2 research outputs found
A computational approach to Conway's thrackle conjecture
A drawing of a graph in the plane is called a thrackle if every pair of edges
meets precisely once, either at a common vertex or at a proper crossing. Let
t(n) denote the maximum number of edges that a thrackle of n vertices can have.
According to a 40 years old conjecture of Conway, t(n)=n for every n>2. For any
eps>0, we give an algorithm terminating in e^{O((1/eps^2)ln(1/eps))} steps to
decide whether t(n)2. Using this approach, we improve the
best known upper bound, t(n)<=3/2(n-1), due to Cairns and Nikolayevsky, to
167/117n<1.428n.Comment: 16 pages, 7 figure
Bridge between worlds: relating position and disposition in the mathematical field
Using ethnographic observations and interview based research I document the
production of research mathematics in four European research institutes,
interviewing 45 mathematicians from three areas of pure mathematics: topology,
algebraic geometry and differential geometry. I use Bourdieu's notions of habitus,
field and practice to explore how mathematicians come to perceive and interact
with abstract mathematical spaces and constructions. Perception of mathematical
reality, I explain, depends upon enculturation within a mathematical discipline. This
process of socialisation involves positioning an individual within a field of
production. Within a field mathematicians acquire certain structured sets of
dispositions which constitute habitus, and these habitus then provide both
perspectives and perceptual lenses through which to construe mathematical
objects and spaces.
I describe how mathematical perception is built up through interactions
within three domains of experience: physical spaces, conceptual spaces and
discourse spaces. These domains share analogous structuring schemas, which are
related through Lakoff and Johnson's notions of metaphorical mappings and image
schemas. Such schemas are mobilised during problem solving and proof
construction, in order to guide mathematicians' intuitions; and are utilised during
communicative acts, in order to create common ground and common reference frames. However, different structuring principles are utilised according to the
contexts in which the act of knowledge production or communication take place.
The degree of formality, privacy or competitiveness of environments affects the
presentation of mathematicians' selves and ideas. Goffman's concept of interaction
frame, front-stage and backstage are therefore used to explain how certain
positions in the field shape dispositions, and lead to the realisation of different
structuring schemas or scripts.
I use Sewell's qualifications of Bourdieu's theories to explore the multiplicity
of schemas present within mathematicians' habitus, and detail how they are given
expression through craftwork and bricolage. I argue that mathematicians'
perception of mathematical phenomena are dependent upon their positions and
relations. I develop the notion of social space, providing definitions of such spaces
and how they are generated, how positions are determined, and how individuals
reposition within space through acquisition of capital