The work of this thesis is to motivate the following: Statement: The Hodge conjecture holds for products of varieties Z = XxC where (i) X is smooth, projective of dimension 2m-l, (ii) C is a smooth curve. The basic setting of this thesis is depicted by the following where (i) k⁻¹ (t) = Zt = Xt xC, {Xt } a Lefschetz pencil of hyperplane sections of X (ii) £ is the singular set of k, i.e., k = k is smooth and proper. Corresponding to this diagram are the extended Hodge bundle U H (Z . C) with integrable connection V , and the family of tt?1 t intermediate Jacobians. U JCZ ) with corresponding normal functions Now V induces an operator (also denoted by V) on the normal functions, and those normal functions v satisfying the differential equation Vv = 0 are labeled horizontal, which includes those normal functions arising from the primitive algebraic cocycles in H²m (Z). Now the known generalization of Lefschetz's techniques state that every primitive integral class of type (m,m) in H²m (Z) comes from a horizontal normal function in some natural way, so that what's needed to prove the above statement is some way of converting a normal function to an algebraic cocycle. We motivate this statement by proving some results about the group of normal functions, in particular our main result: Theorem: The group of normal functions are horizontal. To prove this theorem, we exhibit Vv as a global section of some holomorphic vector bundle over p¹, and then show that there are no non-zero global sections of this vector bundle. The main idea is to compare the quasi-canonical extensions of certain holomorphic vector bundles with integrable connection with those extensions arising from algebra (hypercohomology), by calculating certain periods of growth. Once this comparison is made precise, we apply a vanishing theorem statement about the global sections of the algebraic extensions to our geometric extensions, thus concluding the proof of the theorem. / Science, Faculty of / Mathematics, Department of / Graduate

M.Sc. / The aim of this dissertation will be an investigation into a classical result which asserts the uniqueness of the norm topology on a semi-simple Banach algebra. For a commutative semi-simple Banach algebra, say A, it is relatively simple matter, with the aid of the Closed Graph Theorem, to show that all Banach algebra norms on A must be equivalent. The same result for non-commutative Banach algebras was conjectured by I. Kaplansky in the 1950’s and solved more then a decade later, in 1967, by B E Johnson. However, Johnson’s proof was difficult and relied heavily on representation theory. As a result, the problem remained unsolved for the more difficult situation of Jordan Banach algebras. Fifteen years later in 1982, B. Aupetit succeeded in proving Johnson’s result, using a subharmonic method that was independent of algebra representations. Moreover he could, using these techniques, also settle the problem in the Jordan Banach algebra case. A while later, in 1989, T. Ransford provided a shorter algebraic proof of Johnson’s result using the well-known spectral radius formula. This dissertation will be a comparative study of the three different approaches on the problem for Banach algebras.

No description available.

With plans for advancing into the rest of the solar system in the coming decades, an understanding of how interlinked satellite systems behave as a network will be essential. The relatively recent development of optics as a method of space communication means that inter-satellite networks are more feasible than ever. That said, there are currently no analyses which take into account a planet-wide, largely uncoordinated, optically linked satellite network. To provide a look at the properties of such a network, movement and connections of Earth's currently active satellites were simulated based on real-world data, and their networks modeled via graphs. Ultimately, it was found that many properties of such a network are periodic, fluctuating in sync with the orbital time of low-earth orbit satellites. This, among other data, suggests that the peaks of these waves are caused by a meeting of satellites near the north and south poles.

