Abstract:
Late career workers with lump-sum retirement plans face important financial decisions. Because ability and confidence interact when making financial decisions, this paper asks two related questions: Are workers approaching retirement financially capable of making decisions regarding lump-sum retirement benefits? Do they see themselves as capable? Comparing self-assessed measures of financial capacity to more objective measures reveals patterns of under and overconfidence in financial matters by socio-economic group. Comparing questions correlation to overall score, compound interest concepts are found to be strongly correlated with overall financial literacy. Findings suggest that financial planners may act as partial compensation for a lack of financial literacy, especially among the wealthy. This papers findings support: (1) developing tools that help potential clients engage and assess the work of financial managers, (2) including both self-assessment and objective finance questions on planners’ client intake surveys and (3) facilitating client understanding of compound interest concepts.

Abstract:
Objective: Women with Sickle Cell Disease (SCD) who become pregnant are at risk for serious maternal and fetal complications. Our objective was to determine if pregnancy outcome is dependent on phenotype. Methods: Retrospective cohort study of pregnant women with SCD, including hemoglobin (Hb) SS, Hb SC, and Hb Sβ-thalassemia, between January 1999 and December 2008). Antenatal and neonatal outcomes were compared between pregnancies with painful episodes and those without. The primary outcome was preterm birth (PTB) <37 weeks. Secondary outcomes included maternal medical complications, antenatal complications, delivery outcomes, and neonatal outcomes. Results: 31 women were included (18 (58%) with painful episodes, 13 (42%) without painful episodes). The median number of painful episodes was 2.5 (1 - 19) and these women required a median of 13 total days (1 - 59) of inpatient treatment. At delivery, women who had experienced painful episodes had lower Hb levels and were more likely to be taking chronic narcotic pain medications. The overall incidence of PTB <37wks was 55% and was not significantly different between groups (11 [61%] with painful episodes versus 6 [46%] without painful episodes; p = 0.485). Secondary outcomes were also not significantly different between groups. There was one maternal death. Conclusion: Adverse obstetrical out-comes were more common among women with sickle cell disease who experienced painful crises however, in this small sample, the difference were not statistically significant.

Abstract:
What is the scaling limit of diffusion limited aggregation (DLA) in the plane? This is an old and famously difficult question. One can generalize the question in two ways: first, one may consider the {\em dielectric breakdown model} $\eta$-DBM, a generalization of DLA in which particle locations are sampled from the $\eta$-th power of harmonic measure, instead of harmonic measure itself. Second, instead of restricting attention to deterministic lattices, one may consider $\eta$-DBM on random graphs known or believed to converge in law to a Liouville quantum gravity (LQG) surface with parameter $\gamma \in [0,2]$. In this generality, we propose a scaling limit candidate called quantum Loewner evolution, QLE$(\gamma^2, \eta)$. QLE is defined in terms of the radial Loewner equation like radial SLE, except that it is driven by a measure valued diffusion $\nu_t$ derived from LQG rather than a multiple of a standard Brownian motion. We formalize the dynamics of $\nu_t$ using an SPDE. For each $\gamma \in (0,2]$, there are two or three special values of $\eta$ for which we establish the existence of a solution to these dynamics and explicitly describe the stationary law of $\nu_t$. We also explain discrete versions of our construction that relate DLA to loop-erased random walk and the Eden model to percolation. A certain "reshuffling" trick (in which concentric annular regions are rotated randomly, like slot machine reels) facilitates explicit calculation. We propose QLE$(2,1)$ as a scaling limit for DLA on a random spanning-tree-decorated planar map, and QLE$(8/3,0)$ as a scaling limit for the Eden model on a random triangulation. We propose using QLE$(8/3,0)$ to endow pure LQG with a distance function, by interpreting the region explored by a branching variant of QLE$(8/3,0)$, up to a fixed time, as a metric ball in a random metric space.

Abstract:
We establish existence and uniqueness for Gaussian free field flow lines started at {\em interior} points of a planar domain. We interpret these as rays of a random geometry with imaginary curvature and describe the way distinct rays intersect each other and the boundary. Previous works in this series treat rays started at {\em boundary} points and use Gaussian free field machinery to determine which chordal \SLE_\kappa(\rho_1; \rho_2) processes are time-reversible when \kappa < 8. Here we extend these results to whole-plane \SLE_\kappa(\rho) and establish continuity and transience of these paths. In particular, we extend ordinary whole-plane SLE reversibility (established by Zhan for \kappa \in [0,4]) to all \kappa \in [0,8]. We also show that the rays of a given angle (with variable starting point) form a space-filling planar tree. Each branch is a form of SLE_\kappa for some \kappa \in (0, 4), and the curve that traces the tree in the natural order (hitting x before y if the branch from x is left of the branch from y) is a space-filling form of \SLE_{\kappa'} where \kappa':= 16/\kappa \in (4, \infty). By varying the boundary data we obtain, for each \kappa'>4, a family of space-filling variants of \SLE_{\kappa'}(\rho) whose time reversals belong to the same family. When \kappa' \geq 8, ordinary \SLE_{\kappa'} belongs to this family, and our result shows that its time-reversal is \SLE_{\kappa'}(\kappa'/2 - 4; \kappa'/2 - 4). As applications of this theory, we obtain the local finiteness of \CLE_{\kappa'}, for \kappa' \in (4,8), and describe the laws of the boundaries of \SLE_{\kappa'} processes stopped at stopping times.

