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  • Thesis

  • Amalgamation of inverse semigroups and operator algebras

    • Authors: Haataja, Steven P. (2006)

  • We focus on three constructions: amalgamated free products of inverse semi-groups, C*-algebras of inverse semigroups, and amalgamated free products of C*-algebras. The starting point is an amalgam [S1, S2, U] of inverse semigroups that is full, i.e., the embeddings of U into S1 and S2 are bijective on the semilattice of idempotents. Although the order structure of the amalgamated free product is well-understood, the structure of the maximal subgroups was somewhat mysterious prior to this work. We use Bass-Serre theory to characterize these maximal subgroups and determine which graphs of groups arise in this setting. We obtain necessary and sufficient conditions for the amalgamated free product to have trivial subgroups. One surprising consequence is that an amalgamated free product ...
  • Thesis

  • Decoding algorithms for algebraic geometric codes over rings

    • Authors: Bartley, Katherine (2006)

  • Algebraic geometric codes over rings were defined and studied in the late 1990's by Walker, but no decoding algorithm was given. In this dissertation, we present three decoding algorithms for algebraic geometric codes over rings. The first algorithm presented is a modification of the basic algorithm for algebraic geometric codes over fields, and decodes with respect to the Hamming weight. The second algorithm presented is a modification of the Guruswami-Sudan algorithm, a list decoding algorithm for one-point algebraic geometric codes over fields. This algorithm also decodes with respect to the Hamming weight. Finally, we show how the Koetter-Vardy algorithm, a soft-decision decoding algorithm, can be used to decode one-point algebraic geometric codes over rings of the form Z/prZ, ...
  • Thesis

  • Representations of the symplectic reflection algebras and of the rational Cherednik algebras

    • Authors: Chmutova, Tatyana S. (2006)

  • Given a symplectic reflection group W one can define a symplectic reflec¬tion algebra Hk(W). When W is a complex reflection group, the corresponding algebra is a called rational Cherednik algebra. The representation theory of these algebras is similar to the representation theory of Lie algebras. In par¬ticular, one can define the category 0 of Hk(W)-modules. In the thesis we study the rational Cherednik algebras associated to a sym¬metric group, to a group of the form ST,, x (Z/1Z)n and to a dihedral group. For a symmetric group and for Sn (Z/l7G)n we study the structure of the standard module corresponding to the trivial representation, while for a dihedral group we explicitly describe the structure of all the standard modules. In particular, we compute their Jordan-Holder series...
  • Thesis

  • Some Diophantine Equations with Complex-Valued Exponents

    • Authors: Zuehlke, John Alan (2006)

  • Some Diophantine Equations with Complex-Valued Exponents John Alan Zuehlke This thesis studies solutions to some Diophantine equations with exponents that either are algebraic numbers or can be suitably well approximated by algebraic numbers. For example, it is shown that Fermat's Last Theorem remains valid for Gaussian integer exponents.
  • Thesis

  • Some new non-unimodal level algebras

    • Authors: Weiss, Arthur Jay (2006)

  • In 2005, building on his own recent work and that of F. Zanello, A. Iarrobino discovered some constructions that, he conjectured, would yield level algebras with non-unimodal Hilbert functions. This thesis provides proofs of nonunimodality for Iarrobino’s level algebras, as well as for other level algebras that the author has constructed along similar lines. The key technical contribution is to extend some results published by Iarrobino in 1984. Iarrobino’s results provide insight into some naturally arising vector subspaces of the vector space Rd of forms of fixed degree in a polynomial ring in several variables. In this thesis, the problem is approached by combinatorial methods and results similar to Iarrobino’s are proved for a different class of vector subspaces of Rd. The...
  • Thesis

  • The Algebraic Crossing Number and The Braid Index of Knots and Links

    • Authors: Kawamuro, Keiko (2006)

  • It has been conjectured that the algebraic crossing number of link is uniquely determined in minimal braid representation. This conjecture is true for many classes of knots and links. The Morton-Franks-Williams inequality gives a lower bound for braid index. And sharpness of the inequality on a knot type implies the truth of the conjecture for the knot type. We prove that there are infinitely many examples of knots and links on which the inequality is not sharp, but the conjecture is still true in these cases. We also show that if the conjecture is true for K and G, then it is also true for Kp,q, the (p, q)-cable of K, and for K#G, the connect sum of K and G.
  • Thesis

  • The E(1, 2) cohomology of the Eilenberg-MacLane space K( Z , 3)

    • Authors: Su, Hsin-hao (2006)

  • In this paper, we give new bases of E* (BP (1)4) and E* (BP (1)6) in order to define a basis of E* (BP (1)4) and E* (BP (1)6). With these two new bases, we have a short exact sequence of Hopf algebras E* E* (K (Z, 3)) Ff E* [[y(i) : i > 0] ] 14-1 E* [[x(i) : i 0] ] E*. The action of vi is calculated by using formal group laws and used to compute E* (K (Z, 3)) as a completion of E* [z(o)].
  • Thesis

  • Z-Classes in Central Simple Algebras

    • Authors: Gouraige, Rony (2006)

  • Two elements in a finite-dimensional central simple algebra are said to be z-equivalent if the corresponding centralizers are conjugate. We determine in this thesis the invariants which characterize z-equivalence.