Dr Andrew Tonks

Job Title: Senior Lecturer, Pure Mathematics / STORM

 Office: T9-03d Tower Building Telephone: (+44) (0)20 7133 4635 Internal (79) 4635 e-mail address: a.tonks@londonmet.ac.uk

BA Mathematics, University of Cambridge

MA, University of Cambridge

PhD Mathematics, University of Wales, Bangor

Current Teaching:

Semester 2, 2010-1: MA1040, MA3036, MA3038, (MA3P47)

Research and Publications

Recent papers

• I. Gálvez, A. Tonks, and B. Vallette. Homotopy Batalin-Vilkovisky algebras. J. Noncommut. Geom., 2011.
[Accepted for publication]

• I. Gálvez, V. Gorbounov, and A. Tonks. Homotopy Gerstenhaber structures and vertex algebras. Appl. Categ. Structures, 18(1):1–15, 2010.

• Kathryn Hess and Andrew Tonks. The loop group and the cobar construction. Proc. Amer. Math. Soc., 138(5):1861–1876, 2010.

• Fernando Muro and Andrew Tonks. On $K_1$ of a Waldhausen category. In $K$-theory and noncommutative geometry, EMS Ser. Congr. Rep., pp. 91–115, Eur. Math. Soc., Zürich, 2008.

• Fernando Muro and Andrew Tonks. The 1-type of a Waldhausen $K$-theory spectrum. Adv. Math., 216(1):178–211, 2007.

• K. Worytkiewicz, K. Hess, P. E. Parent, and A. Tonks. A model structure à la Thomason on \bf 2-Cat. J. Pure Appl. Algebra, 208(1):205–236, 2007.

• Carles Casacuberta, Marek Golasi\'nski, and Andrew Tonks. Homotopy localization of groupoids. Forum Math., 18(6):967–982, 2006.

• Kathryn Hess, Paul-Eugène Parent, Jonathan Scott, and Andrew Tonks. A canonical enriched Adams-Hilton model for simplicial sets. Adv. Math., 207(2):847–875, 2006.

• Kathryn Hess, Paul-Eugène Parent, Andrew Tonks, and Krzysztof Worytkiewicz. Simulations as Homotopies. Electronic Notes in Theoretical Computer Science, 100:65–93, 2004. CONCUR 2003: CMCIM and GETCO

• Imma Gálvez and Andrew Tonks. Differential operators and the Witten genus for projective spaces and Milnor manifolds. Math. Proc. Cambridge Philos. Soc., 135(1):123–131, 2003.

• A. P. Tonks. On the Eilenberg-Zilber theorem for crossed complexes. J. Pure Appl. Algebra, 179(1-2):199–220, 2003.

Preprints

• Homotopy Batalin-Vilkovisky algebras (with I Gálvez, B Vallette)
CRM Preprint Number 875, arxiv:0907.2246

Other work (in progress, caveat lector!)

Associahedron diagonal approximation

Older papers

• Cubical groups which are Kan,
Journal of Pure and Applied Algebra 81 (1992) 83-87.
The Kan extension condition is shown to be automatic for cubical groups which have extra connection degeneracies; the cubical theory is then parallel to the simplicial.

• Theory and applications of crossed complexes: the Eilenberg-Zilber theorem and homotopy colimits.
Ph.D. thesis University of Wales (1994) 126pp.
We prove an Eilenberg-Zilber theorem for crossed complexes, and use this to develop the theory of homotopy colimits of (homotopy coherent) diagrams in Crs. An application is given to crossed resolutions of group extensions.

• Calculations with simplicial and cubical groups in Axiom.
Journal of Symbolic Computation 17 (1994) 159-179
(with R. Brown).
We discuss the capabilities of the Axiom symbolic algebra system and our implementation and use of code for conjectures and calculations with near-rings and cubical operators.

• An I-Category structure for crossed chain algebras.
(Max-Planck-Institut Preprint MPIM1995-135)
We define the relative cylinder construction on the category of crossed chain algebras  (monoids in the category of crossed complexes) and show it defines an I-category structure, in the sense of Baues, on this category.

• On sum-normalised cohomology of categories, twisted homotopy pairs and universal Toda brackets.
Oxford Quarterly Journal of Mathematics 47 no.188 (1996) pp.405-433
(with H.-J. Baues)
.
A degree 1 map is given from the cohomology of a category with (co)products to that of its twisted pair category. Applying this to topology, we show how universal Toda brackets determine categories of maps between homotopy (co)fibres.
•
• On the twisted cobar construction.
Mathematical Proceedings of the Cambridge Philosophical Society 121 (1997) 229-245
(with H.-J. Baues)
.
The cobar construction of Adams is extended to twisted coefficients, giving an algebraic model for the universal cover of the loops on a 1-reduced space.
•
• Relating the associahedron and the permutohedron.
In Operads: Proceedings of the Renaissance Conferences (Hartford CT / Luminy Fr 1995)''
Contemporary Mathematics 202 (1997) 33-36
We show how the cells of the associahedron and the permutohedron codify the choices in the order of evaluation of a product of terms; this leads to natural cellular quotient maps from the permutohedra to the associahedra.

• Cohomology of monoids in monoidal categories.
In Operads: Proceedings of the Renaissance Conferences (Hartford CT / Luminy Fr 1995)''
Contemporary Mathematics 202 (1997) 137-165
.
We define cohomology of monoids in general monoidal categories and identify significant simplifications for the abelian and left-distributive cases. We recover as particular examples several cohomology theories in the literature, including those of bimodules, theories, small categories and the motivating example of operads.
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• Spaces of maps into classifying spaces for equivariant crossed complexes.
Indagationes Mathematicae 8 (1997) 157-172
(with R. Brown, M. Golasiński, T. Porter).
An equivariant version of the homotopy theory of crossed complexes is given which generalises recent work of Moerdijk-Svensson on equivariant 2-types and leads to results on homotopy of equivariant function spaces.

• Spaces of maps into classifying spaces for equivariant crossed complexes II - the general topological group case.
K-Theory
23 (2001) 129-155
(with R. Brown, M. Golasiński, T. Porter).
We generalize the previous paper to the case of an arbitrary topological group action. The extra ingredient necessary is an analysis of the crossed complex homotopy coherence arising from the Eilenberg-Zilber theorem, and its relation to simplicial homotopy coherence. Again we are able to define an equivariant classifying space and use it to calculate the weak equivariant homotopy type of certain equivariant function spaces.
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Page last updated : : 08 Apr 2013