We study the power series ring R= K[[x1,x2,x3,...]]on countably infinitely many variables, over a field K, and two particular K-subalgebras of it: the ring S, which is isomorphic to an inverse limit of the polynomial rings in finitely many variables over K, and the ring R', which is the largest graded subalgebra of R. Of particular interest are the homogeneous, finitely generated ideals in R', among them the generic ideals. The definition of S as an inverse limit yields a set of truncation homomorphisms from S to K[x1,...,xn] which restrict to R'. We have that the truncation of a generic I in R' is a generic ideal in K[x1,...,xn]. It is shown in <b>Initial ideals of Truncated Homogeneous Ideals</b> that the initial ideal of such an ideal converge to the initial ideal of the corresponding ideal in R'. This initial ideal need no longer be finitely generated, but it is always locally finitely generated: this is proved in <b>Gröbner Bases in R'</b>. We show in <b>Reverse lexicographic initial ideals of generic ideals are finitely generated</b> that the initial ideal of a generic ideal in R' is finitely generated. This contrast to the lexicographic term order. If I in R' is a homogeneous, locally finitely generated ideal, and if we write the Hilbert series of the truncated algebras K[x1,...,xn] module the truncation of I as qn(t)/(1-t)n, then we show in <b>Generalized Hilbert Numerators </b>that the qn's converge to a power series in t which we call the generalized Hilbert numerator of the algebra R'/I. In <b>Gröbner bases for non-homogeneous ideals in R'</b> we show that the calculations of Gröbner bases and initial ideals in R' can be done also for some non-homogeneous ideals, namely those which have an associated homogeneous ideal which is locally finitely generated. The fact that S is an inverse limit of polynomial rings, which are naturally endowed with the discrete topology, provides S with a topology which makes it into a complete Hausdorff topological ring. The ring R', with the subspace topology, is dense in R, and the latter ring is the Cauchy completion of the former. In <b>Topological properties of R'</b> we show that with respect to this topology, locally finitely generated ideals in R'are closed.

<p>We define a moduli functor parametrizing finite maps from a projective (locally) Cohen-Macaulay curve to a fixed projective space. The definition of the functor includes a number of technical conditions, but the most important is that the map is almost everywhere an isomorphism onto its image. The motivation for this definition comes from trying to interpolate between the Hilbert scheme and the Kontsevich mapping space. The main result is that our functor is represented by a proper algebraic space. As applications we obtain a new proof of the existence of Macaulayfications for varieties, and secondly, interesting compactifications of the spaces of smooth curves in projective space. We illustrate this in the case of rational quartics, where the resulting space appears easier than the Hilbert scheme.</p>

<p>Geometri är ett matematiskt område som visat sig vara svårt för svenska elever. Denna undersökning syftar till att ta reda på vilka aspekter på geometri elever i år nio, med särskilt stöd i matematik, behöver utveckla för att nå godkända resultat. Den söker även svar på vilka faktorer som påverkar eleven i inlärningssituationer.</p><p>Empirin inhämtades genom sju semistrukturerade intervjuer. Resultatet visar att begreppsförståelse, formelhantering och förmåga att kommunicera är aspekter som måste fördjupas för att målen i matematik ska nås. De mest frekventa påverkansfaktorer som angavs i inlärningssituationer var läraren, tiden, kompisar och läromedlet.</p>

We introduce the definition of 1-cotilting object in a Grothendieck category and investigate its relation to the analogue of the standard definition of 1-cotilting module. The 1-cotilting quasi-coherent sheaves on a Noetherian scheme are stud- ied in particular: using the classification of hereditary torsion pairs in the category of quasi-coherent sheaves on a Noetherian scheme X, to each hereditary torsion- free class F that is generating we assign a 1-cotilting quasi-coherent sheaf whose 1-cotilting class is F. This provides a family of pairwise non-equivalent 1-cotilting quasi-coherent sheaves which are parametrized by specialization closed subsets of X avoiding the set of associated points of a chosen generator of the category of quasi-coherent sheaves. In many cases (e.g. for separated schemes), this set of avoided points can be chosen as the set of associated points of the scheme. 1

<p>A compact Riemann surface can be realized as a quotient space $\mathcal{U}/\Gamma$, where $\mathcal{U}$ is the sphere $\Sigma$, the euclidian plane $\mathbb{C}$ or the hyperbolic plane $\mathcal{H}$ and $\Gamma$ is a discrete group of automorphisms. This induces a covering $p:\mathcal{U}\rightarrow\mathcal{U}/\Gamma$.</p><p>For each $\Gamma$ acting on $\mathcal{H}$ we have a polygon $P$ such that $\mathcal{H}$ is tesselated by $P$ under the actions of the elements of $\Gamma$. On the other hand if $P$ is a hyperbolic polygon with a side pairing satisfying certain conditions, then the group $\Gamma$ generated by the side pairing is discrete and $P$ tesselates $\mathcal{H}$ under $\Gamma$.</p>

