Equivariant orthogonal calculus
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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Dr David Barnes
Postgraduate Advisor - Mathematical Sciences Research Centre
Professor Ivan Todorov
Director of Research - Mathematical Sciences Research Centre
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|Northern Ireland (NI)||£4,407|
|England, Scotland or Wales (GB)||£4,407|
|Other (non-UK) EU||£4,407|
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