Mapping class group e complesso delle curve per superfici di Heegaard

In una 3-varietà chiusa è possibile individuare alcune superfici (dette di Heegaard) tali che, tagliando la 3-varietà lungo una di queste, essa si spezza in due corpi con manici che hanno per bordo tale superficie. La tesi propone alcuni recenti risultati circa l'interazione tra la topologia della 3-varietà, il gruppo di automorfismi delle sue superfici di Heegaard e complessi simpliciali costruiti a partire dalle curve su tali superfici.

Twisted conjugacy classes in symplectic groups, mapping class groups and braid groups (with an appendix written jointly with Francois Dahmani)

We prove that the symplectic group Sp(2n, Z) and the mapping class group Mod(S) of a compact surface S satisfy the R(infinity) property. We also show that B(n)(S), the full braid group on n-strings of a surface S, satisfies the R(infinity) property in the cases where S is either the compact disk D, or the sphere S(2). This means that for any automorphism phi of G, where G is one of the above groups, the number of twisted phi-conjugacy classes is infinite.

Braid groups, mapping class groups, and Torelli groups

This thesis discusses subgroups of mapping class groups of particular surfaces. First, we study the Torelli group, that is, the subgroup of the mapping class group that acts trivially on the first homology. We investigate generators of the Torelli group, and we give an algorithm that factorizes elements of the Torelli group into products of particular generators. Furthermore, we investigate normal closures of powers of standard generators of the mapping class group of a punctured sphere. By using the Jones representation, we prove that in most cases these normal closures have infinite index in the mapping class group. We prove a similar result for the hyperelliptic mapping class group, that is, the group that consists of mapping classes that commute with a fixed hyperelliptic involution. As a corollary, we recover an older theorem of Coxeter (with 2 exceptional cases), which states that the normal closure of the m-th power of standard generators of the braid group has infinite index in the braid group. Finally, we study finite index subgroups of braid groups, namely, congruence subgroups of braid groups. We discuss presentations of these groups and we provide a topological interpretation of their generating sets.

The Noether-Lefschetz theorem for the divisor class group

Let X be a normal projective threefold over a field of characteristic zero and vertical bar L vertical bar be a base-point free, ample linear system on X. Under suitable hypotheses on (X, vertical bar L vertical bar), we prove that for a very general member Y is an element of vertical bar L vertical bar, the restriction map on divisor class groups Cl(X) -> Cl(Y) is an isomorphism. In particular, we are able to recover the classical Noether-Lefschetz theorem, that a very general hypersurface X subset of P-C(3) of degree >= 4 has Pic(X) congruent to Z.

Fundamental domains for proper affine actions of Coxeter groups in three dimensions

We study proper actions of groups $G \cong \Z/2\Z \ast \Z/2\Z \ast \Z/2\Z$ on affine space of three real dimensions. Since $G$ is nonsolvable, work of Fried and Goldman implies that it preserves a Lorentzian metric. A subgroup $\Gamma < G$ of index two acts freely, and $\R^3/\Gamma$ is a Margulis spacetime associated to a hyperbolic surface $\Sigma$. When $\Sigma$ is convex cocompact, work of Danciger, Gu{\'e}ritaud, and Kassel shows that the action of $\Gamma$ admits a polyhedral fundamental domain bounded by crooked planes. We consider under what circumstances the action of $G$ also admits a crooked fundamental domain. We show that it is possible to construct actions of $G$ that fail to admit crooked fundamental domains exactly when the extended mapping class group of $\Sigma$ fails to act transitively on the top-dimensional simplices of the arc complex of $\Sigma$. We also provide explicit descriptions of the moduli space of $G$ actions that admit crooked fundamental domains.

Computation of locally free class groups

We show that the locally free class group of an order in a semisimple algebra over a number field is isomorphic to a certain ray class group. This description is then used to present an algorithm that computes the locally free class group. The algorithm is implemented in MAGMA for the case where the algebra is a group ring over the rational numbers.