Thesis advisor: Ying Ran / Recently there has been a surging interest in the topological phases of matter, including the symmetry-protected topological phases, symmetry-enriched topological phases, and topological semimetals. This thesis is aiming at finding new ways of searching and probing these topological phases of matter in order to deepen our understanding of them. The body of the thesis consists of three parts. In the first part, we study the search of filling-enforced topological phases of matter in materials. It shows the existence of symmetry-protected topological phases enforced by special electron fillings or fractional spin per unit-cell. This is an extension of the famous Lieb-Schultz-Mattis theorem. The original LSM theorem states that the symmetric gapped ground state of the system must exhibit topological order when there's fractional spin or fractional electron filling per unit-cell. However, the LSM theorem can be circumvented when commensurate magnetic flux is present in the system, which enlarge the unit-cells to accommodate integer numbers of electrons. We utilize this point to prove that the ground state of the system must be a symmetry-protected topological phase when magnetic translation symmetry is satisfied, which we coin the name “generalized LSM theorem”. The theorem is proved using two different methods. The first proof is to use the tensor network representation of the ground state wave-function. The second proof consists of a physical argument based on the idea of entanglement pumping. As a byproduct of this theorem, a large class of decorated quantum dimer models are introduced, which satisfy the condition of the generalized LSM theorem and exhibit SPT phases as their ground states. In part II, we switch to the nonlinear response study of Weyl semimetals. Weyl semimetals (WSM) have been discovered in time-reversal symmetric materials, featuring monopoles of Berry’s curvature in momentum space. WSM have been distinguished between Type-I and II where the velocity tilting of the cone in the later ensures a finite area Fermi surface.To date it has not been clear whether the two types results in any qualitatively new phenomena. In this part we focus on the shift-current response ($\sigma_{shift}(\omega)$), a second order optical effect generating photocurrents. We find that up to an order unity constant, $\sigma_{shift}(\omega)\sim \frac{e^3}{h^2}\frac{1}{\omega}$ in Type-II WSM, diverging in the low frequency $\omega\rightarrow 0$ limit. This is in stark contrast to the vanishing behavior ($\sigma_{shift}(\omega)\propto \omega$) in Type-I WSM. In addition, in both Type-I and Type-II WSM, a nonzero chemical potential $\mu$ relative to nodes leads to a large peak of shift-current response with a width $\sim |\mu|/\hbar$ and a height $\sim \frac{e^3}{h}\frac{1}{|\mu|}$, the latter diverging in the low doping limit. We show that the origin of these divergences is the singular Berry’s connections and the Pauli-blocking mechanism. Similar results hold for the real part of the second harmonic generation, a closely related nonlinear optical response. In part III, we propose a new kind of thermo-optical experiment: the nonreciprocal directional dichroism induced by a temperature gradient. The nonreciprocal directional dichroism effect, which measures the difference in the optical absorption coefficient between counterpropagating lights, occurs only in systems lacking inversion symmetry. The introduction of temperature-gradient in an inversion-symmetric system will also yield nonreciprocal directional dichroism effect. This effect is then applied to quantum magnetism, where conventional experimental techniques have difficulty detecting magnetic mobile excitations such as magnons or spinons exclusively due to the interference of phonons and local magnetic impurities. A model calculation is presented to further demonstrate this phenomenon. / Thesis (PhD) — Boston College, 2021. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

In 1969, Lutzer proved that a linearly ordered topological space with a Gδ-diagonal is metrizable. This appears to be the first work in the field of metrization of ordered topological spaces. Very little seems to have been done in this direction. This thesis is a study of the various conditions necessary for metrizability of such spaces. One of the earliest papers concerned with ordered topological spaces is that of Eilenberg. Since then, ordered spaces have been considered by various authors, but few considered the conditions under which such spaces would be metrizable. Bennet gave a characterization of metrizability for a linearly ordered topological space with a σ-point finite base. A linearly ordered topological space is a space for which the interval topology coincides with the original topology for the space. We investigate the metrizability of linearly ordered topological space satisfying certain covering properties, countability conditions on the base, certain conditions on the diagonal and spaces which admit a symmetric. We obtain four characterizations of metrizability for linearly ordered topological space in terms of some of the above notions.

