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*Thesis*
Authors: Yuncken, Robert; Co-Author: 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 complex of homogeneous bundles over G/B. The reasons for considering this complex are explained in detail.
For G = SL(3, C), the space G/B admits two canonical fibrations, which play a recurring role in the analysis to follow. The local geometry of G/B can be modeled on the geometry of the three-dimensional complex Heisenberg group H in a very strong way. Consequently, we study the algebra of differential operators on H. We define a two-parameter family H(m,n)(H) of Sobolev-like spaces, using the two fibrations of G/B.
We introduce fibrewise Laplacian operators OX and AY on H. We show that these operators satisfy a kind of directional ellipticity in terms of the spaces H (m,n)(H) for
certain values of (m, n), but also provide a counterexample to this property for another choice of (m, n). This counterexample is a significant obstacle to a pseudodifferential approach to the 'y-element for SL(3, C).
Instead we turn to the harmonic analysis of the compact subgroup K = SU(3). Here, using the simultaneous spectral theory of the K-invariant fibrewise Laplacians on G/B, we construct a C*-category A and ideals KX and KY which are related to the canonical fibrations. We explain why these are likely natural homes for the operators which would appear in a construction of the 'y-element.) |

*Thesis*
Authors: Carter, Andrea C.; Co-Author: 2006 (Let S1 be a Del Pezzo surface of degree one over a number field k, and let S1 denote the base-change of S1 over the algebraic closure k of k. We establish a criterion for the existence of a non-trivial element of order five in the Brauer group of S1 in terms of certain Galois stable configurations of exceptional divisors on this surface.) |

*Thesis*
Authors: Durocher, Stephane; Co-Author: 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 between these by the Euclidean distance metric.
In this thesis, we examine these problems in the mobile setting. A problem instance consists of a set of mobile clients, each following a continuous trajectory through Euclidean space under bounded velocity. The positions of the mobile Euclidean k-centre and k-median are defined as functions of the instantaneous positions of the clients. Since mobile facilities located at the exact Euclidean k-centre or k-median involve either unbounded velocity or discontinuous motion, we explore approximations to these. The goal is to define a set of functions, corresponding to positions for the set of mobile facilities, that provide a good approximation to the Euclidean k-centre or k-median while maintaining motion that is continuous and whose magnitude of velocity has a low fixed upper bound. Thus, the fitness of a mobile facility is determined not only by the quality of its optimization of the objective function but also by the maximum velocity and continuity of its motion. These additional constraints lead to a trade-off between velocity and approximation factor, requiring new approximation strategies quite different from previous static approximations.
We identify existing functions and introduce new functions that provide bounded-velocity approximations of the mobile Euclidean 1-centre, 2-centre, and 1-median. We show that no bounded-velocity approximation of the Eu¬clidean 3-centre or the Euclidean 2-median is possible. Finally, we present kinetic algorithms for maintaining these various functions using both exact and approximate solutions.) |

*Thesis*
Authors: McCarthy, Anne E.; Co-Author: 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 close to the identity. We show that any action of FA on a surface with non-zero Euler characteristic has a global fixed point. Also, we show that for any compact manifold M, there are no faithful actions of the Baumslag-Solitar group generated by diffeomorphisms close to the identity.) |

*Thesis*
Authors: Navilarekallu, Tejaswi; Co-Author: 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: Sakellaridis, Loannis; Co-Author: 2006 (The description of irreducible representations of a group G can be seen as a question in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of G x G by left and right multiplication.
For a split p-adic reductive group G over a local non-archimedean field, unramified irreducible smooth representations are in bijection with semisimple conjugacy classes in the "Langlands dual" group.
In the first part of this work, we generalize this description to an arbitrary spherical variety X of G as follows: Irreducible quotients of the "unramified" Bernstein component of C°°(X) (or, more generally, CC°(X, LT), where L,y is a G-linear line bundle over X) are in natural almost bijection with (a number of copies of) the quotient of a complex torus by a finite reflection group (the "little Weyl group" of X, if LT is trivial). This leads to a weak analog of results of D. Gaitsgory and D. Nadler on the Hecke module of unramified vectors, and an understanding of the phenomenon that representations "distinguished" by certain subgroups are functorial lifts. In the course of the proof, rationality properties of spherical varieties are examined and a new interpretation is given for the action, defined by F. Knop, of the Weyl group on the set of Borel orbits.
In the second part of this work, we examine the dual problem of embedding an irreducible unramified representation 7r into the space C°°(X, LT); the goal here is to compute an explicit formula for the image of the unramified vector. We develop a variant of the Casselman-Shalika method, extending work of Y. Hironaka, in order to compute such a formula for the Shalika model of GLn, which is known to distinguish lifts from odd orthogonal groups.) |

*Thesis*
Authors: Takeda, Shuichiro; Co-Author: 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.) |

*Thesis*
Authors: Ahmadov, Anar; Co-Author: 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 in our proofs are knot surgery and rational-blowdown introduced by Fintushel-Stern.
Next we discuss surface bundles over surfaces with lion-zero signature. Using
the Lefschetz fibration structure on the knot surgered elliptic surfaces E(n)h, recent result by Fintushel-Stern, and a gluing technique, we give a new construction that yields the infinitely many such surface bundles with non-zero signature.) |

*Thesis*
Authors: Clement, Anthony E.; Co-Author: 2006 (-) |

*Thesis*
Authors: Arnold, Trevor S.; Co-Author: 2006 (-) |