Strong shift equivalence, algebraic K-theory, and isolating zero-dimensional dynamics on manifolds

We study the relations of shift equivalence and strong shift equivalence for matrices over a ring $\mathcal{R}$, and establish a connection between these relations and algebraic K-theory. We utilize this connection to obtain results in two areas where the shift and strong shift equivalence relations play an important role: the study of finite group extensions of shifts of finite type, and the Generalized Spectral Conjectures of Boyle and Handelman for nonnegative matrices over subrings of the real numbers. We show the refinement of the shift equivalence class of a matrix $A$ over a ring $\mathcal{R}$ by strong shift equivalence classes over the ring is classified by a quotient $NK_{1}(\mathcal{R}) / E(A,\mathcal{R})$ of the algebraic K-group $NK_{1}(\calR)$. We use the K-theory of non-commutative localizations to show that in certain cases the subgroup $E(A,\mathcal{R})$ must vanish, including the case $A$ is invertible over $\mathcal{R}$. We use the K-theory connection to clarify the structure of algebraic invariants for finite group extensions of shifts of finite type. In particular, we give a strong negative answer to a question of Parry, who asked whether the dynamical zeta function determines up to finitely many topological conjugacy classes the extensions by $G$ of a fixed mixing shift of finite type. We apply the K-theory connection to prove the equivalence of a strong and weak form of the Generalized Spectral Conjecture of Boyle and Handelman for primitive matrices over subrings of $\mathbb{R}$. We construct explicit matrices whose class in the algebraic K-group $NK_{1}(\mathcal{R})$ is non-zero for certain rings $\mathcal{R}$ motivated by applications. We study the possible dynamics of the restriction of a homeomorphism of a compact manifold to an isolated zero-dimensional set. We prove that for $n \ge 3$ every compact zero-dimensional system can arise as an isolated invariant set for a homeomorphism of a compact $n$-manifold. In dimension two, we provide obstructions and examples.

Connecting the dots: Examining the role of parental beliefs and preschoolers’ affect and engagement in predicting parent-child number exploration during a meaningful math experience

The current study examined the frequency and quality of how 3- to 4-year-old children and their parents explore the relations between symbolic and non-symbolic quantities in the context of a playful math experience, as well as the role of both parent and child factors in this exploration. Preschool children’s numerical knowledge was assessed while parents completed a survey about the number-related experiences they share with their children at home, and their math-related beliefs. Parent-child dyads were then videotaped playing a modified version of the card game War. Results suggest that parents and children explored quantity explicitly on only half of the cards and card pairs played, and dyads of young children and those with lower number knowledge tended to be most explicit in their quantity exploration. Dyads with older children, on the other hand, often completed their turns without discussing the numbers at all, likely because they were knowledgeable enough about numbers that they could move through the game with ease. However, when dyads did explore the quantities explicitly, they focused on identifying numbers symbolically, used non-symbolic card information interchangeably with symbolic information to make the quantity comparison judgments, and in some instances, emphasized the connection between the symbolic and non-symbolic number representations on the cards. Parents reported that math experiences such as card game play and quantity comparison occurred relatively infrequently at home compared to activities geared towards more foundational practice of number, such as counting out loud and naming numbers. However, parental beliefs were important in predicting both the frequency of at-home math engagement as well as the quality of these experiences. In particular, parents’ specific beliefs about their children’s abilities and interests were associated with the frequency of home math activities, while parents’ math-related ability beliefs and values along with children’s engagement in the card game were associated with the quality of dyads’ number exploration during the card game. Taken together, these findings suggest that card games can be an engaging context for parent-preschooler exploration of numbers in multiple representations, and suggests that parents’ beliefs and children’s level of engagement are important predictors of this exploration.

A Study of pH Manipulation on Tumor Proliferation and the CTL Response

Adoptive Cell Transfer (ACT) Therapy is a cancer treatment that enhances and utilizes the body’s own immune system. However, this treatment has had limited success in clinical trials. We hypothesized that this is due to the immunosuppressive, acidic microenvironment of cancer tumors. We tested the effects of acidic, neutral, and basic environments in vitro on cytotoxic T lymphocyte (CTL) survival, activation, migration and killing ability and on cancer cell survival. We found that CTLs have most optimum survival, activation, and migration in a neutral environment, while the optimal extracellular conditions for EG-7 lymphoma are slightly acidic and B16-OVA melanoma survives best in physiological conditions. Future research should further study the killing ability of T cells in the three different environments and look to move to in vivo experiments.

Image-Guided Precision Manipulation of Cells and Nanoparticles in Microfluidics

Manipulation of single cells and particles is important to biology and nanotechnology. Our electrokinetic (EK) tweezers manipulate objects in simple microfluidic devices using gentle fluid and electric forces under vision-based feedback control. In this dissertation, I detail a user-friendly implementation of EK tweezers that allows users to select, position, and assemble cells and nanoparticles. This EK system was used to measure attachment forces between living breast cancer cells, trap single quantum dots with 45 nm accuracy, build nanophotonic circuits, and scan optical properties of nanowires. With a novel multi-layer microfluidic device, EK was also used to guide single microspheres along complex 3D trajectories. The schemes, software, and methods developed here can be used in many settings to precisely manipulate most visible objects, assemble objects into useful structures, and improve the function of lab-on-a-chip microfluidic systems.

