The energy of a deterministic Loewner chain: Reversibility and interpretation via SLE0+_{0+}

We study some features of the energy of a deterministic chordal Loewner chain, which is defined as the Dirichlet energy of its driving function. In particular, using an interpretation of this energy as a large deviation rate function for SLE$_\kappa$ as $\kappa$ tends to 0 and the known reversibility of the SLE$_\kappa$ curves for small $\kappa$, we show that the energy of a deterministic curve from one boundary point A of a simply connected domain D to another boundary point B, is equal to the energy of its time-reversal ie. of the same curve but viewed as going from B to A in D.Comment: 28 pages, 5 figures, minor changes in Sec. 2.2., to appear in J. Europ. Math. So

A note on Loewner energy, conformal restriction and Werner's measure on self-avoiding loops

In this note, we establish an expression of the Loewner energy of a Jordan curve on the Riemann sphere in terms of Werner's measure on simple loops of SLE$_{8/3}$ type. The proof is based on a formula for the change of the Loewner energy under a conformal map that is reminiscent of the restriction properties derived for SLE processes.Comment: 11 pages, 3 figure

A new inflationary Universe scenario with inhomogeneous quantum vacuum

We investigate the quantum vacuum and find the fluctuations can lead to the inhomogeneous quantum vacuum. We find that the vacuum fluctuations can significantly influence the cosmological inhomogeneity, which is different from what previously expected. By introducing the modified Green's function, we reach a new inflationary scenario which can explain why the Universe is still expanding without slowing down. We also calculate the tunneling amplitude of the Universe based on the inhomogeneous vacuum. We find that the inhomogeneity can lead to the penetration of the universe over the potential barrier faster than previously thought.Comment: 16 pages, 9 figure

Interplay between Loewner and Dirichlet energies via conformal welding and flow-lines

The Loewner energy of a Jordan curve is the Dirichlet energy of its Loewner driving term. It is finite if and only if the curve is a Weil-Petersson quasicircle. In this paper, we describe cutting and welding operations on finite Dirichlet energy functions defined in the plane, allowing expression of the Loewner energy in terms of Dirichlet energy dissipation. We show that the Loewner energy of a unit vector field flow-line is equal to the Dirichlet energy of the harmonically extended winding. We also give an identity involving a complex-valued function of finite Dirichlet energy that expresses the welding and flow-line identities simultaneously. As applications, we prove that arclength isometric welding of two domains is sub-additive in the energy, and that the energy of equipotentials in a simply connected domain is monotone. Our main identities can be viewed as action functional analogs of both the welding and flow-line couplings of Schramm-Loewner evolution curves with the Gaussian free field.Comment: 28 pages, 3 figures. Minor revision according to referees' repor

Evaluating Digital Math Tools in the Field

Many school districts have adopted digital tools to supplement or replace teacher-led instruction, usually based on the premise that these tools can provide more personalized or individualized experiences for students and at lower cost. Rigorously evaluating whether such initiatives promote better student outcomes in the field is difficult as most schools and teachers are unwilling to enforce rigorous study designs such as randomized control trials. We used study designs that were feasible in practice to assess whether two digital math tools, eSpark and IXL, were associated with improvements in 3rd – 6th grade student test scores in math. We also investigated the resource requirements and costs of implementing eSpark and IXL to assess whether these tools represent a valuable use of resources. We find that while IXL is substantially less costly to implement than eSpark, its use is not significantly associated with students’ math performance