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Syftet med uppsatsen är att försöka ta reda på om andraspråkselever har svårare med matematik på grund av att de inte behärskar språket eller om det finns andra orsaker till detta. Hur kan man förebygga och åtgärda detta och hur kan undervisningen anpassas till dessa elever? Mina frågeställningar är: • Har språket en stor betydelse i matematikundervisning? • Hur arbetar lärare respektive speciallärare med elever som har invandrarbakgrund i matematik? • Finns det andra orsakar till svårigheterna att förstå matematik? Jag har gjort intervjuer med tre lärare som alla har flerårig erfarenhet av att undervisa elever med invandrarbakgrund. Resultat visar att den stora svårigheten för de här eleverna, är bristen av språkförståelse och underliggande orsaker, som man bär med sig från sitt liv i ett krigshärjat land. Allt tillsammans skapar en brist på koncentration. Slutsatsen blir: För elever med invandrarbakgrund är det så viktigt att öka förståelsen och begreppsinlärningen att känna till vad ordet betyder ger en ingång till problemet. Bildens betydelse förstärker tänkandet tillsammans med ordets innebörd och bilden förstärker vi inlärningen.

This thesis consists of three parts: In Part I we study the Burnside ring of the finite group G. This ring has a natural structure of a lambda-ring. However, a priori the images of the G-set S under the lambda-operations can only be computed recursively. We establish an explicit formula, expressing these images as linear combination of classes of G-sets. This formula is derived in two ways: First we give a proof that uses the theory of representation rings in an essential way. We then give an alternative, more intrinsic, proof. This second proof is joint work with Serge Bouc. In Part II we establish a formula for the classes of certain tori in the Grothendieck ring of varieties, in terms of its lambda-structure. More explicitly, we will see that if L* is the torus of invertible elements in the n-dimensional separable k-algebra L, then the class of L* can be expressed as an alternating sum of the images of the spectrum of L under the lambda-operations, multiplied by powers of the Lefschetz class. This formula is suggested from the cohomology of the torus, illustrating a heuristic method that can be used in other situations. To prove the formula will require some rather explicit calculations in the Grothendieck ring. To be able to make these we introduce a homomorphism from the Burnside ring of the absolute Galois group of k, to the Grothendieck ring of varieties over k. In the process we obtain some information about the structure of the subring generated by zero-dimensional varieties. In Part III we give a version of geometric motivic integration that specializes to p-adic integration via point counting. This has been done before for stable sets; we extend this to more general sets. The main problem in doing this is that it requires to take limits, hence the measure will have to take values in a completion of the localized Grothendieck ring of varieties. The standard choice is to complete with respect to the dimension filtration. However, since the point counting homomorphism is not continuous with respect to this topology we have to use a stronger one. We thus begin by defining this stronger topology; we will then see that many of the standard constructions of geometric motivic integration work also in this setting. Using this theory, we are then able to give a geometric explanation of the behavior of certain p-adic integrals, by computing the corresponding motivic integrals.

The first subject we investigate in this thesis deals with optimization problems on graphs. The edges are given costs defined by the values of independent exponential random variables. We show how to calculate some or all moments of the distributions of the costs of some optimization problems on graphs. The second subject that we investigate is 1-error correcting perfect binary codes, perfect codes for short. In most work about perfect codes, two codes are considered equivalent if there is an isometric mapping between them. We call this isometric equivalence. Another type of equivalence is given if two codes can be mapped on each other using a non-singular linear map. We call this linear equivalence. A third type of equivalence is given if two codes can be mapped on each other using a composition of an isometric map and a non-singular linear map. We call this extended equivalence. In Paper 1 we give a new better bound on how much the cost of the matching problem with exponential edge costs varies from its mean. In Paper 2 we calculate the expected cost of an LP-relaxed version of the matching problem where some edges are given zero cost. A special case is when the vertices with probability 1 – p have a zero cost loop, for this problem we prove that the expected cost is given by a formula. In Paper 3 we define the polymatroid assignment problem and give a formula for calculating all moments of its cost. In Paper 4 we present a computer enumeration of the 197 isometric equivalence classes of the perfect codes of length 31 of rank 27 and with a kernel of dimension 24. In Paper 5 we investigate when it is possible to map two perfect codes on each other using a non-singular linear map. In Paper 6 we give an invariant for the equivalence classes of all perfect codes of all lengths when linear equivalence is considered. In Paper 7 we give an invariant for the equivalence classes of all perfect codes of all lengths when extended equivalence is considered. In Paper 8 we define a class of perfect codes that we call FRH-codes. It is shown that each FRH-code is linearly equivalent to a so called Phelps code and that this class contains Phelps codes as a proper subset.