The Real-Quaternionic Indicator of Irreducible Self-Conjugate Representations of Real Reductive Algebraic Groups and A Comment on the Local Langlands Correspondence of GL(2, F )

The real-quaternionic indicator, also called the $\delta$ indicator, indicates if a self-conjugate representation is of real or quaternionic type. It is closely related to the Frobenius-Schur indicator, which we call the $\varepsilon$ indicator. The Frobenius-Schur indicator $\varepsilon(\pi)$ is known to be given by a particular value of the central character. We would like a similar result for the $\delta$ indicator. When $G$ is compact, $\delta(\pi)$ and $\varepsilon(\pi)$ coincide. In general, they are not necessarily the same. In this thesis, we will give a relation between the two indicators when $G$ is a real reductive algebraic group. This relation also leads to a formula for $\delta(\pi)$ in terms of the central character. For the second part, we consider the construction of the local Langlands correspondence of $GL(2,F)$ when $F$ is a non-Archimedean local field with odd residual characteristics. By re-examining the construction, we provide new proofs to some important properties of the correspondence. Namely, the construction is independent of the choice of additive character in the theta correspondence.

MANIPULATION OF IONS, ELECTRONS, AND PHOTONS IN 2D MATERIALS BY ION INTERCALATION

2D materials have attracted tremendous attention due to their unique physical and chemical properties since the discovery of graphene. Despite these intrinsic properties, various modification methods have been applied to 2D materials that yield even more exciting results. Among all modification methods, the intercalation of 2D materials provides the highest possible doping and/or phase change to the pristine 2D materials. This doping effect highly modifies 2D materials, with extraordinary electrical transport as well as optical, thermal, magnetic, and catalytic properties, which are advantageous for optoelectronics, superconductors, thermoelectronics, catalysis and energy storage applications. To study the property changes of 2D materials, we designed and built a planar nanobattery that allows electrochemical ion intercalation in 2D materials. More importantly, this planar nanobattery enables characterization of electrical, optical and structural properties of 2D materials in situ and real time upon ion intercalation. With this device, we successfully intercalated Li-ions into few layer graphene (FLG) and ultrathin graphite, heavily dopes the graphene to 0.6 x 10^15 /cm2, which simultaneously increased its conductivity and transmittance in the visible range. The intercalated LiC6 single crystallite achieved extraordinary optoelectronic properties, in which an eight-layered Li intercalated FLG achieved transmittance of 91.7% (at 550 nm) and sheet resistance of 3 ohm/sq. We extend the research to obtain scalable, printable graphene based transparent conductors with ion intercalation. Surfactant free, printed reduced graphene oxide transparent conductor thin film with Na-ion intercalation is obtained with transmittance of 79% and sheet resistance of 300 ohm/sq (at 550 nm). The figure of merit is calculated as the best pure rGO based transparent conductors. We further improved the tunability of the reduced graphene oxide film by using two layers of CNT films to sandwich it. The tunable range of rGO film is demonstrated from 0.9 um to 10 um in wavelength. Other ions such as K-ion is also studied of its intercalation chemistry and optical properties in graphitic materials. We also used the in situ characterization tools to understand the fundamental properties and improve the performance of battery electrode materials. We investigated the Na-ion interaction with rGO by in situ Transmission electron microscopy (TEM). For the first time, we observed reversible Na metal cluster (with diameter larger than 10 nm) deposition on rGO surface, which we evidenced with atom-resolved HRTEM image of Na metal and electron diffraction pattern. This discovery leads to a porous reduced graphene oxide sodium ion battery anode with record high reversible specific capacity around 450 mAh/g at 25mA/g, a high rate performance of 200 mAh/g at 250 mA/g, and stable cycling performance up to 750 cycles. In addition, direct observation of irreversible formation of Na2O on rGO unveils the origin of commonly observed low 1st Columbic Efficiency of rGO containing electrodes. Another example for in situ characterization for battery electrode is using the planar nanobattery for 2D MoS2 crystallite. Planar nanobattery allows the intrinsic electrical conductivity measurement with single crystalline 2D battery electrode upon ion intercalation and deintercalation process, which is lacking in conventional battery characterization techniques. We discovered that with a “rapid-charging” process at the first cycle, the lithiated MoS2 undergoes a drastic resistance decrease, which in a regular lithiation process, the resistance always increases after lithiation at its final stage. This discovery leads to a 2- fold increase in specific capacity with with rapid first lithiated MoS2 composite electrode material, compare with the regular first lithiated MoS2 composite electrode material, at current density of 250 mA/g.