Fractals, largely characterized as structures that look the same at different scales, are ubiquitous in nature and the sciences. They are intriguing for both their aesthetic appeal and for the fact that despite their high complexity they often arise as the result of very simple mechanisms. On the mathematical side, such structures arise when repeatedly applying a simple rule such as a quadratic polynomial. In nature and in statistical physics models, very similar patterns and shapes emerge from randomness. The Principal Investigator’s research is mostly concerned with two-dimensional fractals that possess some form of scale and rotation invariance, both in the deterministic and in the random setting. Using methods from complex analysis, the principal investigator and his students will work towards answering foundational questions and an understanding of why the deterministic and random patterns are so similar. The project also supports the training of graduate students.
Conformally self-similar two-dimensional sets can often be described through conformal welding. Conformal welding is the process of gluing two Riemann surfaces along their boundaries to form a new surface. It is instrumental in Teichmueller theory, of independent interest in geometric function theory, and has recently been intensely studied in the context of the Schramm-Loewner Evolution (SLE) and Liouville Quantum Gravity (LQG). The PI will explore the recently found connections between Teichmueller theory and Loewner theory. He will apply his new conditions on weldability to the setting of SLE and LQG, aiming at an analytic construction of SLE by gluing quantum discs. He will also apply this technique to obtain conformal spheres from matings of trees, both in the deterministic setting of conformal mating of Julia sets and in the probabilistic setting of the Brownian map as the gluing of two continuum random trees.
This award reflects NSF’s statutory mission and has been deemed worthy of support through evaluation using the Foundation’s intellectual merit and broader impacts review criteria.