The various results presented in this thesis are naturallysubdivided into three different topics, namely combinatorialcomplexes, Bruhat intervals and expected reflection distances.Each topic is made up of one or several of the altogether sixpapers that constitute the thesis. The following are some of ourresults, listed by topic: Combinatorial complexes: Using a shellability argument, we compute the cohomologygroups of the complements of polygraph arrangements. These arethe subspace arrangements that were exploited by Mark Haiman inhis proof of the n! theorem. We also extend these results toDowling generalizations of polygraph arrangements. We consider certainB- andD-analogues of the quotient complex Δ(Πn)=Sn, i.e. the order complex of the partition latticemodulo the symmetric group, and some related complexes.Applying discrete Morse theory and an improved version of knownlexicographic shellability techniques, we determine theirhomotopy types. Given a directed graphG, we study the complex of acyclic subgraphs ofGas well as the complex of not strongly connectedsubgraphs ofG. Known results in the case ofGbeing the complete graph are generalized. We list the (isomorphism classes of) posets that appear asintervals of length 4 in the Bruhat order on some Weyl group. Inthe special case of symmetric groups, we list all occuringintervals of lengths 4 and 5. Expected reflection distances:Consider a random walk in the Cayley graph of the complexreflection groupG(r, 1,n) with respect to the generating set of reflections. Wedetermine the expected distance from the starting point aftertsteps. The symmetric group case (r= 1) has bearing on the biologist’s problem ofcomputing evolutionary distances between different genomes. Moreprecisely, it is a good approximation of the expected reversaldistance between a genome and the genome with t random reversalsapplied to it. / QC 20100616