Abstract:
Liouville quantum gravity (LQG) and the Brownian map (TBM) are two distinct models of measure-endowed random surfaces. LQG is defined in terms of a real parameter $\gamma$, and it has long been believed that when $\gamma = \sqrt{8/3}$, the LQG sphere should be equivalent (in some sense) to TBM. However, the LQG sphere comes equipped with a conformal structure, and TBM comes equipped with a metric space structure, and endowing either one with the other's structure has been an open problem for some time. This paper is the first in a three-part series that unifies LQG and TBM by endowing each object with the other's structure and showing that the resulting laws agree. The present work uses a form of the quantum Loewner evolution (QLE) to construct a metric on a dense subset of a $\sqrt{8/3}$-LQG sphere and to establish certain facts about the law of this metric, which are in agreement with similar facts known for TBM. The subsequent papers will show that this metric extends uniquely and continuously to the entire $\sqrt{8/3}$-LQG surface and that the resulting measure-endowed metric space is TBM.

Abstract:
The Brownian map is a random sphere-homeomorphic metric measure space obtained by "gluing together" the continuum trees described by the $x$ and $y$ coordinates of the Brownian snake. We present an alternative "breadth-first" construction of the Brownian map, which produces a surface from a certain decorated branching process. It is closely related to the peeling process, the hull process, and the Brownian cactus. Using these ideas, we prove that the Brownian map is the only random sphere-homeomorphic metric measure space with certain properties: namely, scale invariance and the conditional independence of the inside and outside of certain "slices" bounded by geodesics. We also formulate a characterization in terms of the so-called L\'evy net produced by a metric exploration from one measure-typical point to another. This characterization is part of a program for proving the equivalence of the Brownian map and Liouville quantum gravity with parameter $\gamma= \sqrt{8/3}$.

Abstract:
We show that the unit area Liouville quantum gravity sphere can be constructed in two equivalent ways. The first, which was introduced by the authors and Duplantier, uses a Bessel excursion measure to produce a Gaussian free field variant on the cylinder. The second uses a correlated Brownian loop and a "mating of trees" to produce a Liouville quantum gravity sphere decorated by a space-filling path. In the special case that $\gamma=\sqrt{8/3}$, we present a third equivalent construction, which uses the excursion measure of a $3/2$-stable L\'evy process (with only upward jumps) to produce a pair of trees of quantum disks that can be mated to produce a sphere decorated by SLE$_6$. This construction is relevant to a program for showing that the $\gamma=\sqrt{8/3}$ Liouville quantum gravity sphere is equivalent to the Brownian map.

Abstract:
Given a simply connected planar domain D, distinct points x,y \in \partial D, and \kappa >0, the Schramm-Loewner evolution SLE_\kappa is a random continuous non-self-crossing path in the closure of D from x to y. The SLE_\kappa(\rho_1;\rho_2) processes, defined for \rho_1, \rho_2 > -2, are in some sense the most natural generalizations of SLE_\kappa. When \kappa \leq 4, we prove that the law of the time-reversal of an \SLE_\kappa(\rho_1;\rho_2) from x to y is, up to parameterization, an SLE_\kappa(\rho_2;\rho_1) from y to x. This assumes that the "force points" used to define SLE_\kappa(\rho_1;\rho_2) are immediately to the left and right of the SLE seed. A generalization to arbitrary (and arbitrarily many) force points applies whenever the path does not (or is conditioned not to) hit the boundary of D except at the endpoints. The time-reversal symmetry has a particularly natural interpretation when the paths are coupled with the Gaussian free field and viewed as rays of a random geometry. It allows us to couple two instances of the Gaussian free field (with different boundary conditions) so that their difference is almost surely constant on either side of the path. In a fairly general sense, adding appropriate constants to the two sides of a ray reverses its orientation.

Abstract:
Suppose that D is a planar Jordan domain and x and y are distinct boundary points of D. Fix \kappa \in (4,8) and let \eta\ be an SLE_\kappa process from x to y in D. We prove that the law of the time-reversal of \eta is, up to reparameterization, an SLE_\kappa process from y to x in D. More generally, we prove that SLE_\kappa(\rho_1;\rho_2) processes are reversible if and only if both \rho_i are at least \kappa/2-4, which is the critical threshold at or below which such curves are boundary filling. Our result supplies the missing ingredient needed to show that for all \kappa \in (4,8) the so-called conformal loop ensembles CLE_\kappa\ are canonically defined, with almost surely continuous loops. It also provides an interesting way to couple two Gaussian free fields (with different boundary conditions) so that their difference is piecewise constant and the boundaries between the constant regions are SLE_\kappa curves.

Abstract:
Fix constants \chi >0 and \theta \in [0,2\pi), and let h be an instance of the Gaussian free field on a planar domain. We study flow lines of the vector field e^{i(h/\chi+\theta)} starting at a fixed boundary point of the domain. Considering all \theta \in [0,2\pi), one obtains a family of curves that look locally like SLE_\kappa, with \kappa \in (0,4), where \chi = 2/\kappa^{1/2} - \kappa^{1/2}/2, which we interpret as the rays of a random geometry with purely imaginary curvature. We extend the fundamental existence and uniqueness results about these paths to the case that the paths intersect the boundary. We also show that flow lines of different angles cross each other at most once but (in contrast to what happens when h is smooth) may bounce off of each other after crossing. Flow lines of the same angle started at different points merge into each other upon intersecting, forming a tree structure. We construct so-called counterflow lines (SLE_{16/\kappa}) within the same geometry using ordered "light cones" of points accessible by angle-restricted trajectories and develop a robust theory of flow and counterflow line interaction. The theory leads to new results about SLE. For example, we prove that SLE_\kappa(\rho) processes are almost surely continuous random curves, even when they intersect the boundary, and establish Duplantier duality for general SLE_{16/\kappa}(\rho) processes.