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<p> We study two objects: an ideal of a Hecke algebra, and a pairing in Galois cohomology.</p> <p> Let h be the Hecke algebra of cusp forms of weight 2, level n, and a fixed Dirichlet character modulo n generated by all Hecke operators, where n is an odd prime p or a product of two distinct odd primes N and p. We study the Eisenstein I ideal of h. We wrote a computer program to test whether Up - 1 generates this ideal, where Up is the pth Hecke operator in h. We found many cases of n and the character so that Up - 1 alone generates I. On the other hand, we found one example with N = 3 and p = 331 where Up - 1 does not generate I.</p> <p> Let K = Q(μn) be the nth cyclotomic field. Let S be the set of primes above p in K, and let G_K,S be the Galois group of the maximal extension of K unramified outside S. We study a pairing on cyclotomic p-units that arises from the cup product on H1(G_K,S, μp). This pairing takes values in a Gal(K/Q)-eigenspace of the p-part of the class group of K. Sharifi has conjectured that this pairing is surjective. We studied this pairing in detail by imposing linear relations on the possible pairing values. We discovered many values of n and the character such that these relations single out a unique nontrivial possibility for the pairing, up to a possibly zero scalar.</p> <p> Sharifi showed in [S2] that, under an assumption on Bernoulli numbers, the element Up - 1 generates the Eisenstein ideal I if and only if pairing with the single element p is surjective. In particular, in the instances for which we found a unique nontrivial possibility for the pairing, then if Up - 1 generates I, we know that the scalar up to which it is determined cannot be zero.</p> / Thesis / Master of Science (MSc)

We prove the positivity of Lusztig's q-analogue of weight multiplicity in a purely combinatorial way for rank 2 Lie algebras. Each summand in the polynomial can be interpreted as a linear combination of positive roots. We prove that all negative coefficients are cancelled in the polynomial. Further, the analysis of the root systems allows us to state formulae for every coefficient in Lusztig's q-analogue for rank 2 Lie algebras. / Ph. D.

A software package has been written to compute the Hopf bifurcation structure in functional differential equations. The package is modular, and consists of several routines which perform one or more tasks. In conjunction with the routines available in this package, the user is required to provide a few routines which describe the specific system under analysis. Three example systems (from epidemiology, biochemistry and aerospace engineering) have been analyzed to illustrate the use of this package. / M.S.

A number of significant problems require the solution of a system of linear equations Ax = b in which A is large, highly sparse, nonsymmetric, and unstructured. Several iterative methods which are applicable to nonsymmetric and indefinite problems are applied to a suite of test problems derived from simulations of actual bipolar circuits and to a viscous flow problem. Methods tested include Craig’s method, GMRES(k), BiCGSTAB, QMR, KACZ (a row-projection method) and LSQR. The convergence rates of these methods may be improved by use of a suitable preconditioner. Several such techniques are considered, including incomplete LU factorization (ILU), sparse submatrix ILU, and ILU allowing restricted fill in bands or blocks. Timings and convergence statistics are given for each iterative method and preconditioner. / Master of Science

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<p dir="ltr">A group is said to be matricially (Frobenius) stable if every function from the group to unitary matrices that is "almost multiplicative" in the point operator (Frobenius) norm topology is "close" to a genuine unitary representation in the same topology. A result of Dadarlat shows that for a large class of groups, non-torsion even cohomology obstructs matricial stability. However, the proof doesn't generate explicit almost multiplicative maps that are far from genuine representations. In this paper, we compute explicit almost homomorphisms for all finitely generated groups with a non-torsion 2-cohomology class with a residually finite central extension. We use similar techniques to show that finitely generated nilpotent groups are Frobenius stable if and only if they are virtually cyclic, and that a finitely generated group with a non-torsion 2-cohomology class that can be written as a cup product of two 1-cohomology classes is not Frobenius stable.</p><p><br></p>

Fix R = k[[x]]. Let Q_n be the category whose objects are ((M₁,...,M_n),(f₁,...,f_n-1)) where each M_i is a free R-module and f_i:M_i⟶M_i+1 for each i=1,...,n-1, and in which the morphisms are the obvious ones. Let β_n be the full subcategory of Ω_n in which each map f_i is a monomorphism whose cokernel is a torsion module. It is shown that there is a full dense functor Ω_n⟶β_n. If X is an object of β_n, we say that X <u>diagonalizes</u> if X is isomorphic to a direct sum of objects ((M₁,...,M_n),(f₁,...,f_n-1)) in which each M_i is of rank one. We establish an algorithm which diagonalizes any diagonalizable object X of β_n, and which fails only in case X is not diagonalizable. Let Λ be an artin algebra of finite type. We prove that for a fixed C in mod(Λ) there are only finitely many modules A in mod(Λ) (up to isomorphism) for which a short exact sequence of the form 0⟶A⟶B⟶C⟶0 is indecomposable. / Ph. D. / incomplete_metadata

Let (Γ,ρ) be a directed graph with relations. Let F: Γ’ → Γ be a topological covering. It is proved in this thesis that there is a set of relations ρ̅ on Γ such that the category of K-respresentations of Γ’ whose images under the covering functor satisfy ρ is equivalent to the category of finite-dimensional, grades KΓ/<ρ̅>-modules. If Γ’ is the universal cover of Γ, then this category is called the category of unwindable KΓ/<ρ>-modules. For arrow unique graphs it is shown that the category of unwindable KΓ/<ρ>-modules does not depend on <ρ>. Also, it is shown that for arrow unique graphs the finite dimensional uniserial KΓ/<ρ>-modules are unwindable. Let Γ be an arrow unique graph with commutativity relations, ρ. In Section 2, the concept of unwindable modules is used to determine whether a certain factor ring of KΓ/<ρ> is of finite representation type. In a different vein, the relationship between almost split sequences over Artin algebras and the almost split sequences over factor rings of such algebras is studied. Let Λ be an Artin algebra and let Λ̅ be a factor ring of Λ. Two sets of necessary and sufficient conditions are obtained for determining when an almost split sequence of Λ̅-modules remains an almost split sequence when viewed as a sequence of Λ-modules. / Ph. D.