Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1992. / Includes bibliographical references (p. 91-93).

Thesis (doctoral)--Universität Bonn, 1993. / Includes bibliographical references (p. 68-71).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1983. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 52-55).

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We propose a coherent analogue of the non-archimedean case of Venkatesh's conjecture on the cohomology of locally symmetric spaces for Shimura varieties coming from unitary similitude groups. Let G be a unitary similitude group with an indefinite signature at at least one archimedean place. Let Π be an automorphic cuspidal representation of G whose archimedean component Π∞ is a non-degenerate limit of discrete series and let 𝑊 be an automorphic vector bundle such that Π contributes to the coherent cohomology of its canonical extension. We produce a natural action of the derived Hecke algebra of Venketesh with torsion coefficients via cup product coming from étale covers and show that under some standard assumptions this action coincides with the conjectured action of a certain motivic cohomology group associated to the adjoint representation Ad𝜌π of the Galois representation attached to Π. We also prove that if the rank of G is greater than two, then the classes arising from the \'etale covers do not admit characteristic zero lifts, thereby showing that previous work of Harris-Venkatesh and Darmon-Harris-Rotger-Venkatesh is exceptional.

In this dissertation, we study the isotopy problem for a certain three-dimensional contactomorphism which is supported in a neighbourhood of an embedded 2-sphere with standard characteristic foliation. The diffeomorphism which underlies it is the Dehn twist on the sphere, and therefore its square becomes smoothly isotopic to the identity. The main result of this dissertation gives conditions under which any iterate of the Dehn twist along a non-trivial sphere is not contact isotopic to the identity. This provides the first examples of exotic contactomorphisms with infinite order in the contact mapping class group, as well as the first examples of exotic contactomorphisms of 3-manifolds with b_1 = 0. The proof crucially relies on the construction of an invariant for families of contact structures in monopole Floer homology which generalises the Kronheimer--Mrowka--Ozsváth--Szabó contact invariant, together with the nice interaction between this families invariant and the U map in Floer homology.

Embedded contact homology is an invariant of closed oriented contact 3-manifolds first defined by Hutchings, and is isomorphic to both Heegard Floer homology (by the work of Colin, Ghiggini and Honda) and Seiberg-Witten Floer cohomology (by the work of Taubes). The embedded contact chain complex is defined by counting closed orbits of the Reeb vector field and certain pseudoholomorphic curves in the symplectization of the manifold. As part of their proof that ECH=HF, Colin, Ghiggini and Honda showed that if the contact form is suitably adapted to an open book decomposition of the manifold, then embedded contact homology can be computed by considering only orbits and differentials in the complement of the binding of the open book; this fact was then in turn used to define a knot version of embedded contact homology, denoted ECK, where the (null-homologous) knot in question is given by the binding. In this thesis we start by generalizing these results to the case of rational open book decompositions, allowing us to define ECK for rationally null-homologous knots. In its most general form this is a bi-filtered chain complex whose homology yields ECH of the closed manifold. There is also a hat version of ECK in this situation which is equipped with an Alexander grading equivalent to that in the Heegaard Floer setting, categorifies the Alexander polynomial, and is conjecturally isomorphic to the hat version of knot Floer homology. The main result of this thesis is a large negative $n$-surgery formula for ECK. Namely, we start with an (integral) open book decomposition of a manifold with binding $K$ and compute, for all $n$ greater than or equal to twice the genus of $K$, ECK of the knot $K(-n)$ obtained by performing ($-n$)-surgery on $K$. This formula agrees with Hedden's large $n$-surgery formula for HFK, providing supporting evidence towards the conjectured equivalence between the two theories. Along we the way, we also prove that ECK is, in many cases, independent of the choices made to define it, namely the almost complex structure on the symplectization and the homotopy type of the contact form. We also prove that, in the case of integral open book decompositions, the hat version of ECK is supported in Alexander gradings less than or equal to twice the genus of the knot.

The theory of persistent homology (PH), introduced by Edelsbrunner, Letscher, and Zomorodian in [1], provides a framework for extracting topological information from experimental data. This framework was then expanded by Carlsson and Zomorodian in [2] to allow for multiple parameters of analysis with the theory of multidimensional persistent homology (MPH). This particular generalization is considerably more difficult to compute and to apply than its predecessor. We introduce an intermediate theory, coordinated persistent homology (CPH), that allows for multiple parameters while still preserving the clarity and coherence of PH. In addition to introducing the basic theory, we provide a polynomial time algorithm to compute CPH for time series and prove several important theorems about the nature of CPH. We also describe an application of the theory to a problem in seismology.

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