n this thesis we count the number of points defined over finite fields of certain moduli spaces of pointed curves. The aim is primarily to gain cohomological information. Paper I is joint work with Orsola Tommasi. Here we present details of the method of finding cohomological information on moduli spaces of curves by counting points. Another method of determining the cohomology of moduli spaces of curves is also presented. It is by stratifying them into pieces that are quotients of complements of discriminants in complex vector spaces. Results obtained by these two methods allow us to compute the Hodge structure of the cohomology of $\overline{\mathcal{M}}_4$. In Paper II we consider the moduli space $\mathcal{H}_{g,n}$ of $n$-pointed smooth hyper-elliptic curves of genus $g$. We find that there are recursion formulas in the genus that the numbers of points of $\mathcal{H}_{g,n}$ fulfill. Thus, if we can make $\mathbb{S}_n$-equivariant counts of $\mathcal{H}_{g,n}$ for low genus, then we can do this for every genus. Information about curves of genus zero and one is then found to be sufficient to compute the answers for hyperelliptic curves of all genera and with up to seven points. These results are applied to $\overline{\mathcal{M}}_{2,n}$ for $n$ up to seven, and give us the $\mathbb{S}_n$-equivariant Hodge structure of their cohomology. Moreover, we find that the $\mathbb{S}_n$-equivariant counts of $\mathcal{H}_{g,n}$ depend upon whether the characteristic is even or odd, where the first instance of this dependence is for six-pointed curves of genus three. In Paper III we consider the moduli space $\mathcal{Q}_{n}$ of smooth $n$-pointed nonhyperelliptic curves of genus three. Using the canonical embedding of these curves as plane quartics, we make $\mathbb{S}_n$-equivariant counts of the numbers of points of $\mathcal{Q}_{n}$ for $n$ up to seven. We also count pointed plane cubics. This gives us $\mathbb{S}_n$-equivariant counts of the moduli space $\mathcal{M}_{1,n}$ for $n$ up to ten. We can then determine the $\mathbb{S}_n$-equivariant Hodge structure of the cohomology of $\overline{\mathcal{M}}_{3,n}$ for $n$ up to five. n this thesis we count the number of points defined over finite fields of certain moduli spaces of pointed curves. The aim is primarily to gain cohomological information. Paper I is joint work with Orsola Tommasi. Here we present details of the method of finding cohomological information on moduli spaces of curves by counting points. Another method of determining the cohomology of moduli spaces of curves is also presented. It is by stratifying them into pieces that are quotients of complements of discriminants in complex vector spaces. Results obtained by these two methods allow us to compute the Hodge structure of the cohomology of $\overline{\mathcal{M}}_4$. In Paper II we consider the moduli space $\mathcal{H}_{g,n}$ of $n$-pointed smooth hyper-elliptic curves of genus $g$. We find that there are recursion formulas in the genus that the numbers of points of $\mathcal{H}_{g,n}$ fulfill. Thus, if we can make $\mathbb{S}_n$-equivariant counts of $\mathcal{H}_{g,n}$ for low genus, then we can do this for every genus. Information about curves of genus zero and one is then found to be sufficient to compute the answers for hyperelliptic curves of all genera and with up to seven points. These results are applied to $\overline{\mathcal{M}}_{2,n}$ for $n$ up to seven, and give us the $\mathbb{S}_n$-equivariant Hodge structure of their cohomology. Moreover, we find that the $\mathbb{S}_n$-equivariant counts of $\mathcal{H}_{g,n}$ depend upon whether the characteristic is even or odd, where the first instance of this dependence is for six-pointed curves of genus three. In Paper III we consider the moduli space $\mathcal{Q}_{n}$ of smooth $n$-pointed nonhyperelliptic curves of genus three. Using the canonical embedding of these curves as plane quartics, we make $\mathbb{S}_n$-equivariant counts of the numbers of points of $\mathcal{Q}_{n}$ for $n$ up to seven. We also count pointed plane cubics. This gives us $\mathbb{S}_n$-equivariant counts of the moduli space $\mathcal{M}_{1,n}$ for $n$ up to ten. We can then determine the $\mathbb{S}_n$-equivariant Hodge structure of the cohomology of $\overline{\mathcal{M}}_{3,n}$ for $n$ up to five. n this thesis we count the number of points defined over finite fields of certain moduli spaces of pointed curves. The aim is primarily to gain cohomological information. Paper I is joint work with Orsola Tommasi. Here we present details of the method of finding cohomological information on moduli spaces of curves by counting points. Another method of determining the cohomology of moduli spaces of curves is also presented. It is by stratifying them into pieces that are quotients of complements of discriminants in complex vector spaces. Results obtained by these two methods allow us to compute the Hodge structure of the cohomology of $\overline{\mathcal{M}}_4$. In Paper II we consider the moduli space $\mathcal{H}_{g,n}$ of $n$-pointed smooth hyper-elliptic curves of genus $g$. We find that there are recursion formulas in the genus that the numbers of points of $\mathcal{H}_{g,n}$ fulfill. Thus, if we can make $\mathbb{S}_n$-equivariant counts of $\mathcal{H}_{g,n}$ for low genus, then we can do this for every genus. Information about curves of genus zero and one is then found to be sufficient to compute the answers for hyperelliptic curves of all genera and with up to seven points. These results are applied to $\overline{\mathcal{M}}_{2,n}$ for $n$ up to seven, and give us the $\mathbb{S}_n$-equivariant Hodge structure of their cohomology. Moreover, we find that the $\mathbb{S}_n$-equivariant counts of $\mathcal{H}_{g,n}$ depend upon whether the characteristic is even or odd, where the first instance of this dependence is for six-pointed curves of genus three. In Paper III we consider the moduli space $\mathcal{Q}_{n}$ of smooth $n$-pointed nonhyperelliptic curves of genus three. Using the canonical embedding of these curves as plane quartics, we make $\mathbb{S}_n$-equivariant counts of the numbers of points of $\mathcal{Q}_{n}$ for $n$ up to seven. We also count pointed plane cubics. This gives us $\mathbb{S}_n$-equivariant counts of the moduli space $\mathcal{M}_{1,n}$ for $n$ up to ten. We can then determine the $\mathbb{S}_n$-equivariant Hodge structure of the cohomology of $\overline{\mathcal{M}}_{3,n}$ for $n$ up to five. / QC 20100701

