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This paper was published online on 13 February 2018 by the
Glasgow
Mathematical Journal, and calculates the integral cohomology
rings of the symmetric squares of the infinite and finite dimensional
complex and quaternionic projective spaces. The solution provides
generators and relations, by considering the quotients of the symmetric
squares by their diagonals; these are identified with the Thom spaces of
certain *Pin*^{c}*(2)*- and *Pin*^{-}*(4)*-bundles respectively.

This document appears as Tohoku Mathematical Journal 68(4) (2016) 487--513, and calculates the equivariant complex oriented cohomology algebras (in the topological, rather than algebraic, sense) of certain singular algebraic varieties, namely those weighted projective spaces whose weights form a divisor chain. The answers agree with the predictions of GKM theory in the smooth situation, and involve piecewise algebraic objects, such as Laurent polynomials in the K-theoretic case, over the defining fan.

This work appears in the
Osaka Journal of Mathematics 51(1) (2014) 89--121, and is a
sequel to our article on equivariant cohomology;
it studies weighted projective spaces by considering their restrictions
to specific primes *p* in terms of iterated Thom spaces, and
computes their complex oriented cohomology algebras and homology
coalgebras in terms of the resulting Thom isomorphisms. Particular
attention is paid to the universal example, and the interpretation of
certain top dimensional complex bordism classes as desingularisations.

This article appears in the
Journal of Pure and Applied Algebra 218(2) (2014) 303--320; it
adopts several complementary viewpoints (ranging from the Adams spectral
sequence to CAT(K)-diagrams) in considering extensions to real
*K*-theory of various complex *K*-theoretic computations in
toric topology. In particular, it builds on Adrian Dobson's Manchester
PhD thesis to compute *KO*(BT ^{m})* for

This article appears in Fundamenta Mathematicae, 220 (2013) 217--226, and describes the classification of weighted projective spaces up to homeomorphism and up to homotopy equivalence. The first turns out to coincide with Al~Amrani's classification up to isomorphism of algebraic varieties, and the second follows from the fact that the Mislin genus of any weighted projective space is rigid; they are both expressed in terms of permissible operations on weight vectors.

This work appears in the
International Mathematical Research Notices 2010(16) (2010)
3207--3262, and considers torus actions on stably complex manifolds
from the viewpoint of our article below;
it discusses formal group laws and Hirzebruch genera, Krichever's
generalised elliptic genus, special unitary structures, and the
combinatorial evaluation of genera on omnioriented quasitoric
manifolds. It also proves that Krichever's generalised elliptic genus
is universal amongst genera that are rigid on SU-manifolds,
and conjectures that every omnioriented quasitoric *SU*-manifold
is a stably complex boundary.

This article appears
in
Algebraic & Geometric
Topology 10 (2010) 1747--1780, and builds on our rational
investigation of spaces *X* whose integral cohomology ring
realises the Stanley-Reisner algebra of a chosen simplicial
complex *K*, begun in Part I. Here we
consider the problem one prime at a time, and utilise Lannes' *T*
functor and Bousfield-Kan type obstruction theory to study
the *p*-completion of *X* by confirming uniqueness for joins
of skeleta of simplices. We combine the rational
and *p*-complete results by appealing to Sullivan's arithmetic
square, and deduce integral uniqueness whenever the Stanley-Reisner
algebra is a complete intersection. The version currently available
here is slightly out of date ...

This article appears in the Mathematical Proceedings of the Cambridge Philosophical Society 146(02) (2009), 395--405. It computes the equivariant integral cohomology ring of an arbitrary weighted projective space; the answer is given by generators and relations amongst piecewise polynomials, and is shown to distinguish between all cases (unlike the non-equivariant ring). By way of application, a solution is given to a problem of Al Amrani concerning the weighted projectivisation of a split complex vector bundle. A sequel now exists that places the computations firmly within the context of toric topology, by developing more explicit links with lens spaces, localisations, Thom complexes, homotopy colimits, and complex cobordism.

This invitation is not so much a survey, more a collection of ideas and examples that have driven the development of toric topology from four distinct viewpoints; it also aims to highlight several of their many interdisciplinary connections. It appears as the preface to the proceedings of the 2006 Osaka international conference, which were published in 2008 as volume 460 of the American Mathematical Society Contemporary Mathematics series.

