The Equivariant Cobordism Category

For a finite group $G$, we define an equivariant cobordism category $\mathcal{C}_d^G$. Objects of the category are $(d-1)$-dimensional closed smooth $G$-manifolds and morphisms are smooth $d$-dimensional equivariant cobordisms. We identify the homotopy type of its classifying space (i.e. geometric realization of its simplicial nerve) as the fixed points of the infinite loop space of an equivariant spectrum.Comment: 44 pages; v2 has more details and many improvement

Singularities and stable homotopy groups of spheres II

We establish a connection between Morin singularities and stable homotopy groups of spheres. This connection allows us to describe how the images of singularity strata behave around the image of a more complicated stratum.Comment: 31 pages, submitted to Journal of Singularitie

Creative processes in Western art music performance practice with reference to the journey of a professional cellist

This practice-based research undertaking investigates the creative and decision-making processes pursued within expert-professional performance practice, in the Western art music performance tradition, through the work of the individual signature practitioner. The thesis examines the practice in terms of a complex, relationally defined, knowledge-practical system and aims to demonstrate how this knowledge is acquired, shared, communicated and disseminated through an elaborative process of articulation. The research explores the professional identity of a concert cellist (in the first person) as a creative decision-maker, by revealing parts of the practice, which are rarely accessible to spectators or even to theorists or musicologists. It aims to show the habitus of the expert performer through the stages of development as well as through specific and documented accounts of professional practice, including studio work, preparation, rehearsals, pedagogy, research, and performance events in a variety of conditions and environments. This is an investigation into expertise itself as an epistemic category, by exploring questions of professional judgement, the use of expert-intuitive processes, models of intelligibility, the artist’s signature and the notion of qualitative transformation, as they appear in actual practice. The audio-video documentation of rehearsals, performances and discussions provide an opportunity to consider questions concerning technique, style, interpretation, communication methods between performers, the performer’s relationship with the notation of the score, and the experience and conceptualisation of performance events from the performer’s point of view, representing the seldom-heard voice of the practitioner. My work in this research context is highly experimental in terms of the relationship between research methods and expert-creative practice, where the ‘immersed’ researcher is also the research subject, with many of the problematic implications noted by social sciences. This research is presented in a mixed-mode heuristic framework, where the focus is on the practice itself, while the text and documentation serves to illuminate that practice, as I propose to write, demonstrate, transfer and validate non-conceptual and non-discursive knowledge through the mediation of the paradoxes inherent in ‘theorising performance’. The critical engagement relates to my claim as to the lack of effective treatment in published research of issues specific to expert-professional performance practice, and the new knowledge emerges from the new questions I am asking with reference to my own practice

Singularities and stable homotopy groups of spheres I

We establish an interesting connection between Morin singularities and stable homotopy groups of spheres. We apply this connection to computations of cobordism groups of certain singular maps. The differentials of the spectral sequence computing these cobordism groups are given by the composition multiplication in the stable homotopy groups of spheres.Comment: 39 pages, 1 figure, revised again for Journal of Singularitie

Multiplicative properties of Morin maps

In the first part of the paper we construct a ring structure on the rational cobordism classes of Morin maps (i. e. smooth generic maps of corank 1). We show that associating to a Morin map its singular strata defines a ring homomorphism to \Omega_* \otimes \Q, the rational oriented cobordism ring. This is proved by analyzing multiple-point sets of product immersion. Using these homomorphisms we are able to identify the ring of Morin maps. In the second part of the paper we compute the oriented Thom polynomial of the $\Sigma^2$ singularity type with \Q coefficients. Then we provide a product formula for the $\Sigma^2$ and the $\Sigma^{1,1}$ singularities.Comment: Corrected some small misprints and made lot of minor (mainly grammatical) alterations. 10 page