruoyu Wang works in analysis and PDE on manifolds where global dyanimcs of geodesics quantitatively determine analytic properties of solutions.

For more information on the activities of the group please click here .  

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The research interests of the group focus mainly on PDEs, operator theory and spectral theory.

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Outreach at king's maths school.

King's Maths School has a vision to promote the mathematical sciences at A Level and beyond. We run programmes for students aged 13-16 supporting their interest in mathematics and physics, and activities for teachers to enable them to provide the very best of mathematical educations.

We are currently accepting applications for GCSE+ Maths, GCSE+ Physics, MESME and REVISION+. Follow the link below to find out more.

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King’s Maths School is for students aged 16-19 with a particular aptitude and enthusiasm for mathematics, and aims to widen participation in mathematical degrees and careers at the very best universities and institutions.

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