The English version of this article appears in
Russian Mathematical Surveys 62(1) (2007), 178--180. It
studies the universal equivariant genus for tangentially stably complex
circle-manifolds *M*, and proposes connected representatives
for the coefficients of the resulting power series (in contrast to the
constructions of Sinha). It also extends a formula of Krichever
concerning the case of isolated fixed points; his formula applies only
to almost complex examples, and we stabilise by introducing
appropriate signs. The longer article below,
develops and extensively generalises these results, in collaboration
with Taras Panov.

This article appears in the
American Mathematical Society
Contemporary Mathematics Series Volume 460 (2008), 298 -322;
it is based on the slides of my talk in Osaka on 2 June 2006, and contains results that
we obtained over the previous six years (some of which were mentioned at
the Keldysh Memorial
Conference in Moscow on 26 August 2004). These include the rational
formality of quasitoric manifolds, and the description of the rational
Pontrjagin ring of loops on *DJ(K)* as the quadratic dual of the
Stanley-Reisner algebra for any flag complexes
*K*. The central theme is that category theory, both in the large
and in the small, provides a valuable framework for the pursuit of
algebraic and geometric insight into toric topology.

This article appears in the Moscow Mathematical Journal 7 (2007) 219--242, and a version is lodged at arXiv as math/0609346. It applies geometric methods to the interpretation of quasitoric manifolds in terms of combinatorial data, and discusses the relationship with non-singular projective toric varieties. In particular, it applies Khovanskii's theory of analogous polytopes, and constructs representatives for every complex cobordism class as the quotient of a free torus action on a real quadratic complete intersection. The opportunity is taken to simplify and correct certain proofs of the first and third author from their 2001 article below (whose inconsistencies were identified by Kostya Feldman and Neil Strickland), which involves modifying our definition of the connected sum of polytopes.

This article appears in
*K*-Theory 34 (2005) 1-33, and a version is lodged at
arXiv as math/0408261.
It has evolved from the first author's Manchester PhD thesis, and
considers the algebraic topology of certain sequences of toric
manifolds, known as Bott towers. A simple decomposition is given of
their suspensions into a wedge of Thom complexes, and the
multiplicative structure of their real *K*-theory is deduced;
the results are compared with those of Bahri and Bendersky (who used
the Adams spectral sequence for additive calculations), and applied to
the study of Bott towers in complex cobordism theory.

This article appears in
Algebraic & Geometric
Topology 5 (2005) 31-51, and a version is lodged at arXiv as math/0311167. It is the
first of a pair that investigate the homotopy
type of Davis and Januszkiewicz's spaces *DJ(K)*, whose
integral cohomology ring realises the Stanley-Reisner algebra of a
chosen simplicial complex
*K*. It is couched in the language of model category theory, and shows
that the singular cochain algebra of *DJ(K)* is formal as a
differential graded noncommutative algebra; on specialising to the
rationals, the corresponding property is confirmed for Sullivan's
commutative cochain algebra, and the rationalisation of *DJ(K)*
is proven to be unique in a family of special cases.

This paper is now in its final version (after ten years of work) and appears in the Annals of Combinatorics 7 (2003) 55-88. It studies the enumeration of permutations with specific forbidden subsequences in terms of a certain ranked poset of permutation matrices, and enumerates the saturated chains of length two; closed polynomial formulae related to the Robinson-Schensted correspondence are obtained, along with simple asymptotic estimates for some longer chains.

This article grew out of a talk at the International Conference on
Algebraic Topology, held on the Isle of Skye in June 2001, and is
lodged at arXiv as math/0202081; it appears in
Algebraic Topology: Categorical Decomposition Techniques,
Birkhauser Progress in Mathematics Volume 215 (2003) 261-291.
We construct models for loops on Davis and Januszkiewicz's
spaces *DJ(K)* (defined for finite simplicial complexes
*K*), which rely on the development of homotopy colimits in the
category of topological monoids and a proof that they commute with the
classifying space functor. Whenever *K* is *flag*, the
construction reduces to a standard colimit, which is a right-angled
Artin or Coxeter group in the real or exterior case; in the complex
case, it is a continuous analogue, which we call a *circulation
group*.

