We analyze Loewner traces driven by functions asymptotic to κ √ 1 − t. We prove a stability result when κ = 4 and show that κ = 4 can lead to non locally connected hulls. As a consequence, we obtain a driving term λ(t) so that the hulls driven by κλ(t) are generated by a continuous curve for all κ > 0 with κ = 4 but not when κ = 4, so that the space of driving terms with continuous traces is not convex. As a byproduct, we obtain an explicit construction of the traces driven by κ √ 1 − t and a conceptual proof of the corresponding results of Kager, Nienhuis and Kadanoff.

for κ near 0. In this paper, we give improved lower bounds on the optimal Hölder exponent for two natural parametrizations of the SLE trace. Our estimates give a Hölder exponent near 1 for κ near 0, as expected. The work of I. Binder and B. Duplantier (2002) suggests that our results are optimal for the two parametrizations considered.

Similar to the well-known phases of SLE, the Loewner differential equation with Lip(1/2) driving terms is known to have a phase transition at norm 4, when traces change from simple to non-simple curves. We establish the deterministic analog of the second phase transition of SLE, where traces change to space-filling curves: There is a constant C > 4 such that a Loewner driving term whose trace is space filling has Lip(1/2) norm at least C. We also provide a geometric criterion for traces to be driven by Lip(1/2) functions, and show that for instance the Hilbert space filling curve and the Sierpinski gasket fall into this class.

The regularity of Loewner curves Huy V. Tran Chair of the Supervisory Committee:Professor Steffen Rohde MathematicsThe Loewner differential equation, a classical tool that has attracted recent attention due to Schramm-Loewner evolution (SLE), provides a unique way of encoding a simple 2-dimensional curve into a continuous 1-dimensional driving function. In this thesis we study the curve in three cases according to the regularity of driving function: weakly Hölder-1/2, Hölder-1/2 with norm less than 4 and C α with α ∈ (1/2, ∞). In the first case, given the existence of the curve we show that the standard algorithm simulating the curve converges in a strong sense. One direct application is for simulating SLE. In the second case, wegive another proof of Marshall, Rohde [26] and Lind [22] in which the curve exists and is a quasi-arc. A sufficient condition for the rectifiability of the curve is also given. In the final case, we show that the Loewner curve is in C α+1/2 . The thesis is a combination of three projects [39], [32] and [21] which are joint work with

Through the Loewner equation, real-valued driving functions generate sets called Loewner hulls. We analyze driving functions that approach 0 at least as fast as a