Row-Column Mirror Symmetry for Colored Torus Knot Homology

We give a recursive construction of the categorified Young symmetrizer introduced by Abel-Hogancamp in arXiv:1510.05330 corresponding to the single-column partition. As a consequence, we obtain new expressions for the uncolored $y$-ified HOMFLYPT homology of positive torus links and the $y$-ified column-colored HOMFLYPT homology of positive torus knots. In the latter case, we compare with the row-colored homology of positive torus knots computed by Hogancamp-Mellit in arXiv:1909.00418, verifying the mirror symmetry conjectures of arXiv:1112.0030 and arXiv:1304.3481 in this case.Comment: 86 pages; many figures. Comments welcome

Thermalization and Quantum Information in Conformal Field Theory

The consequences of the constraints of conformal symmetry are far-reaching withintheoretical physics. In this dissertation we address a series of questions in conformalfield theory: 1) We calculate the spectrum of qKdV charges in a large central chargeexpansion. 2) We determine the corrections to bulk information geometry from 1/Ncontributions to holographic correlators. 3) We study the higher genus partitionsfunctions of CFTs associated with classical and quantum error-correcting codes

Pariteettirikko molekyylien magneettisissa ominaisuuksissa

The Standard Model of particle physics consists of the quantum electrodynamics (QED) and the weak and strong nuclear interactions. The QED is the basis for molecular properties, and thus it defines much of the world we see. The weak nuclear interaction is responsible for decays of nuclei, among other things, and in principle, it should also effects at the molecular scale. The strong nuclear interaction is hidden in interactions inside nuclei. From the high-energy and atomic experiments it is known that the weak interaction does not conserve parity. Consequently, the weak interaction and specifically the exchange of the Z^0 boson between a nucleon and an electron induces small energy shifts of different sign for mirror image molecules. This in turn will make the other enantiomer of a molecule energetically favorable than the other and also shifts the spectral lines of the mirror image pair of molecules into different directions creating a split. Parity violation (PV) in molecules, however, has not been observed. The topic of this thesis is how the weak interaction affects certain molecular magnetic properties, namely certain parameters of nuclear magnetic resonance (NMR) and electron spin resonance (ESR) spectroscopies. The thesis consists of numerical estimates of NMR and ESR spectral parameters and investigations of the effects of different aspects of quantum chemical computations to them. PV contributions to the NMR shielding and spin-spin coupling constants are investigated from the computational point of view. All the aspects of quantum chemical electronic structure computations are found to be very important, which makes accurate computations challenging. Effects of molecular geometry are also investigated using a model system of polysilyene chains. PV contribution to the NMR shielding constant is found to saturate after the chain reaches a certain length, but the effects of local geometry can be large. Rigorous vibrational averaging is also performed for a relatively small and rigid molecule. Vibrational corrections to the PV contribution are found to be only a couple of per cents. PV contributions to the ESR g-tensor are also evaluated using a series of molecules. Unfortunately, all the estimates are below the experimental limits, but PV in some of the heavier molecules comes close to the present day experimental resolution.Hiukkasfysiikan standardimalli koostuu kvanttielektrodynamiikasta, heikosta vuorovaikutuksesta ja vahvasta vuorovaikutuksesta. Kvanttielektrodynamiikka kuvaa sähkömagneettista vuorovaikutusta kappaleiden välillä ja näin ollen molekyylitason ilmiöt johtuvat pääasiassa siitä. Vahva vuorovaikutus tapahtuu pääasiassa ydinten sisällä, joten sen suora vaikutus molekyylien ominaisuuksiin on olematon. Heikko vuorovaikutus on osallisena mm. ydinten radioaktiivisessa hajoamisessa mutta periaatteessa se vaikuttaa myös stabiilien ytimien ja elektronien välillä, joten sillä on vaikutusta myös molekyylitasolla. Hiukkasfysiikan ja atomitason kokeista tiedetään, että heikko vuorovaikutus ei säilytä peilisymmetriaa. Molekyylitasolla tämä ilmenee siten, että saman molekyylin erikätisillä muodoilla on hiukan erilaiset ominaisuudet. Näiden eroavaisuuksien teoreettinen tutkiminen ydinmagneettisessa resonanssispektroskopiassa (NMR) ja elektronin spin resonanssispektroskopiassa on tämän väitöskirjan aihe. Väitöskirjassa on tutkittu eri menetelmien vaikutusta näiden ominaisuuksen laskemiseen ja havaittu, että eroavaisuuksien tarkka ennustaminen on laskennallisesti erittäin haastavaa. Valitettavasti lasketut ennusteet ovat kuitenkin kokeellisen tarkkuuden alapuolella. Ero ei kuitenkaan ole kaikissa tapauksissa kovin suuri, joten on mahdollista, että peilisymmetrian rikko havaitaan molekyylitasolla tulevaisuudessa. Jos eroavaisuus molekyylien ominaisuuksissa havaitaan NMR spektroskopiassa, voidaan yhdistämällä tulokset tarkkoihin laskuihin tehdä myös johtopäätöksiä ydinten ominaisuuksista

