Mathematical Sciences Research Institute

Home » AGRS Research Seminar Series: "A Deterministic Approach to Loewner-Energy Minimizers" & "The Combinatorial Method and Applications"

Seminar

AGRS Research Seminar Series: "A Deterministic Approach to Loewner-Energy Minimizers" & "The Combinatorial Method and Applications" March 02, 2022 (02:00 PM PST - 03:15 PM PST)
Parent Program: The Analysis and Geometry of Random Spaces MSRI: Simons Auditorium, Online/Virtual
Speaker(s) Adi Glucksam (Northwestern University), Tim Mesikepp (Peking University)
Description No Description
Video
Abstract/Media

To participate in this seminar, please register HERE.

This seminar is presented in 2 talks.

2 - 2:30 PM: "A Deterministic Approach to Loewner-Energy Minimizers" - Tim Mesikepp

Abstract: In her foundational work on the conformal invariance of Loewner energy, Yilin Wang showed that the family of curves from 0 to a fixed $z_0 \in \mathbb{H}$ admits a unique minimizer of Loewner energy, with explicit formula for the energy expenditure.  Wang determined the latter through SLE techniques, but we revisit these curves from a purely deterministic point of view, showing one can derive the energy formula as well other results, such as an explicit computation of the driving function, through deterministic symmetries.  We also introduce a dual'' family of curves which minimize the Loewner energy among all curves which weld together given  $x<y$.  Both families, it turns out, have related properties of universality, algebraicity, and energy usage, which we explain.  The talk will feature plenty of pictures, several open questions, and a surprise appearance of the square root of a circle.

2:45 - 3:15 PM: "The Combinatorial Method and Applications" - Adi Glucksam

Abstract: Let Q be a large cube and let G be a subset of Q of large relative measure. In this talk I will present a refinement of the combinatorial technique used by Jones and Makarov in 95', which allows one to find points with high density of the set G. I will motivate the technique, present some applications, and show this technique is optimal.