Metal-Organic Frameworks (MOFs), an emerging class of porous crystalline materials, have shown promising properties for diverse applications such as catalysis, gas storage and separation. The high degree of tunability of MOFs vs other solid materials enable the assembly of advanced materials with fascinating properties for specific applications. Nevertheless, the precise control in the construction of MOFs at the molecular level remains challenging. Particularly, the formation of pre-targeted multi-nuclear Molecular Building Block (MBB) precursors to unveil materials with targeted structural characteristics is captivating. The aim of my master project in the continuous quest of the group of Prof. Eddaoudi in exploring different synthetic pathways to control the assembly of Rare Earth (RE) based MOF. After giving a general overview about MOFs, I will discuss in this thesis the results of my work on the use of tri-topic oriented organic carboxylate building units with the aim to explore the assembly/construction of new porous RE based MOFs. In chapter 2 will discuss the assembly of 3-c linkers with RE metals was then evaluated based on symmetry and angularity of the three connected linkers. The focus of chapter 3 is cerium based MOFs and heterometallic system, based on 3-c ligands with different length and symmetry. Overall, the incompatibility of 3-c ligands with the 12-c cuo MBB did not allow to any formation of higher neuclearity (˃6), but it has resulted in affecting the connectivity of the cluster.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第19469号 / 理博第4129号 / 新制||理||1594(附属図書館) / 32505 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 岸本 大祐, 教授 加藤 毅, 教授 藤原 耕二 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Thesis advisor: Kenneth S. Burch / In condensed matter physics we study the behavior of crystals at finite density and low temperatures. By tuning and breaking the various materials, symmetries, and the topology of a crystal one can bring about brand new quantum phases of matter. These new phases of matter in turn produce emergent quasiparticles such as the cooper pair in superconductivity, the spinon in magnetic systems, and the Fermi arcs in Weyl semimetals. Of particular interest are systems in which superconductivity interacts with topology. These systems have been theoretically predicted to produce anyonic quasiparticles which may be used as qubits in a future fault-tolerant quantum computer. However, these ideas usually require the use of the superconducting proximity effect to inject cooper pairs into the topological system. This in turn requires interfacing two different materials which not only requires extremely clean interfaces, but also matching Fermi surfaces, comparable Fermi velocities, and more. The ideal candidate for topological superconductivity would therefore be a material that is both superconducting and topologically non-trivial. One promising candidate is the iron-based superconductor FeTe(1−x)Sex, specifically at the FeTe0.55Se0.45 (FTS) doping which also has non-trivial topology. In this dissertation, we address the fabrication of pristine interfaces using a new tool as well as new probes into the topology of FTS. In Chapter II we discuss the motivation, construction, and use of the “cleanroomin-a-glovebox”. This tool places an entire nanofabrication workflow into an inert argon atmosphere which has allowed us access to study a myriad of new materials and systems. A delightful offshoot of this glovebox is that it is a useful tool in training new scientists in fabrication techniques. The photolithography, Physical Vapor Deposition (PVD), and characterization tools in the glovebox are designed to be easy to use and thus afford new users a low-risk method of learning new techniques. In chapter III we discuss a specific example of a new quantum phase of matter e.g. topological superconductivity in FTS. There, I discuss the fabrication requirements to probe this elusive phase as well as the unique measurement technique used to provide evidence that FTS is a higher-order topological superconductor. The characterization of FTS continues in Chapter IV where we reveal some exciting new results in the FTS system. These new results are direct evidence for the topological nature of FTS, a feat which has only been shown in Angle-Resolved Photo Emission Spectroscopy (ARPES) and Scanning Tunneling Microscopy (STM). Chapter V concludes the dissertation with a summary of Chapters II, III, and IV. In addition, we give suggestions for future experiments to investigate the FTS system further as well as suggestions for insightful teaching programs with the cleanroom-in-a-glovebox. / Thesis (PhD) — Boston College, 2021. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

Transformation of Beckholmen through topological sitemapping of hidden layers.