Heegaard Floer homology of L-space links with two components

We compute different versions of link Floer homology $HFL^{-}$ and $\widehat{HFL}$ for any $L$-space link with two components. The main approach is to compute the $h$-function of the filtered chain complex which is determined by the Alexander polynomials of every sublink of the $L$-space link. As an application, Thurston polytope and Thurston norm of any 2-component $L$-space link are explicitly determined by Alexander polynomials of the link and the link components.Comment: 23 page

Geometric finiteness in negatively pinched Hadamard manifolds

In this paper, we generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, $\mathbb{C}$) to geometrically infinite discrete subgroups $\Gamma$ of isometries of negatively pinched Hadamard manifolds $X$. We then generalize a theorem of Bishop to prove that every discrete geometrically infinite isometry subgroup $\Gamma$ has a set of nonconical limit points with the cardinality of the continuum.Comment: We improved our results to any discrete geometrically infinite isometry subgroup of a negatively pinched Hadamard manifold, and we sharpened our main results to deal with limit sets of ends of the convex cor

Dynamical rearrangement of super-Earths during disk dispersal II. Assessment of the magnetospheric rebound model for planet formation scenarios

Context.The Kepler mission has provided a large sample to statistically analyze the orbital properties of the super-Earth planets. We hypothesize that these planets formed early and consider the problem of matching planet formation theory to the current observations. Two scenarios, disk migration and in-situ formation, have been proposed to explain their origin. In the migration scenario planets migrate inward due to planet-disk interaction, whereas in the in-situ scenario planets assemble locally. Therefore, planets formed by migration are expected to end up in resonances, whereas those formed in-situ are expected to stay in short period ratios and in non-resonant orbits. Both predictions are at odds with observations. Aims. We investigate whether a preferred formation scenario can be identified through a comparison between the magnetospheric rebound model and the Kepler data. Methods. We conduct N-body simulations of two-planet systems during the disk dispersal phase, and make a statistical comparison between the simulations and the Kepler observations. Results. Comparing the two scenarios, we find that magnetospheric rebound tends to erase the difference in the orbital configuration that was initially presented. After disk dispersal, not all planets are in resonance in the migration scenario, whereas planets do not remain in compact configurations in the in-situ scenario. In both scenarios, the orbits of planets increase with the cavity expansion, and their period ratios have a wider distribution. Conclusions. From a statistical perspective, the magnetospheric rebound model reproduces several observed properties of Kepler planets, such as the significant number of planets are not in resonances and planet pairs can end up at large period ratios. The disparity in orbital configuration between the two formation scenarios is substantially reduced after disk dispersal.Comment: 8 pages, 4 figures, accepted for publication in A&

Formation of TRAPPIST-1 and other compact systems

TRAPPIST-1 is a nearby 0.08 M M-star, which was recently found to harbor a planetary system of at least seven Earth-mass planets, all within 0.1 au. The configuration confounds theorists as the planets are not easily explained by either in situ or migration models. In this Paper we present a scenario for the formation and orbital architecture of the TRAPPIST-1 system. In our model, planet formation starts at the H2O iceline, where pebble-size particles -- whose origin is the outer disk -- concentrate to trigger streaming instabilities. After their formation, planetary embryos quickly mature by pebble accretion. Planet growth stalls at Earth masses, where the planet's gravitational feedback on the disk keeps pebbles at bay. Planets are transported by Type I migration to the inner disk, where they stall at the magnetospheric cavity and end up in mean motion resonances. During disk dispersal, the cavity radius expands and the inner-most planets escape resonance. We argue that the model outlined here can also be applied to other compact systems and that the many close-in super-Earth systems are a scaled-up version of TRAPPIST-1. We also hypothesize that few close-in compact systems harbor giant planets at large distances, since they would have stopped the pebble flux from the outer disk.Comment: 8 pages, accepted for publication in A&

DNA-binding Small Molecules as Drug Agents that Interfere with Transcription Factors: the Development, the Potential and the Future

