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  • Thesis

  • Authors: Yuncken, Robert (2006)

  • If G is a connected Lie group, the Kasparov representation ring KKG(C, C) contains a singularly important element—the 'y-element—which is an idempotent relating the Kasparov representation ring of G with the representation ring of its maximal compact subgroup K. In the proofs of the Baum-Connes conjecture with coefficients for the groups G = SO0(n, 1) ([Kas84]) and G = SU(n, 1) ([JK95]), a key component is an explicit construction of the 'y-element as an element of G-equivariant K-homology for the space G/B, where B is the Borel subgroup of G. In this thesis, we describe some analytical constructions which may be useful for such a construction of -y in the case of the rank-two Lie group G = SL(3, C). The inspiration is the Bernstein-Gel’fand-Gel’fand complex—a natural differential ...

  • Thesis

  • Authors: Durocher, Stephane (2006)

  • The traditional problems of facility location are defined statically; a set (or multiset) of n points is given as input, corresponding to the positions of clients, and a solution is returned consisting of set of k points, corresponding to the positions of facilities, that optimizes some objective function of the input set. In the k-centre problem, the objective is to select k points for locating facilities such that the maximum distance from any client to its nearest facility is mini¬mized. In the k-median problem, the objective is to select k points for locating facilities such that the average distance from each client to its nearest facility is minimized. A common setting for these problems is to model clients and facilities as points in Euclidean space and to measure distances b...

  • Thesis

  • Authors: Howse, Samuel (2006)

  • NummSquared Explained is the thesis version of the comprehensive formal docu¬ment NummSquared 2006a0 Done Formally, which is available at http / /nummist . com/poohbist/. Set theory is the standard foundation for mathematics, but often does not include rules of reduction for function calls. Therefore, for computer science, the untyped lambda calculus or type theory is usually preferred. The untyped lambda calculus (and several improvements on it) make functions fundamental, but suffer from non-terminating reductions and have partially non-classical logics. Type theory is a good foundation for logic, mathematics and computer science, except that, by making both types and functions fundamental, it is more complex than either set theory or the un¬typed lambda calculus. This document p...

  • Thesis

  • Authors: Navilarekallu, Tejaswi (2006)

  • For a finite Galois extension K/Q of number fields with Galois group G and a motive M = M' h°(Spec(K))(O) with coefficients in Q[G], the equivariant Tamagawa number conjecture relates the special value L* (M, 0) of the motivic L-function to an element of Ko(Z[G];R) constucted via complexes associated to M. The conjecture for nonabelian groups G is very much unexplored. In this thesis, we will develop some techniques to verify the conjecture for Artin motives and motives attached to elliptic curves. In particular, we consider motives h°(Spec(K))(0) for an A4-extension K/Q and, hi (E x Spec(L))(1) for an S3-extension L/Q and an elliptic curve E/Q.

  • Thesis

  • Authors: Pasol, Vicentiu (2006)

  • In this thesis I study the relation between the (CM) Elliptic Construction and the Mod¬ular Construction of p-adic L-functions in two variables which interpolate special values of Hecke L-functions. The first construction, due to N. Katz, and then R. Yager, is very explicit in terms of elliptic units using Iwasawa theory on formal groups. However, this construction does not imply the construction of a modular symbol for which the attached L-function is exactly this p-adic L-function. The Modular construction, developed by R. Greenberg and G. Stevens, is more general. They attach to any Hida family of p-stabilized modular forms a p-adic modular symbol with values in the space of two variable p-adic distributions. However, this construction is not explicit. We use p-adic methods to...

  • Thesis

  • Authors: McCarthy, Anne E. (2006)

  • This thesis investigates dynamical properties of actions of abelian-by-cyclic groups on compact manifolds. For a non-singular integer matrix A, let FA be the fundamental group of the mapping cylinder of the induced map fA on the torus T n. The standard actions p), of FA on the circle RP 1 are generated by maps f(x) = Ax and gi(x) = x + bi, where A is a real-valued eigenvalue for A, and (01, ..., 0n) is the associated eigenvector. It is known that any analytic action of FA on the circle is a ramified lift of one of the standard actions p),. This thesis shows that for each analytic action, p, there exists R > 2 such that p is Cr locally rigid for all r > R. We then consider actions of the groups FA on compact manifolds of higher dimension that are generated by C1 diffeomorphisms clos...

  • Thesis

  • Authors: Ahmadov, Anar (2006)

  • We investigate the exotic smooth structures on 4-manifolds with small Euler characteristic. Using the surgical techniques of knot surgery and fiber sum, we construct new symplectic 4-manifolds with zero signature and cohomology equivalent to S2 x S2. In fact, a more general construction provides exotic manifolds with zero signature and cohomology equivalent to #(29_1)(S2 x S2) for any g. Applying the surgical methods of Fintushel-Stern [FS4], we also construct simply connected nonsymplectic 4-manifolds starting from the elliptic surfaces E(n) for n > 3. Seiberg-Witten invariants are used to distinguish their smooth structures. «'e also give a uniform technique that yields non-symplectic manifolds homeomorphic CP 2#k CP 2 for k = 5, 6 and also 3CP 2#8 CP 2. The main construction used...

  • Thesis

  • Authors: Takeda, Shuichiro (2006)

  • We, firstly, improve a theorem of B. Roberts which characterizes non-vanishing of a global theta lift from 0(X) to Sp(n) in terms of non-vanishing of local theta lifts. In particular, we will remove all the archimedean conditions imposed upon his theorem. Secondly, we will apply our theorem to theta lifting of low rank similitude groups as Roberts did so. Namely we characterize the non-vanishing condition of a global theta lift from GO(4) to GSp(2) in our improved setting. Also we consider non-vanishing conditions of a global theta lift from GO(4) to GSp(1) and explicitly compute the lift when it exists.