Geometri är ett matematiskt område som visat sig vara svårt för svenska elever. Denna undersökning syftar till att ta reda på vilka aspekter på geometri elever i år nio, med särskilt stöd i matematik, behöver utveckla för att nå godkända resultat. Den söker även svar på vilka faktorer som påverkar eleven i inlärningssituationer. Empirin inhämtades genom sju semistrukturerade intervjuer. Resultatet visar att begreppsförståelse, formelhantering och förmåga att kommunicera är aspekter som måste fördjupas för att målen i matematik ska nås. De mest frekventa påverkansfaktorer som angavs i inlärningssituationer var läraren, tiden, kompisar och läromedlet.

A compact Riemann surface can be realized as a quotient space $\mathcal/\Gamma$, where $\mathcal$ is the sphere $\Sigma$, the euclidian plane $\mathbb$ or the hyperbolic plane $\mathcal$ and $\Gamma$ is a discrete group of automorphisms. This induces a covering $p:\mathcal\rightarrow\mathcal/\Gamma$. For each $\Gamma$ acting on $\mathcal$ we have a polygon $P$ such that $\mathcal$ is tesselated by $P$ under the actions of the elements of $\Gamma$. On the other hand if $P$ is a hyperbolic polygon with a side pairing satisfying certain conditions, then the group $\Gamma$ generated by the side pairing is discrete and $P$ tesselates $\mathcal$ under $\Gamma$.

This thesis is concerned with the existence of pushouts in two different settings of algebraic geometry. At first, we study the pushouts in the cat- egory of affine algebraic sets over an infinite field. We show that this can be regarded as an instance of much general problem whether the pullback of finitely generated algebras over a commutative Noetherian ring is finitely generated. We give a partial solution to this problem and study some ex- amples. Secondly, we examine the existence of pushouts in the category of schemes with an emphasis on diagrams of affine schemes. We use the methods of Ferrand [2003] and Schwede [2004] and generalise some of their results. We conclude by giving some examples and suggest another approach to the problem.

Let G be a finite graph with vertex set V and edge set E. A graph complex on G is an abstract simplicial complex consisting of subsets of E. In particular, we may interpret such a complex as a family of subgraphs of G. The subject of this thesis is the topology of graph complexes, the emphasis being placed on homology, homotopy type, connectivity degree, Cohen-Macaulayness, and Euler characteristic. We are particularly interested in the case that G is the complete graph on V. Monotone graph properties are complexes on such a graph satisfying the additional condition that they are invariant under permutations of V. Some well-studied monotone graph properties that we discuss in this thesis are complexes of matchings, forests, bipartite graphs, disconnected graphs, and not 2-connected graphs. We present new results about several other monotone graph properties, including complexes of not 3-connected graphs and graphs not coverable by p vertices. Imagining the vertices as the corners of a regular polygon, we obtain another important class consisting of those graph complexes that are invariant under the natural action of the dihedral group on this polygon. The most famous example is the associahedron, whose faces are graphs without crossings inside the polygon. Restricting to matchings, forests, or bipartite graphs, we obtain other interesting complexes of noncrossing graphs. We also examine a certain "dihedral" variant of connectivity. The third class to be examined is the class of digraph complexes. Some well-studied examples are complexes of acyclic digraphs and not strongly connected digraphs. We present new results about a few other digraph complexes, including complexes of graded digraphs and non-spanning digraphs. Many of our proofs are based on Robin Forman's discrete version of Morse theory. As a byproduct, this thesis provides a loosely defined toolbox for attacking problems in topological combinatorics via discrete Morse theory. In terms of simplicity and power, arguably the most efficient tool is Forman's divide and conquer approach via decision trees, which we successfully apply to a large number of graph and digraph complexes. / QC 20100622

On the category of representations of a given quiver we define a tensor product point-wise and arrow-wise. The corresponding Clebsch-Gordan problem of how the tensor product of indecomposable representations decomposes into a direct sum of indecomposable representations is the topic of this thesis. The choice of tensor product is motivated by an investigation of possible ways to modify the classical tensor product from group representation theory to the case of quiver representations. It turns out that all of them yield tensor products which essentially are the same as the point-wise tensor product. We solve the Clebsch-Gordan problem for all Dynkin quivers of type A, D and E6, and provide explicit descriptions of their respective representation rings. Furthermore, we investigate how the tensor product interacts with Galois coverings. The results obtained are used to solve the Clebsch-Gordan problem for all extended Dynkin quivers of type Ãn and the double loop quiver with relations βα=αβ=αn=βn=0.