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.

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.

Automorphisms of irreducible holomorphic symplectic manifolds and related problems

We study automorphisms and the mapping class group of irreducible holomorphic symplectic (IHS) manifolds. We produce two examples of manifolds of K3[2] type with a symplectic action of the alternating group A7. Our examples are realized as double EPW-sextics, the large cardinality of the group allows us to prove the irrationality of the associated families of Gushel-Mukai threefolds. We describe the group of automorphisms of double EPW-cubes. We give an answer to the Nielsen realization problem for IHS manifolds in analogy to the case of K3 surfaces, determining when a finite group of mapping classes fixes an Einstein (or Kähler-Einstein) metric. We describe, for some deformation classes, the mapping class group and its representation in second cohomology. We classify non-symplectic involutions of manifolds of OG10 type determining the possible invariant and coinvariant lattices. We study non-symplectic involutions on LSV manifolds that are geometrically induced from non-symplectic involutions on cubic fourfolds.

Introduzione al Gruppo delle Trecce

L'obiettivo di questa tesi è fornire un'introduzione al gruppo delle trecce. In particolare, tale gruppo verrà presentato prima come gruppo astratto, poi tramite l'ausilio di trecce geometriche, mostrandone successivamente alcune proprietà algebriche. Viene infine trattato il legame tra il gruppo delle trecce e altri due gruppi, il mapping class group del disco bidimensionale puntato e il gruppo degli automorfismi treccia del gruppo libero. Nell'appendice si può trovare una breve introduzione al calcolo del gruppo fondamentale (teorema di Van Kampen).

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)).

Aprender com os outros : uma estratégia para a inclusão de um aluno com Autismo

Este Projecto de Intervenção, “Aprender com os outros - uma estratégia para a inclusão de um aluno com autismo”, fundamentado nos pressupostos e nos procedimentos da investigação-acção, centrou-se nas acções em áreas de maior e menor sucesso do aluno, de nome fictício “Francisco”, no âmbito da língua portuguesa e da socialização, numa perspectiva inclusiva. Este aluno considerado com necessidades educativas especiais (NEE) apresentava perturbações do espectro do autismo (PEA), o que, à partida, se repercutia no seu défice de atenção, na autonomia para a realização das tarefas escolares, na área da linguagem e da comunicação e na interacção social. Como as interacções na turma e com a turma são essenciais para a aprendizagem, propusemo-nos implementar actividades específicas para o desenvolvimento das competências sociais e cognitivas, com abordagem comportamentalista, numa turma do 3º ano de escolaridade, onde estava incluído um aluno diagnosticado com PEA. Também procurámos desenvolver as suas competências académicas, através do trabalho realizado no grupo e com o grupo-turma, criando as condições que favorecessem a socialização do aluno e a sua autonomia. Para atingirmos aqueles objectivos, iniciámos um trabalho a pares e depois em pequenos grupos, para desta forma incluir o “Francisco” na dinâmica das aulas, para que participasse nas actividades propostas, obtendo o respeito dos colegas na valorização das suas intervenções e do seu ritmo de trabalho. Os objectivos definidos, bem como as actividades realizadas e avaliadas, implicando todos os intervenientes no processo, permitiram que o “Francisco” fizesse aprendizagens significativas nas áreas, académica, social, da autonomia e da comunicação. Segundo Silva (2009), a inclusão dos alunos considerados com necessidades educativas especiais no ensino regular implica mudanças ao nível das atitudes e das práticas pedagógicas de todos os intervenientes no processo ensino e aprendizagem, da organização e da gestão na sala de aula e na própria escola enquanto instituição. Acreditamos que só desta forma se pode proporcionar aos alunos marcados pela diferença, que é um valor em si mesma (Rodrigues, 2006; Leitão, 2006; Sanches & Teodoro, 2006; Silva, 2009), as mesmas experiências, aprendizagens e vivências que são proporcionadas aos restantes colegas.

Inclusão e currículo - as práticas curriculares dos professores dos 2º e 3º ciclos do ensino básico, decorrentes da inclusão de alunos com N.E..E.

Dissertação de mestrado em ciências da educação especialidade educação especial

Conceções alternativas de alunos do 2.º ano de escolaridade sobre a reprodução humana

Relatório de Estágio apresentado à Escola Superior de Educação de Lisboa para obtenção de grau de mestre em Ensino do 1.º e 2.º ciclo do Ensino Básico

Growing up in Ireland. Key findings: infant cohort (at 3 years).

