Generalised eta invariants, end-periodic manifolds, and their applications to positive scalar curvature

This thesis studies the applications of index theory to positive scalar curvature (PSC), in particular questions of existence and number of path components of the moduli space of PSC metrics. After Atiyah-Singer proved their legendary index theorem [AS63, AS68a, AS68b], many fruitful applications to positive scalar curvature were discovered, for instance Lichnerowicz's obstruction to positive scalar curvature metrics on spin manifolds with nonvanishing A-hat genus [Lic63]. The theorem of Lichnerowicz relies notably on the existence of a spin Dirac operator on any spin manifold|a self-adjoint, elliptic, first order differential operator having marvellous connections to the geometry of the underlying Riemannian manifold. In 1975, Atiyah, Patodi and Singer [APS75a, APS75b, APS76] proved their index theorem for a Dirac operator D on a manifold Z with boundary @Z = Y . This takes a similar form to the Atiyah-Singer index theorem but notably has a correction term n(A) = (nA(0)+h)=2, called the eta invariant, appearing for the boundary. The eta invariant is defined solely in terms of the spectrum of the Dirac operator A on the boundary, so is a spectral invariant. As it stands, this invariant is not at all robust; if one slightly perturbs the metric on Y then most likely one will produce a change in the eta invariant. A notable exception to this is conformal deformations, which leave the Dirac operator unchanged. There is, however, a more robust invariant which can be procured from the eta invariant. If one twists the Dirac operator A on Y by two unitary representations δ1; δ2 : π1(Y ) → U(N) of the fundamental group of Y , one obtains two twisted Dirac operators A1 and A2 on Y which are locally isomorphic. Subtracting the eta invariants of these twisted Dirac operators yields a modified invariant p(δ1; δ2;A), called the rho invariant. The rho invariant still isn't quite robust, but upon taking the mod Z reduction of the rho invariant, many striking invariance properties emerge. For example, writing = 1(Y ), the rho invariant descends to a well-defined map on geometric K-homology [HR10]: p(δ1; δ2) : K1(Bπ) → R=/Z. The rho invariant can be used to further study the properties of PSC metrics on manifolds. Whereas the Atiyah-Singer theorem is mostly useful for even-dimensional manifolds, the APS index theorem allows one to obtain results for odd-dimensional manifolds by considering them as boundaries of even-dimensional manifolds. In particular, one can obtain obstructions to PSC [HR10], and study the number of path components of the moduli space of PSC metrics on a manifold [BG95]. More recently, Mrowka, Ruberman and Saveliev [MRS16] discovered and proved a new index theorem for manifolds with periodic ends. Roughly speaking, these are manifolds Z∞ which have a compact piece Z, attached to which are one or more ends which repeat themselves periodically off to infinity. Such manifolds were first studied by Taubes [Tau87], who used them to prove (following work of Donaldson and Freedman) that R4 admits an uncountable family of mutually non-diffeomorphic smooth structures. The index theorem of MRS involves, like the APS index theorem, a correction term Nep(D) appearing for the periodic ends. The main contribution of this thesis is the development of a new analogue of geometric K-homology that is tailored to the setting of manifolds with periodic ends. The group is called Kep 1 (Bπ), and as in the APS case there is an analogous rho invariant descending to a well-defined map: ...Thesis (MPhil) -- University of Adelaide, School of Mathematical Sciences, 201

Topological Groups: Yesterday, Today, Tomorrow

In 1900, David Hilbert asked whether each locally euclidean topological group admits a Lie group structure. This was the fifth of his famous 23 questions which foreshadowed much of the mathematical creativity of the twentieth century. It required half a century of effort by several generations of eminent mathematicians until it was settled in the affirmative. These efforts resulted over time in the Peter-Weyl Theorem, the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups, and finally the solution of Hilbert 5 and the structure theory of locally compact groups, through the combined work of Andrew Gleason, Kenkichi Iwasawa, Deane Montgomery, and Leon Zippin. For a presentation of Hilbert 5 see the 2014 book “Hilbert’s Fifth Problem and Related Topics” by the winner of a 2006 Fields Medal and 2014 Breakthrough Prize in Mathematics, Terence Tao. It is not possible to describe briefly the richness of the topological group theory and the many directions taken since Hilbert 5. The 900 page reference book in 2013 “The Structure of Compact Groups” by Karl H. Hofmann and Sidney A. Morris, deals with one aspect of compact group theory. There are several books on profinite groups including those written by John S. Wilson (1998) and by Luis Ribes and ‎Pavel Zalesskii (2012). The 2007 book “The Lie Theory of Connected Pro-Lie Groups” by Karl Hofmann and Sidney A. Morris, demonstrates how powerful Lie Theory is in exposing the structure of infinite-dimensional Lie groups. The study of free topological groups initiated by A.A. Markov, M.I. Graev and S. Kakutani, has resulted in a wealth of interesting results, in particular those of A.V. Arkhangelʹskiĭ and many of his former students who developed this topic and its relations with topology. The book “Topological Groups and Related Structures” by Alexander Arkhangelʹskii and Mikhail Tkachenko has a diverse content including much material on free topological groups. Compactness conditions in topological groups, especially pseudocompactness as exemplified in the many papers of W.W. Comfort, has been another direction which has proved very fruitful to the present day

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal