Where Tori fear to tread : hypermassive neutron star remnants and absolute event horizons Or topics in computational general relativity

Computational general relativity is a field of study which has reached maturity only within the last decade. This thesis details several studies that elucidate phenomena related to the coalescence of compact object binaries. Chapters 2 and 3 recounts work towards developing new analytical tools for visualizing and reasoning about dynamics in strongly curved spacetimes. In both studies, the results employ analogies with the classical theory of electricity and magnitism, first (Ch. 2) in the post-Newtonian approximation to general relativity and then (Ch. 3) in full general relativity though in the absence of matter sources. In Chapter 4, we examine the topological structure of absolute event horizons during binary black hole merger simulations conducted with the SpEC code. Chapter 6 reports on the progress of the SpEC code in simulating the coalescence of neutron star-neutron star binaries, while Chapter 7 tests the effects of various numerical gauge conditions on the robustness of black hole formation from stellar collapse in SpEC. In Chapter 5, we examine the nature of pseudospectral expansions of non-smooth functions motivated by the need to simulate the stellar surface in Chapters 6 and 7. In Chapter 8, we study how thermal effects in the nuclear equation of state effect the equilibria and stability of hypermassive neutron stars. Chapter 9 presents supplements to the work in Chapter 8, including an examination of the stability question raised in Chapter 8 in greater mathematical detail.

Valley evolution by meandering rivers

Fluvial systems form landscapes and sedimentary deposits with a rich hierarchy of structures that extend from grain- to valley scale. Large-scale pattern formation in fluvial systems is commonly attributed to forcing by external factors, including climate change, tectonic uplift, and sea-level change. Yet over geologic timescales, rivers may also develop large-scale erosional and depositional patterns that do not bear on environmental history. This dissertation uses a combination of numerical modeling and topographic analysis to identify and quantify patterns in river valleys that form as a consequence of river meandering alone, under constant external forcing. Chapter 2 identifies a numerical artifact in existing, grid-based models that represent the co-evolution of river channel migration and bank strength over geologic timescales. A new, vector-based technique for bank-material tracking is shown to improve predictions for the evolution of meander belts, floodplains, sedimentary deposits formed by aggrading channels, and bedrock river valleys, particularly when spatial contrasts in bank strength are strong. Chapters 3 and 4 apply this numerical technique to establishing valley topography formed by a vertically incising, meandering river subject to constant external forcing—which should serve as the null hypothesis for valley evolution. In Chapter 3, this scenario is shown to explain a variety of common bedrock river valley types and smaller-scale features within them—including entrenched channels, long-wavelength, arcuate scars in valley walls, and bedrock-cored river terraces. Chapter 4 describes the age and geometric statistics of river terraces formed by meandering with constant external forcing, and compares them to terraces in natural river valleys. The frequency of intrinsic terrace formation by meandering is shown to reflect a characteristic relief-generation timescale, and terrace length is identified as a key criterion for distinguishing these terraces from terraces formed by externally forced pulses of vertical incision. In a separate study, Chapter 5 utilizes image and topographic data from the Mars Reconnaissance Orbiter to quantitatively identify spatial structures in the polar layered deposits of Mars, and identifies sequences of beds, consistently 1-2 meters thick, that have accumulated hundreds of kilometers apart in the north polar layered deposits.

Some constructions, related to noncommutative tori; Fredholm modules and the Beilinson-Bloch regulator

A noncommutative 2-torus is one of the main toy models of noncommutative geometry, and a noncommutative n-torus is a straightforward generalization of it. In 1980, Pimsner and Voiculescu in [17] described a 6-term exact sequence, which allows for the computation of the K-theory of noncommutative tori. It follows that both even and odd K-groups of n-dimensional noncommutative tori are free abelian groups on 2^n-1 generators. In 1981, the Powers-Rieffel projector was described [19], which, together with the class of identity, generates the even K-theory of noncommutative 2-tori. In 1984, Elliott [10] computed trace and Chern character on these K-groups. According to Rieffel [20], the odd K-theory of a noncommutative n-torus coincides with the group of connected components of the elements of the algebra. In particular, generators of K-theory can be chosen to be invertible elements of the algebra. In Chapter 1, we derive an explicit formula for the First nontrivial generator of the odd K-theory of noncommutative tori. This gives the full set of generators for the odd K-theory of noncommutative 3-tori and 4-tori.

In Chapter 2, we apply the graded-commutative framework of differential geometry to the polynomial subalgebra of the noncommutative torus algebra. We use the framework of differential geometry described in [27], [14], [25], [26]. In order to apply this framework to noncommutative torus, the notion of the graded-commutative algebra has to be generalized: the "signs" should be allowed to take values in U(1), rather than just {-1,1}. Such generalization is well-known (see, e.g., [8] in the context of linear algebra). We reformulate relevant results of [27], [14], [25], [26] using this extended notion of sign. We show how this framework can be used to construct differential operators, differential forms, and jet spaces on noncommutative tori. Then, we compare the constructed differential forms to the ones, obtained from the spectral triple of the noncommutative torus. Sections 2.1-2.3 recall the basic notions from [27], [14], [25], [26], with the required change of the notion of "sign". In Section 2.4, we apply these notions to the polynomial subalgebra of the noncommutative torus algebra. This polynomial subalgebra is similar to a free graded-commutative algebra. We show that, when restricted to the polynomial subalgebra, Connes construction of differential forms gives the same answer as the one obtained from the graded-commutative differential geometry. One may try to extend these notions to the smooth noncommutative torus algebra, but this was not done in this work.

A reconstruction of the Beilinson-Bloch regulator (for curves) via Fredholm modules was given by Eugene Ha in [12]. However, the proof in [12] contains a critical gap; in Chapter 3, we close this gap. More specifically, we do this by obtaining some technical results, and by proving Property 4 of Section 3.7 (see Theorem 3.9.4), which implies that such reformulation is, indeed, possible. The main motivation for this reformulation is the longer-term goal of finding possible analogs of the second K-group (in the context of algebraic geometry and K-theory of rings) and of the regulators for noncommutative spaces. This work should be seen as a necessary preliminary step for that purpose.

For the convenience of the reader, we also give a short description of the results from [12], as well as some background material on central extensions and Connes-Karoubi character.

Ghosts of order on the frontier of chaos

What kinds of motion can occur in classical mechanics? We address this question by looking at the structures traced out by trajectories in phase space; the most orderly, completely integrable systems are characterized by phase trajectories confined to low-dimensional, invariant tori. The KAM theory examines what happens to the tori when an integrable system is subjected to a small perturbation and finds that, for small enough perturbations, most of them survive.

The KAM theory is mute about the disrupted tori, but, for two-dimensional systems, Aubry and Mather discovered an astonishing picture: the broken tori are replaced by "cantori," tattered, Cantor-set remnants of the original invariant curves. We seek to extend Aubry and Mather's picture to higher dimensional systems and report two kinds of studies; both concern perturbations of a completely integrable, four-dimensional symplectic map. In the first study we compute some numerical approximations to Birkhoff periodic orbits; sequences of such orbits should approximate any higher dimensional analogs of the cantori. In the second study we prove converse KAM theorems; that is, we use a combination of analytic arguments and rigorous, machine-assisted computations to find perturbations so large that no KAM tori survive. We are able to show that the last few of our Birkhoff orbits exist in a regime where there are no tori.