This thesis concerns the study of chiral algebras over schemes of arbitrary dimension n. In Chapter I, we construct a chiral algebra over each smooth variety X of dimension n. We do this via the Hilbert scheme of points of X, which we use to build a factorisation space over X. Linearising this space produces a factorisation algebra over X, and hence, by Koszul duality, the desired chiral algebra. We begin the chapter with an overview of the theory of factorisation and chiral algebras, before introducing our main constructions. We compute the chiral homology of our factorisation algebra, and show that the D-modules underlying the corresponding chiral algebras form a universal D-module of dimension n. In Chapter II, we discuss the theory of universal D-modules and OO- modules more generally. We show that universal modules are equivalent to sheaves on certain stacks of étale germs of n-dimensional varieties. Furthermore, we identify these stacks with the classifying stacks of groups of automorphisms of the n-dimensional disc, and hence obtain an equivalence between the categories of universal modules and the representation categories of these groups. We also define categories of convergent universal modules and study them from the perspectives of the stacks of étale germs and the representation theory of the automorphism groups.

The purpose of the research study is to investigate the effect of peer instruction method on the 8th grade students&rsquo / mathematics achievement and mathematics attitudes in transformation geometry (fractals, rotation, reflection, translation) in crowded classrooms (more than 50 students). Besides, in this study it was aimed to investigate the gender differences regarding mathematics achievement and mathematics attitude. The study was conducted during the academic year 2009-2010. The sample was consisted of 112 eighth grade students from a public elementary school in K&uuml / &ccedil / &uuml / k&ccedil / ekmece district in Istanbul. Two classes, instructed by the researcher, were randomly assigned as experimental and control groups. The experimental group students were taught the subject transformation geometry through peer instruction method, while the control group students were taught the subject transformation geometry conventionally. Mathematics Achievement Test (MAT) and Attitude towards Mathematics Scale (ATMS) were administered to students as measuring instruments. The two-way ANCOVA and two-way ANOVA statistical techniques were performed in order to answer to the research questions. Results indicated that the peer instruction method has significant positive effects on students&rsquo / mathematics achievement and attitudes towards mathematics. Also, it was shown that there is not a significant difference between the female and male students&rsquo / mathematics achievement and mathematics attitudes.

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