One-Dimensional Fermi liquids

I attempt to give a pedagogical overview of the progress which has occurred during the past decade in the description of one-dimensional correlated fermions. Fermi liquid theory based on a quasi-particle picture, breaks down in one dimension because of the Peierls divergence and because of charge-spin separation. It is replaced by a Luttinger liquid whose elementary excitations are collective charge and spin modes, based on the exactly solvable Luttinger model. I review this model and various solutions with emphasis on bosonization (and its equivalence to conformal field theory), and its physical properties. The notion of a Luttinger liquid implies that all gapless 1D systems share these properties at low energies. Chapters 1 and 2 of the article contain an introduction and a discussion of the breakdown of Fermi liquid theory. Chapter 3 describes in detail the solution of the Luttinger model both by bosonization and by Green's functions methods and summarizes the properties of the model, expressed thorugh correlation functions. The relation to conformal field theory is discussed. Chapter 4 of the article introduces the notion of a Luttinger liquid. It describes in much detail the various mappings applied to realistic models of 1D correlated fermions, onto the Luttinger model, as well as important corrections to the Luttinger model properties discussed in Ch.3. Chapter 5 describes situations where the Luttinger liquid is not a stable fixed point, and where spin or charge gaps open in at least one channel. Chapter 6 discusses multi-band and multichain problems, in particular the stability of a Luttinger liquid with respect to interchain hopping. Ch. 7 gives a brief summary of experimental efforts to uncover Luttinger liquid correlations in quasi-1D materials.Comment: uuencoded Latex files and postscript figures, one Readme-file approx 160 pages + 13 figures; to be published by Reports on Progress in Physic

Computational Geometric Mechanics and Control of Rigid Bodies.

This dissertation studies the dynamics and optimal control of rigid bodies from two complementary perspectives, by providing theoretical analyses that respect the fundamental geometric characteristics of rigid body dynamics and by developing computational algorithms that preserve those geometric features. This dissertation is focused on developing analytical theory and computational algorithms that are intrinsic and applicable to a wide class of multibody systems. A geometric numerical integrator, referred to as a Lie group variational integrator, is developed for rigid body dynamics. Discrete-time Lagrangian and Hamiltonian mechanics and Lie group methods are unified to obtain a systematic method for constructing numerical integrators that preserve the geometric properties of the dynamics as well as the structure of a Lie group. It is shown that Lie group variational integrators have substantial computational advantages over integrators that preserve either one of none of these properties. This approach is also extended to mechanical systems evolving on the product of two-spheres. A computational geometric approach is developed for optimal control of rigid bodies on a Lie group. An optimal control problem is discretized at the problem formulation stage by using a Lie group variational integrator, and discrete-time necessary conditions for optimality are derived using the calculus of variations. The discrete-time necessary conditions inherit the desirable computational properties of the Lie group variational integrator, as they are derived from a symplectic discrete flow. They do not exhibit the numerical dissipation introduced by conventional numerical integration schemes, and consequently, we can efficiently obtain optimal controls that respect the geometric features of the optimality conditions. The approach that combines computational geometric mechanics and optimal control is illustrated by various examples of rigid body dynamics, which include a rigid body pendulum on a cart, pure bending of an elastic rod, and two rigid bodies connected by a ball joint. Since all of the analytical and computational results developed in this dissertation are coordinate-free, they are independent of a specific choice of local coordinates, and they completely avoid any singularity, ambiguity, and complexity associated with local coordinates. This provides insight into the global dynamics of rigid bodies.Ph.D.Aerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/60804/1/tylee_1.pd

