An Introduction To Mathematical Reasoning Peter J Eccles Pdf

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Peter J Eccles
School of Mathematics
The University of Manchester

(combining the Victoria University of Manchester and UMIST)

Oxford Road
Manchester M13 9PL
Great Britain

Email address: pjeccles@manchester.ac.uk

The University of Manchester coat of arms

Research interests

My main research interest is in the application of algebraic topology to differential topology. Its main focus in recent years has been the study of multiple point sets of immersions of smooth manifolds. If the immersion is in `general position' then the multiple point sets will themselves be the the images of immersions of other immersed manifolds. We can ask about the properties of these multiple point manifolds, which manifolds can arise and to what extent the original manifold determines the multliple point manifolds.

An example of a result in this area is the one I started with. For a codimension 1 immersion of a closed n dimensional manifold in Euclidean (n+1)-space the set of (n+1) fold points (points in the image of (n+1) distinct points in the manifold) is a discrete set. It turns out that if the manifold is orientable then the number of (n+1) fold points can be odd if and only if n = 1 or n = 3. In these cases we can take the manifold to be the n-sphere. This result was obtained by translating the problem into a problem in algebraic topology by means of cobordism theory and the homotopy theory of Thom complexes. Other problems of the same type can be explored by the same means and tackling these brings into play a whole range of techniques from algebraic topology.

The same question for non-orientable manifolds is closely connected to classical problems in the homotopy groups of spheres: the Hopf invariant problem and the Kervaire invariant problem.

Recent papers:

Double point manifolds of immersions of spheres in Euclidean space, Proceedings of the William Browder Sixtieth Birthday Conference, March 1994, Annals of Math. Studies, 138, Princeton University Press, 1997, 125-137.
Postscript file

(with Paul R Turner and W Stephen Wilson) On the Hopf ring for the sphere, Math. Z. 224 (1997), 229-233.
Mathematische Zeitschrift

(with Pyotr M Akhemt'ev) A geometrical proof of Browder's result on the vanishing of the Kervaire invariant, (in Russian) Tr. Math. Inst. Steklova 225 (1999), 46-51, (in English) Proc. Steklov Inst. Math. 225 (1999), 40-44.
Postscript file

(with Mohammad A Asadi-Golmankhaneh) Determining the characteristic numbers of self-intersection manifolds, J. London Math. Soc. (2) 62 (2000), 278-290.
Postscript file

(with Mohammad A Asadi-Golmankhaneh) Double point self-intersection surfaces of immersions, Geometry and Topology 4 (2000), 149-170.
Geometry and Topology

(with Mark Grant) Bordism classes represented by multiple point manifolds of immersed manifolds, Proceedings of the Keldysh Centenary Conference, Moscow, August 2004.
pdf file

(with Mark Grant) Bordism groups of immerersions and classes represented by self-intersections, Algebr. Geom. Topol., 7 (2007) 1081-2097.
files on arXiv.org

(with Pyotr Akhmet'ev) The relationship between framed bordism and skew-framed bordism, to appear.
pdf file

(with Mark Grant) Self-intersections of immersions and Steenrod operations, Acta Math. Hungar. 137 (4) 2012, 272-281.
pdf file

Book:

Introduction to Mathematical Reasoning: numbers, sets and functions, Cambridge University Press, December 1997, reprinted 1998, 2000(twice), 2001, 2002, 2004.

Table of contents
Preface

For further information and ordering details see Great Britain or United States of America.

Book (not mathematics):

The Presence in the Midst: reflections on discernment (Swarthmore Lecture 2009), Quaker Books, London, 2009.

Editorial responsibilities:

Journal of Homotopy and Related Structures.

Graduate students:

Sadoon M Al-Musawe, Multiplications on certain H-spaces, MSc 1974
Salih Karadag, Intersection points of immersed manifolds, MSc 1980
Khalid Felali, Intersection points of immersed manifolds, PhD 1982
William P R Mitchell, An application of and the derivation of a certain splitting, MSc 1984
William P R Mitchell, p-fold intersection points and their relation with MU(n), PhD 1986
Richard J Morris, Immersions of manifolds, MSc 1988
Adam J Harrison, Self-intersection points of immersed manifolds, MSc 1990
Paul R Turner, A filtration of the homotopy groups of spheres, MSc 1991
Paul R Turner, On the homology of certain infinite loop spaces, PhD 1993
Mohammad A Asadi-Golmankhaneh, Self-intersection manifolds of immersions, PhD 1998
Christopher I Hazell, Immersions up to bordism, PhD 1998
Jonathan M A Burgess, On self-intersections and the stable homotopy of D₂MO(2), PhD 2002
Mark Grant, Bordism of immersions, PhD 2006
Hadi Zare, On spherical classes in H_*QSⁿ, PhD 2009
Azzah Al-Shehry, The double point surfaces of immersions in complex projective spaces, PhD 2010
Alex Longdon, Stably complex structures on self-intersection manifolds of immersions, PhD 2015

There won't be any more now that I have retired. I much enjoyed working with each of these students.

Last revised 8 September 2016