Equivariant orthogonal calculusSchool of Mathematics and Physics | PHD
Applications are now CLOSED
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 . 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  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 
and modern categories of spectra .
 D. Barnes and P. Oman. Model categories for orthogonal calculus. Algebr. Geom. Topol., 13(2):959–999,
 W. G. Dwyer and J. Spali´nski. Homotopy theories and model categories. In Handbook of algebraic topology,
pages 73–126. North-Holland, Amsterdam, 1995.
 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.
 M. Weiss. Orthogonal calculus. Trans. Amer. Math. Soc., 347(10):3743–3796, 1995.
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People teaching you
Dr David Barnes
Postgraduate Advisor - Mathematical Sciences Research Centre
Professor Ivan Todorov
Director of Research - Mathematical Sciences Research Centre
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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.
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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.
|Northern Ireland (NI)||£4,407|
|England, Scotland or Wales (GB)||£4,407|
|Other (non-UK) EU||£4,407|
There are no specific additional course costs associated with this programme.
Additional course costs
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.
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