‘Did Ronald McDonald also tend to scare you as a child?’: working to emplace consumption, commodities, and citizen-students in a large classroom setting

So-called ‘radical’ and ‘critical’ pedagogy seems to be everywhere these days on the landscapes of geographical teaching praxis and theory. Part of the remit of radical/critical pedagogy involves a de-centring of the traditional ‘banking’ method of pedagogical praxis. Yet, how do we challenge this ‘banking’ model of knowledge transmission in both a large-class setting and around the topic of commodity geographies where the banking model of information transfer still holds sway? This paper presents a theoretically and pedagogically driven argument, as well as a series of practical teaching ‘techniques’ and tools—mind-mapping and group work—designed to promote ‘deep learning’ and a progressive political potential in a first-year large-scale geography course centred around lectures on the Geographies of Consumption and Material Culture. Here students are not only asked to place themselves within and without the academic materials and other media but are urged to make intimate connections between themselves and their own consumptive acts and the commodity networks in which they are enmeshed. Thus, perhaps pedagogy needs to be emplaced firmly within the realms of research practice rather than as simply the transference of research findings.

The classification and the conjugacy classes of the finite subgroups of the sphere braid groups

Let n >= 3. We classify the finite groups which are realised as subgroups of the sphere braid group B(n)(S(2)). Such groups must be of cohomological period 2 or 4. Depending on the value of n, we show that the following are the maximal finite subgroups of B(n)(S(2)): Z(2(n-1)); the dicyclic groups of order 4n and 4(n - 2); the binary tetrahedral group T*; the binary octahedral group O*; and the binary icosahedral group I(*). We give geometric as well as some explicit algebraic constructions of these groups in B(n)(S(2)) and determine the number of conjugacy classes of such finite subgroups. We also reprove Murasugi`s classification of the torsion elements of B(n)(S(2)) and explain how the finite subgroups of B(n)(S(2)) are related to this classification, as well as to the lower central and derived series of B(n)(S(2)).

Youth theatre and the articulation of cultural capital : refocusing Bourdieu through ethnography

This study of working-class and middle-class youth theatre workshops examines the processes through which this cultural form is appropriated by different class groups. Whereas the middle-class workshop proceeded efficiently and harmoniously, the working-class group resisted a number of institutional constraints traditionally associated with play rehearsal and performance. The processes of such symbolic struggle in the working-class group appeared to differ from Bourdieu's account of cultural domination. The article explores the explanatory contribution of the ethnographic case study to the analysis of the class basis of cultural tastes and practices and suggest that Bourdieu's account of class relations would gain from inclusion of this level of analysis. The situated study of the youth theatre workshops suggests that at this level, there is possibly more scope for symbolic struggle between the classes than was found by Bourdieu.

The flying doctors

This book is a reader for primary school students, stage 24-26 (fluent), incorporating mathematics themes. There is a fictional narrative, entitled "A Flying Visit", that describes Tess' and Alex's encounter with the Flying Doctor. A non-fiction recount, entitled "Fundraising for the Flying Doctors", describes the activities of a class group in raising money for the Flying Doctors. Accompanying the book is a "building comprehension card" to assist teachers in their classroom use of the reader.

Understanding QC and EQA : an authentic learning model for undergraduate students (Conference Abstract)

Introduction QC and EQA are integral to good pathology laboratory practice. Medical Laboratory Science students undertake a project exploring internal QC and EQA procedures used in chemical pathology laboratories. Each student represents an individual lab and the class group represents the peer group of labs performing the same assay using the same method. Methods Using a manual BCG assay for serum albumin, normal and abnormal controls are run with a patient sample over 7 weeks. The QC results are assessed each week using calculated z-scores and both 2S & 3S control rules to determine whether a run is ‘in control’. At the end of the 7 weeks a completed LJ chart is assessed using the Westgard Multirules. Students investigate causes of error and the implications for both lab practice and patient care if runs are not ‘in control’. Twice in the 7 weeks two EQA samples (with target values unknown) are assayed alongside the weekly QC and patient samples. Results from each student are collated and form the basis of an EQA program. ALP are provided and students complete a Youden Plot, which is used to analyse the performance of each ‘lab’ and the method to identify bias. Students explore the concept of possible clinical implications of a biased method and address the actions that should be taken if a lab is not in consensus with the peer group. Conclusion This project is a model of ‘real world’ practice in which student demonstrate an understanding of the importance of QC procedures in a pathology laboratory, apply and interpret statistics and QC rules and charts, apply critical thinking and analytical skills to quality performance data to make recommendations for further practice and improve their technical competence and confidence.