This article appears in
International
Mathematics Research Notices 4 (2001) 193-219, and is also
lodged at arXiv as math/0010025. It answers the "Hirzebruch question" raised by the 1998 paper below, by proving that every complex cobordism
class contains a toric manifold, which is necessarily
*connected*; in addition, it confirms that the appropriate stably
complex structure is associated with the torus action, and preserved
by it. The concept of omniorientation is defined for the purpose, and
an operation of connected sum is developed for simple convex polytopes
endowed with specific combinatorial data. [*Corrections,
clarifications and extensions appear in work with Victor Buchstaber
and Taras Panov above; none of the main
results are affected*]

Many early versions of this work were developed throughout the 90s,
and it now appears in the
Annales de l'Institut Fourier 51(2), 297--336 (2001).
The paper develops the theory of double delta operators, with
applications to the study of Hopf algebroids of homology cooperations
for complex oriented theories; it addresses the number theoretic
questions which arise, and describes related generalisations of the
Hattori-Stong theorem of complex *K*-theory.

This paper was originally written in Russian, and appears in Uspheki Mat Nauk 53 (1998) 2, 139-140; the translation is in Russian Mathematical Surveys 53 (1998) 2, 371-373. It proves that every complex cobordism class contains a disjoint union of suitably oriented toric manifolds (in the sense of Davies and Januszkiewicz) by introducing certain projective bundles over bounded flag manifolds; these provide alternatives to the Milnor hypersurfaces, which are shown not to be toric for cohomological reasons.

This paper appears in Advances in Mathematics 138 (1998), 211-262, after eight years of gestation. It introduces the combinatorial category of cell sets, and employs them to generalise notions such as incidence coalgebra and Hopf algebra; steps are taken towards investigating the category as some sort of discrete analogue for topological spaces and spectra with cells in even dimensions.

This paper appears in Contemporary Mathematics 220 (1998), 293-311. It expands on my talk at the 1997 Northwestern Conference, and describes computations with formal power series over a graded ring in terms of cobordism theory; the results suggest a deeper layer of structure than is normally evident, and pose new problems in cobordism theory.

This paper appears in Discrete Mathematics 180 (1998), 255-280. It establishes a procedure for creating Hopf algebras out of certain set systems, which may be equipped with group actions; applications are described to the theory of delta operators, formal group laws, and chromatic polynomials.

This paper appears in Discrete Mathematics 167/168 (1997), 419-444. It extends the domain of definition of the chromatic polynomial to a class of set systems known as partition systems, which may also be equipped with group actions; examples include the normalised conjugate Bell polynomials.

This 1996 preprint has never been published, but much of the material is incorporated into the following two documents:

This paper appears in Homology, Homotopy and Applications 1 (1999), 169-185. It develops geometrical background to the quantum algebraic description of the algebra of cohomology operations in double complex cobordism, and interprets the constructions in the context of Boardman's eightfold way.

This paper appears in Geometry and Topology 2 (1998), 79-101. It introduces the complex bounded flag manifolds (as toric varieties, even), and discusses their combinatorial and geometrical structure; their complex cobordism is computed, and used to describe a geometrical interpretation of the dual of the Landweber-Novikov algebra.

This paper summarises my talk during the 1996 Rotafest at MIT, in which I described algebraic foundations for umbral calculus over a graded ring; it appears in the 1998 Birkhauser volume "Mathematical Essays in Honor of Gian-Carlo Rota", edited by Bruce Sagan and Richard Stanley.

This paper appears in the Journal of Pure and Applied Algebra 101 (1995), 313-333 (a minor error has been corrected). It describes how Hopf ring computations which normally use the p-series of a formal group law may profitably be recast over the rationals by using the exponential series; applications are made to Landweber exact theories.

This paper appears in the American Journal of Mathematics 117 (1995), 1063-1088. It makes considerable headway with computations in the appropriate mod 2 Hopf rings, emphasising the role played by the phi elements and proposing a model for the universal example.

This paper appears in Discrete Mathematics 125 (1994), 329-341. It describes the ultimate version of our enriched chromatic polynomial as an element of an appropriate free abelian group.

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