DNA-minor groove binding small molecules have been extensively developed to achieve higher binding affinity and specificity. Polyamides are a class of small molecules that can be programmed to target any predetermined DNA sequence. The development of hairpin polyamides along with introduction of β-alanine substituents, has greatly enhanced the DNA binding properties of these molecules. Yet the correlation between β-insert and binding properties remains unclear. On the other hand, the design of small-size, fluorescent hybrid polyamides has facilitated cell studies due to their ease of observation. There is a strong need to expand the DNA recognition sites of such molecules and extend their biological applications. This dissertation has explored the systematic design and synthesis of eight-ring hairpin polyamides as well as the modified Pyr-AzaHx hybrid polyamides. Comprehensive biophysical and biochemical tools were employed to evaluate their binding properties. The effects of β-alanine and N-terminal cationic groups on hairpin polyamides-DNA binding have been discussed. The binding properties of modified Pyr-AzaHx polyamides were explored. Altogether, the work provided fundamental guidance for the prediction of binding properties of similar molecules as well as strategies for the design of more competitive molecules. Transcription factors bind to specific DNA sequences in the major groove and regulate gene expression. Abnormal expression of transcription factors is involved in the development of many serious diseases. Precise control of gene expression by targeting transcription factors can be an alternative therapeutic approach. Polyamides bind to DNA with affinities comparable to proteins, empowering them with the ability to interfere with transcription factors at specific DNA binding site and consequently altering the gene expression level. In this dissertation, the effect of polyamides on the binding of transcription factor PU.1 was studied. Abnormal expression of PU.1 is involved in the development of acute myeloid leukemia (AML). A positive correlation was established between eight-ring polyamide binding affinity and inhibition efficacy for PU.1. A non-inhibitor polyamide FH1024 was identified and the mechanism of action among polyamide, DNA and PU.1 was explored. The studies showed strong evidence of the capability of polyamides serving as drug agents. This work also established solid basis for the further cell studies

Surgery on links of linking number zero and the Heegaard Floer dd-invariant

We study Heegaard Floer homology and various related invariants (such as the $h$-function) for two-component L-space links with linking number zero. For such links, we explicitly describe the relationship between the $h$-function, the Sato-Levine invariant and the Casson invariant. We give a formula for the Heegaard Floer $d$-invariants of integral surgeries on two-component L-space links of linking number zero in terms of the $h$-function, generalizing a formula of Ni and Wu. As a consequence, for such links with unknotted components, we characterize L-space surgery slopes in terms of the $\nu^{+}$-invariants of the knots obtained from blowing down the components. We give a proof of a skein inequality for the $d$-invariants of $+1$ surgeries along linking number zero links that differ by a crossing change. We also describe bounds on the smooth four-genus of links in terms of the $h$-function, expanding on previous work of the second author, and use these bounds to calculate the four-genus in several examples of links.Comment: This version accepted for publication in Quantum Topolog

Dynamical rearrangement of super-Earths during disk dispersal I. Outline of the magnetospheric rebound model

The Kepler mission has discovered that multiple close-in super-Earth planets are common around solar-type stars, but their period ratios do not show strong pile-ups near mean motion resonances (MMRs). One scenario is that super-Earths form in a gas-rich disk, and they interact gravitationally with the surrounding gas, inducing their orbital migration. Disk migration theory predicts, however, that planets would end up at resonant orbits due to their differential migration speed. Motivated by the discrepancy between observation and theory, we seek for a mechanism that moves planets out of resonances. We examine the orbital evolution of planet pairs near the magnetospheric cavity during the gas disk dispersal phase. Our study determines the conditions under which planets can escape resonances. We perform two-planet N-body simulations, varying the planet masses, stellar magnetic field strengths, disk accretion rates and gas disk depletion timescales. As planets migrate outward with the expanding magnetospheric cavity, their dynamical configurations can be rearranged. Migration of planets is substantial (minor) in a massive (light) disk. When the outer planet is more massive than the inner planet, the period ratio of two planets increases through outward migration. On the other hand, when the inner planet is more massive, the final period ratio tends to remain similar to the initial one. Larger stellar magnetic field strengths result in planets stopping their migration at longer periods. We highlight \textit{magnetospheric rebound} as an important ingredient able to reconcile disk migration theory with observations. Even when planets are trapped into MMR during the early gas-rich stage, subsequent cavity expansion would induce substantial changes to their orbits, moving them out of resonance.Comment: 10 pages, 5 figures, accepted for publication in A&