"New" Veneziano amplitudes from "old" Fermat (hyper) surfaces

The history of discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler's beta function. Such an analogue had in fact been known in mathematics literature at least in 1922 and was studied subsequently by mathematicians such as Selberg, Weil and Deligne among others. The mathematical interpretation of this multidimensional beta function is markedly different from that described in physics literature. This paper aims to bridge the gap between the existing treatments. Preserving all results of conformal field theories intact, developed formalism employing topological, algebro-geometric, number-theoretic and combinatorial metods is aimed to provide better understanding of the Veneziano amplitudes and, thus, of string theories.Comment: 92 pages LaTex, some typos removed, discussion section is added along with several additional latest reference

Angles and devices for quantum approximate optimization

A potential application of emerging Noisy Intermediate-Scale Quantum (NISQ) devices is that of approximately solving combinatorial optimization problems. This thesis investigates a gate-based algorithm for this purpose, the Quantum Approximate Optimization Algorithm (QAOA), in two major themes. First, we examine how the QAOA resolves the problems it is designed to solve. We take a statistical view of the algorithm applied to ensembles of problems, first, considering a highly symmetric version of the algorithm, using Grover drivers. In this highly symmetric context, we find a simple dependence of the QAOA state’s expected value on how values of the cost function are distributed. Furthering this theme, we demonstrate that, generally, QAOA performance depends on problem statistics with respect to a metric induced by a chosen driver Hamiltonian. We obtain a method for evaluating QAOA performance on worst-case problems, those of random costs, for differing driver choices. Second, we investigate a QAOA context with device control occurring only via single-qubit gates, rather than using individually programmable one- and two-qubit gates. In this reduced control overhead scheme---the digital-analog scheme---the complexity of devices running QAOA circuits is decreased at the cost of errors which are shown to be non-harmful in certain regimes. We then explore hypothetical device designs one could use for this purpose.Eine mögliche Anwendung für “Noisy Intermediate-Scale Quantum devices” (NISQ devices) ist die näherungsweise Lösung von kombinatorischen Optimierungsproblemen. Die vorliegende Arbeit untersucht anhand zweier Hauptthemen einen gatterbasierten Algorithmus, den sogenannten “Quantum Approximate Optimization Algorithm” (QAOA). Zuerst prüfen wir, wie der QAOA jene Probleme löst, für die er entwickelt wurde. Wir betrachten den Algorithmus in einer Kombination mit hochsymmetrischen Grover-Treibern für statistische Ensembles von Probleminstanzen. In diesem Kontext finden wir eine einfache Abhängigkeit von der Verteilung der Kostenfunktionswerte. Weiterführend zeigen wir, dass die QAOA-Leistung generell von der Problemstatistik in Bezug auf eine durch den gewählten Treiber-Hamiltonian induzierte Metrik abhängt. Wir erhalten eine Methode zur Bewertung der QAOA-Leistung bei schwersten Problemen (solche zufälliger Kosten) für unterschiedliche Treiberauswahlen. Zweitens untersuchen wir eine QAOA-Variante, bei der sich die Hardware- Kontrolle nur auf Ein-Qubit-Gatter anstatt individuell programmierbare Ein- und Zwei-Qubit-Gatter erstreckt. In diesem reduzierten Kontrollaufwandsschema—dem digital-analogen Schema—sinkt die Komplexität der Hardware, welche die QAOASchaltungen ausführt, auf Kosten von Fehlern, die in bestimmten Bereichen als ungefährlich nachgewiesen werden. Danach erkunden wir hypothetische Hardware- Konzepte, die für diesen Zweck genutzt werden könnten

Weighted Tree Automata -- May it be a little more?

This is a book on weighted tree automata. We present the basic definitions and some of the important results in a coherent form with full proofs. The concept of weighted tree automata is part of Automata Theory and it touches the area of Universal Algebra. It originated from two sources: weighted string automata and finite-state tree automata