# Graded rings and toric methods in algebra

School of Mathematics and Physics | PHD
Funding
Unfunded
Reference Number
MAP/2023/002
28 February 2023
Start Date
1 October 2023

### Overview

There is a fascinating and deep connection between partitions of Rn by cone-shaped subsets into “fans”, and so-called toric varieties (geometric objects that can be described as solutions of polynomial equations of a certain type). By working with fans, toric varieties can be analysed using diagram-theoretic methods eschewing most of the complications of algebraic geometry. More importantly, the approach via diagrams of rings and modules makes it possible to study toric varieties in generalised contexts, for example, over non-commutative rings. The basic notion underlying the theory is that of a LAURENT polynomial ring in several variables L = R[x1, x−11, · · · , xn, x−1n], the collection of polynomials (with coefficients in thering R) where negative and positive powers of the variables may occur. The combinatorial data associated with a toric variety can be used to construct other rings from L, and to inter-relate these rings in a meaningful manner.

For many purposes one can often work in much greater generality: the LAURENT polynomial ring can be replaced by a graded ring, that is, a ring in which elements come equipped with a notion of “degree” similar to the usual notion of degree of a polynomial. The PhD project is about analysing this replacement step. Once the necessary machinery for graded rings has been set up many results of a formal nature carry over from the classical to the generalised setting without major problems. However, a closer look reveals surprising and subtle differences and complications, reflecting the more varied nature of graded rings as opposed to LAURENT polynomial rings.

The project may develop in various different directions, depending on the interests of the student. Possible topics include:
• an analysis of the algebraic K-theory of toric varieties in this generalised context (for example, it is expected that the K-theory of a projective generalised toric variety contains certain “trivial” summands);
• using toric methods to obtain results on tameness and finiteness conditions for chain complexes (for example, consider the “jump loci” for finite domination in the spirit of Sigma invariants from geometric group theory),
• extending results from affine toric algebra to the graded setting (for example, investigate how results connected with the “fundamental theorem for algebraic K-theory” transfer to the generalised setting).

The project is strongly algebraic in nature. The student wishing to undertake this kind of research must be willing to learn and work with the abstract mathematical machinery of graded algebra, homological algebra and algebraic K-theory.

###### Project Summary
Supervisor

Dr Thomas Huettemann

Mode of Study

Full-time: 3 years

#### Mathematics overview

The Mathematical Research Centre conducts world-class research in the following areas: Algebra, Analysis, Operator Algebras, Algebraic Topology, Topological Data Analysis, PDEs, Survival Analysis, Bayesian Networks, Data Analytics and Operational Research. It maintains vibrant international links with a large number of researchers around the globe and regularly hosts international conferences and research visitors.

Mathematics at Queen's is joint 1st for Research Intensity in the UK. Queen’s is ranked in the top 170 in the world for graduate prospects (QS Graduate Employability Rankings 2022)

List of researchers, their interests and upcoming PhD projects can be found at:
https://web.am.qub.ac.uk/wp/msrc/.

Mode of study / duration
Registration is on a full-time or part-time basis, under the direction of a supervisory team appointed by the University. You will be expected to submit your thesis at the end of three years of full-time registration for PhD, or two years for MPhil (or part-time equivalent).

##### Mathematics Highlights
• The School has many industry links, some of which are with Seagate Technology R&D, Andor Technology and AVX Ltd. Many of our graduates take up positions with these companies in posts such as Statistical Analysis Programmer, Trainee Accountant, Financial Engineer and Business Analyst.
##### Career Development
• Queen’s is ranked in the top 170 in the world for graduate prospects (QS Graduate Employability Rankings 2022). Graduates from the School take up employment through a number of companies such as Allstate, AquaQ Analytics, Citigroup, Deloitte, PwC, Randox, Seagate and UCAS.
##### World Class Facilities
• Since 2014, the School has invested over £12 million in new world-class student and staff facilities. Maths and Physics students now have their own teaching centre that opened in 2016 housing experimental physics laboratories, two large computer rooms for mathematical simulations and student study plus a student interaction area.
In addition, Belfast is one of the lowest student cost of living in the UK (Which? University, 2018).
##### Key Facts

• Students will have access to our facilities, resources and our dedicated staff. The School of Mathematics and Physics is one of the largest Schools in the University. Staff are involved in cutting-edge research that spans a multitude of fields.
After an undergraduate degree in QUB, I started my PhD journey in pure mathematics research. In the four years of study, there are two invaluable features. Firstly I received ideal supervision which helped me to solve my research problems and to touch the contemporary research work. Secondly the system is very free whereby I can study, consider and discuss the relevant maths knowledge and problems which are not directly related to my research but very helpful for my future research work.