QC 20100706

One way to construct noncommutative analogues of a Riemannian manifold Σ is to make use of the Toeplitz quantization procedure. In Paper III and IV, we construct C-algebras for a continuously deformable class of spheres and tori, and by introducing the directed graph of a representation, we can completely characterize the representation theory of these algebras in terms of the corresponding graphs. It turns out that the irreducible representations are indexed by the periodic orbits and N-strings of an iterated map s:(reals) 2→(reals)2 associated to the algebra. As our construction allows for transitions between spheres and tori (passing through a singular surface), one easily sees how the structure of the matrices changes as the topology changes. In Paper II, noncommutative analogues of minimal surface and membrane equations are constructed and new solutions are presented -- some of which correspond to minimal tori embedded in S7. Paper I is concerned with the problem of finding differential equations for the eigenvalues of a symmetric N × N matrix satisfying Xdd=0. Namely, by finding N(N-1)/2 suitable conserved quantities, the time-evolution of X (with arbitrary initial conditions), is reduced to non-linear equations involving only the eigenvalues of Χ. / QC 20100630

We consider different aspects of free boundary problems that have financial applications. Papers I–III deal with American option pricing, in which case the boundary is called the early exercise boundary and separates the region where to hold the option from the region where to exercise it. In Papers I–II we obtain boundary regularity results by local analysis of the PDEs involved and in Paper III we perform local analysis of the corresponding stochastic representation. The last paper is different in its character as we are dealing with an optimal switching problem, where a switching of state occurs when the underlying process crosses a free boundary. Here we obtain existence and regularity results of the viscosity solutions to the involved system of variational inequalities. / QC 20100630

This thesis on Topological Combinatorics contains 7 papers. All of them but paper Bare published before.In paper A we prove that!i dim ˜Hi(Ind(G);Q) ! |Ind(G[D])| for any graph G andits independence complex Ind(G), under the condition that G\D is a forest. We then use acorrespondence between the ground states with i+1 fermions of a supersymmetric latticemodel on G and ˜Hi(Ind(G);Q) to deal with some questions from theoretical physics.In paper B we generalize the topological Tverberg theorem. Call a graph on the samevertex set as a (d + 1)(q − 1)-simplex a (d, q)-Tverberg graph if for any map from thesimplex to Rd there are disjoint faces F1, F2, . . . , Fq whose images intersect and no twoadjacent vertices of the graph are in the same face. We prove that if d # 1, q # 2 is aprime power, and G is a graph on (d+1)(q −1)+1 vertices such that its maximal degreeD satisfy D(D + 1) < q, then G is a (d, q)–Tverberg graph. It was earlier known that thedisjoint unions of small complete graphs, paths, and cycles are Tverberg graphs.In paper C we study the connectivity of independence complexes. If G is a graphon n vertices with maximal degree d, then it is known that its independence complex is(cn/d + !)–connected with c = 1/2. We prove that if G is claw-free then c # 2/3.In paper D we study when complexes of directed trees are shellable and how one canglue together independence complexes for finding their homotopy type.In paper E we prove a conjecture by Björner arising in the study of simplicial polytopes.The face vector and the g–vector are related by a linear transformation. We prove thatthis matrix is totaly nonnegative. This is joint work with Michael Björklund.In paper F we introduce a generalization of Hom–complexes, called set partition complexes,and prove a connectivity theorem for them. This generalizes previous results ofBabson, Cukic, and Kozlov, and questions from Ramsey theory can be described with it.In paper G we use combinatorial topology to prove algebraic properties of edge ideals.The edge ideal of G is the Stanley-Reisner ideal of the independence complex of G. Thisis joint work with Anton Dochtermann. / QC 20100712