Equivariant orthogonal calculus

School of Mathematics and Physics | PHD

Applications are now CLOSED
Funding
Funded
Reference Number
MAP/2020/19
14 February 2020
Start Date
1 October 2020

Overview

Homotopy theory is the study of topological spaces up to homotopy equivalence. One useful method to understand a space X is to look at its Postnikov tower. This tower is a collection of increasing good approximations to X Pn(x) → Pn−1X → · · · → P0(X). The difference between Pn−1X and Pn(X) is an Eilenberg–Mac lane space, a particularly well– behaved space which homotopy theorists now a lot about. Hence we have broken X down into a collection of familiar pieces and to understand X all we need to do is see how these pieces fit together to make X.

An important generalisation of this idea is orthogonal homotopy calculus [4]. We start with
an I –space X: a countable collection of spaces (X0, X1, X2, . . .) along with a set of maps
from Xn to Xn+1 (one map for each linear map R
n → R
n+1). For example we may take Xn = Sn, the n–sphere and the maps Sn → Sn+1 corresponding to the choices of an equator of Sn+1 that passes through the north pole.

The orthogonal homotopy calculus provides us with a tower of approximations to X
Tn(x) → Tn−1X → · · · → T0(X)

where each Tn(X) is an I –space. Interestingly, the difference between Tn−1X and Tn(X)
is classified by a topological space DnX (rather than an I –space). The space DnX has a
symmetry group given by O(n) (the group of n-by-n orthogonal matrices). It is also an infinite
loop space — another kind of much–studied space that homotopy theorists understand well.
An interesting question is: what happens when the topological spaces Xk have symmetries
and the maps Xn → Xn+1 respect those symmetries? This leads to the notion of equivariant orthogonal calculus. The project will be based around setting up the definitions in terms of model categories and extending the results of [1] to this setting. With the foundations complete, there a number of directions for the project to take, based around extending currently–known examples and applications.
The student should have attended courses on algebraic topology and topology. This project
will require the student to become familiar with the abstract language of model categories [2]
and modern categories of spectra [3].
References
[1] D. Barnes and P. Oman. Model categories for orthogonal calculus. Algebr. Geom. Topol., 13(2):959–999,
2013.
[2] W. G. Dwyer and J. Spali´nski. Homotopy theories and model categories. In Handbook of algebraic topology,
pages 73–126. North-Holland, Amsterdam, 1995.
[3] M. A. Mandell, J. P. May, S. Schwede, and B. Shipley. Model categories of diagram spectra. Proc. London
Math. Soc. (3), 82(2):441–512, 2001.
[4] M. Weiss. Orthogonal calculus. Trans. Amer. Math. Soc., 347(10):3743–3796, 1995.

Funding Information

Project Summary
Supervisor
Dr David Barnes
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.

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).

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Course content

Research Information

Research Themes
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.

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

Email: d.barnes@qub.ac.uk

Professor Ivan Todorov
Director of Research - Mathematical Sciences Research Centre

Email: i.todorov@qub.ac.uk

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Facilities
Students will enjoy the benefits of modern practical laboratories, extensive computer facilities and interactive spaces.

Entrance requirements

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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.

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