Weijiao Hu, PhD Mathematics, 2019

#### Course content

##### Research Information

Research Themes
Overall, 90% of research submitted to the REF 2021 by the School of Mathematics and Physics was judged as internationally excellent or world-leading.

Information on the research interests and activities of academics in the Mathematical Science Research Centre can be found at https://web.am.qub.ac.uk/wp/msrc/. These interests fit into the themes: Algebra, Analysis, Data Science, Optimization and Operational Research, Partial Differential Equations, Statistics, Topology and Geometry.

Research Impact
Research Impact in Mathematics at Queen’s is gained through having wide interactions with Industry and the Public Sector which enhances and accelerates societal and economic impact. In the REF 2021 results, 75% of impact in Mathematics was graded as 4* which demonstrates that the quality of impact is world-leading in terms of originality, significance and rigour.

##### Career Prospects

Introduction
Mathematical and statistical skills are in great demand in the economy, particularly the advanced skills developed at the PhD level.

Employment after the Course
As well as continuing in research careers, our PhD graduates have also gone on to work in finance, computing, data analysis, management and teaching. Our advisors will be happy to provide further information on the career prospects arising from your chosen research area. Further information on careers can be obtained from the School and the Faculty.

##### People teaching you

Dr David Barnes
Head of Centre - Mathematical Sciences Research Centre

Email: d.barnes@qub.ac.uk

Dr Ying-Fen Lin

Email: y.lin@qub.ac.uk

##### Course structure
There is no specific course content as such. A PhD programme runs for 3-4 years full-time or 6-8 years part-time. Students can register for a writing up year should it be required.

The PhD is open to both full and part time candidates and is often a useful preparation for a career within academia or consultancy.

Application Process
Please review the eligibility criteria on the webpages. If you believe that you meet these criteria then follow the steps below:

Select ONE potential supervisor from our list of Academic Staff: https://www.qub.ac.uk/courses/postgraduate-research/find-a-phd-supervisor/ and send an email to that supervisor advising that you are interested in studying for a PhD, stating when you would start, and how you would plan to fund the research. It would be helpful to provide a a brief statement of the research question or interest, and how you think the question could be investigated. The potential supervisor may invite you to meet with them or they may invite you to apply formally.
##### Assessment

Assessment processes for the Research Degree differ from taught degrees. Students will be expected to present drafts of their work at regular intervals to their supervisor who will provide written and oral feedback; a formal assessment process takes place annually.

This Annual Progress Review requires students to present their work in writing and orally to a panel of academics from within the School. Successful completion of this process will allow students to register for the next academic year.

The final assessment of the doctoral degree is both oral and written. Students will submit their thesis to an internal and external examining team who will review the written thesis before inviting the student to orally defend their work at a Viva Voce.

##### Feedback

Supervisors will offer feedback on draft work at regular intervals throughout the period of registration on the degree.

##### Facilities

Students will enjoy the benefits of modern practical laboratories, extensive computer facilities and interactive spaces. Students will be provided with their individual workspace.

#### Entrance requirements

The minimum academic requirement for admission to a research degree programme is normally an Upper Second Class Honours degree from a UK or ROI HE provider, or an equivalent qualification acceptable to the University. Further information can be obtained by contacting the School.

##### International Students

For information on international qualification equivalents, please check the specific information for your country.

##### English Language Requirements

Evidence of an IELTS* score of 6.0, with not less than 5.5 in any component, or an equivalent qualification acceptable to the University is required. *Taken within the last 2 years.