 The latest results from the study paint a picture of how these families are faring across a range of areas in their lives including their health, family life and financial and economic circumstances. In general the findings show that three-year-olds in Ireland are in good health with a few notable public health and related issues (including overweight and obesity), there is overall stability in family structures over the short term and that the recession has had a substantial effect on families with young children over the last number of years. These are the first longitudinal findings from the study. The first wave of fieldwork with the families of the Infant Cohort included approximately 11,100 nine-month-olds, their parents and carers. Interviews began in September 2008 and were completed in March 2009. Interviews for the second round of interviews with this cohort took place between January and August 2011. A total of 90% of the original sample of nine-month-olds were successfully re-interviewed. (A full download of the results released today, presented in three briefing documents can be found by clicking here. Key findings include: Health â?¢ Most of the children were described as being in good health; 75% were rated as very healthy and a further 23% were rated as healthy, but a few minor problems. Girls were more likely to be reported as very healthy (78%) compared with boys (72%). â?¢ One in four or almost one quarter of three-year-old children were overweight (19%) or obese (6%). â?¢ Childrenâ?Ts weight was related to household social class. 5% of children in families in the professional/managerial group were classified as obese at three years of age compared with 9% of those in the most disadvantaged social class group. However, at least one-fifth of children in every social class were overweight. â?¢ Childrens consumption of energy-dense foods such as crisps, sweets, chips, and non-diet fizzy drinks increased as parental education fell. 63% of children whose mother had a lower secondary education or less ate at least one portion of crisps compared with 36% of those from degree-level backgrounds, although consumption of biscuits/chocolates was over 70% for both groups of children. â?¢ Two-thirds (66%) of three-year-olds had received at least one course of antibiotics in the 12 months preceding the interview. Children with a full medical card (35% of the sample) or a GP-only medical card (5% of the sample) were more likely to have received a course of antibiotics than â?¢ Children with a full medical card received a higher number of antibiotic courses on average (2.6) compared with those without a medical card (2.1). â?¢ Just under 16% of three-year-old children were reported as having at least one longstanding illness, condition or disability. The most commonly reported illness types included Asthma (5.8%), Eczema/Skin allergies (3.9%) and Food/digestive allergies (1.2%) Family Life and Childcare â?¢ While the overall distribution of family structure was stable, there have been transitions from one-parent families to two-parent families and vice-versa over the 27 months between interview â?" approximately 2 to 3 percent in each direction. â?¢ 50% of three year olds were in some form of non-parental childcare for eight or more hours a week. The most common form used was centre-based childcare which almost tripled between nine months and three years, from 11% to 30%. â?¢ A similar percentage of grandparents were caring for children at both nine months and three years, 12% and 11% respectively. A total of 10% of three-year-olds were being minded by a childminder, an increase of 3 percentage points from when the children were nine months of age. â?¢ Children who were in some form of non-parental childcare were spending an average of 23 hours a week in their main type of childcare. â?¢ At time of interview the vast majority of mothers reported that they had regular contact with the Study Childâ?Ts grandparents (91%). In offering support to parents, grandparents were most likely to babysit (50%), and buy clothes (40%) at least on a monthly basis. One-parent families were more likely than two-parent families to receive financial support from grandparents with just under one-third (66%) of one-parent families receiving financial support from grandparents at least once every three months. â?¢ The most frequently used discipline technique was â?~discussing or explaining why the behaviour was wrongâ?T, with 63% of mothers saying they always did this. â?¢ 12% of mothers said they used â?~smackingâ?T as a form of discipline now and again and less than 1% used â?~smackingâ?T as a form of discipline more frequently. Over half reported that they never smacked the Study Child. Financial and Economic Circumstances â?¢ Just over half (53%) of mothers of three-year-olds worked outside the home, 38% said they were on home duties and 6% said they were unemployed. â?¢ The biggest change in terms of the work status of three-year-oldsâ?T parents was an increase in the percentage of unemployed fathers â?" 6% when the child was nine months rising to almost 14% when s/he was three years of age. â?¢ 61% of families of three-year-olds reported experiencing difficulties in making â?~ends meetâ?T. This was a substantial increase from 44% in the first round of interviews when the children were nine-months-old. â?¢ Almost two thirds (63%) of all families with three-year-olds reported that the recession had had a very significant or significant effect on them. â?¢ The most frequently recorded effects were: a reduction in wages (63%); canâ?Tt afford luxuries (54%), social welfare reduction (53%) and canâ?Tt afford/cut back on basics (32%). Growing Up in Ireland is a Government funded study tracking the development of two nationally representative cohorts of children: an Infant Cohort which was interviewed initially at nine months and subsequently at three years of age; and a Child Cohort which was interviewed initially at nine years and subsequently at 13 years of age. The study is being conducted by a consortium of researchers led by the Economic and Social Research Institute (ESRI) and Trinity College Dublin. For Further Information Please Contact: Jillian Heffernan Communications Officer, Growing Up in Ireland Tel: 01 896 3378 Mobile: 087 9016880This resource was contributed to our repository by the National Documentation Centre on Drug Use.