Recording primate vocalizations

Ornithologists have been exploring the possibilities and the methodology of recording and archiving animal sounds for many decades. Primatologists, however, have only relatively recently become aware that recordings of primate sound may be just as valuable as traditional scientific specimens such as skins or skeletons, and should be preserved for posterity (Fig. 16.1). Audio recordings should be fully documented, archived and curated to ensure proper care and accessibility. As natural populations disappear, sound archives will become increasingly important (Bradbury et al., 1999). Studying animal vocal communication is also relevant from the perspective of behavioural ecology. Vocal communication plays a central role in animal societies. Calls are believed to provide various types and amounts of information. These may include, among other things: (1) information about the sender's identity (e.g. species, sex, age class, group membership or individual identity); (2) information about the sender's status andmood (e.g. dominance, fear or aggressive motivation, fitness); and (3) information about relevant events or discoveries in the sender's environment (e.g. predators, food location). When studying acoustic communication, sound recordings are usually required to analyse the spectral and temporal structure of vocalizations or to perform playback experiments (Chapter 11)...

HLA-DR/DQ haplotype in rheumatoid arthritis: Novel allelic associations in UK Caucasians

Objective. To elucidate the relative importance of the HLA-DR and HLA-DQ loci in conferring genetic predisposition to rheumatoid arthritis (RA). Methods. HLA-DRB1 and HLA-DQB1 alleles were typed in a set of 685 patients with RA using sequence-specific polymerase chain reaction. Allele and phenotype frequencies were compared with those in 2 large sets of historical, ethnically matched healthy controls, using the relative predispositional effect method. Results. Positive association was confirmed with the shared epitope positive HLA-DRB1 alleles associated with RA in Caucasians. A significant susceptibility effect was observed with HLA-DRB1*09, described in other ethnically diverse populations but not in Caucasians. A significant underrepresentation of the HLA-DRB1*0103 variant was noted among the RA cases, supporting the proposed protective role of the DERAA motif at residues 70-74 of the DRβ molecule. No HLA-DRB1 independent association of the HLA-DQB1 alleles, implicated in predisposing to RA, was evident. Conclusion. These data corroborate the shared epitope hypothesis of susceptibility to RA and provide strong evidence for the DRB1 locus as the primary RA susceptibility factor in the HLA region.

Arithmetic of cyclotomic fields and Fermat-type equations

Let l be any odd prime, and ζ a primitive l-th root of unity. Let C_l be the l-Sylow subgroup of the ideal class group of Q(ζ). The Teichmüller character w : Z_l → Z^*_l is given by w(x) = x (mod l), where w(x) is a p-1-st root of unity, and x ∈ Z_l. Under the action of this character, C_l decomposes as a direct sum of C^((i))_l, where C^((i))_l is the eigenspace corresponding to w^i. Let the order of C^((3))_l be l^h_3). The main result of this thesis is the following: For every n ≥ max( 1, h_3 ), the equation x^(ln) + y^(ln) + z^(ln) = 0 has no integral solutions (x,y,z) with l ≠ xyz. The same result is also proven with n ≥ max(1,h_5), under the assumption that C_l^((5)) is a cyclic group of order l^h_5. Applications of the methods used to prove the above results to the second case of Fermat's last theorem and to a Fermat-like equation in four variables are given.

The proof uses a series of ideas of H.S. Vandiver ([Vl],[V2]) along with a theorem of M. Kurihara [Ku] and some consequences of the proof of lwasawa's main conjecture for cyclotomic fields by B. Mazur and A. Wiles [MW]. In [V1] Vandiver claimed that the first case of Fermat's Last Theorem held for l if l did not divide the class number h^+ of the maximal real subfield of Q(e^(2πi/i)). The crucial gap in Vandiver's attempted proof that has been known to experts is explained, and complete proofs of all the results used from his papers are given.