International students wishing to apply to Queen's University Belfast (and for whom English is not their first language), must be able to demonstrate their proficiency in English in order to benefit fully from their course of study or research. Non-EEA nationals must also satisfy UK Visas and Immigration (UKVI) immigration requirements for English language for visa purposes.

For more information on English Language requirements for EEA and non-EEA nationals see: www.qub.ac.uk/EnglishLanguageReqs.

If you need to improve your English language skills before you enter this degree programme, INTO Queen's University Belfast offers a range of English language courses. These intensive and flexible courses are designed to improve your English ability for admission to this degree.

### Tuition Fees

 Northern Ireland (NI) 1 £4,596 Republic of Ireland (ROI) 2 £4,596 England, Scotland or Wales (GB) 1 £4,596 EU Other 3 £18,900 International £18,900

1 EU citizens in the EU Settlement Scheme, with settled or pre-settled status, are expected to be charged the NI or GB tuition fee based on where they are ordinarily resident, however this is provisional and subject to the publication of the Northern Ireland Assembly Student Fees Regulations. Students who are ROI nationals resident in GB are expected to be charged the GB fee, however this is provisional and subject to the publication of the Northern Ireland Assembly student fees Regulations.

2 It is expected that EU students who are ROI nationals resident in ROI will be eligible for NI tuition fees. The tuition fee set out above is provisional and subject to the publication of the Northern Ireland Assembly student fees Regulations.

3 EU Other students (excludes Republic of Ireland nationals living in GB, NI or ROI) are charged tuition fees in line with international fees.

All tuition fees quoted are for the academic year 2021-22, and relate to a single year of study unless stated otherwise. Tuition fees will be subject to an annual inflationary increase, unless explicitly stated otherwise.

Mathematics costs

There are no specific additional course costs associated with this programme.

##### All Students

Depending on the programme of study, there may also be other extra costs which are not covered by tuition fees, which students will need to consider when planning their studies . Students can borrow books and access online learning resources from any Queen's library. If students wish to purchase recommended texts, rather than borrow them from the University Library, prices per text can range from £30 to £100. Students should also budget between £30 to £100 per year for photocopying, memory sticks and printing charges. Students may wish to consider purchasing an electronic device; costs will vary depending on the specification of the model chosen. There are also additional charges for graduation ceremonies, and library fines. In undertaking a research project students may incur costs associated with transport and/or materials, and there will also be additional costs for printing and binding the thesis. There may also be individually tailored research project expenses and students should consult directly with the School for further information.

##### Bench fees

Some research programmes incur an additional annual charge on top of the tuition fees, often referred to as a bench fee. Bench fees are charged when a programme (or a specific project) incurs extra costs such as those involved with specialist laboratory or field work. If you are required to pay bench fees they will be detailed on your offer letter. If you have any questions about Bench Fees these should be raised with your School at the application stage. Please note that, if you are being funded you will need to ensure your sponsor is aware of and has agreed to fund these additional costs before accepting your place.

#### How do I fund my study?

1.PhD Opportunities

Find PhD opportunities and funded studentships by subject area.

2.Funded Doctoral Training Programmes

We offer numerous opportunities for funded doctoral study in a world-class research environment. Our centres and partnerships, aim to seek out and nurture outstanding postgraduate research students, and provide targeted training and skills development.

3.PhD loans

The Government offers doctoral loans of up to £26,445 for PhDs and equivalent postgraduate research programmes for English- or Welsh-resident UK and EU students.

4.International Scholarships

Information on Postgraduate Research scholarships for international students.

###### Funding and Scholarships

The Funding & Scholarship Finder helps prospective and current students find funding to help cover costs towards a whole range of study related expenses.

#### How to Apply

Apply using our online Postgraduate Applications Portal and follow the step-by-step instructions on how to apply.

##### Find a supervisor

If you're interested in a particular project, we suggest you contact the relevant academic before you apply, to introduce yourself and ask questions.

To find a potential supervisor aligned with your area of interest, or if you are unsure of who to contact, look through the staff profiles linked here.

You might be asked to provide a short outline of your proposal to help us identify potential supervisors.