Interplay of structures in conformal and universal random geometry

Introduction

My overall goal is to provide novel conceptual understanding of persistent challenges in mathematical physics, in light of recent discoveries of myself and others. The emphasis is especially in finding connections between different areas, making use of my expertise at their crossroads.

Aims Overview

The first two aims concern statistical mechanics (SM) and mathematical formulations of (logarithmic) conformal field theory (CFT), on the one hand algebraically and on the other hand probabilistically. The last two aims focus on connections and interplay of structures arising in SM, such as Schramm-Loewner evolutions (SLE), with algebro-geometric formulations of CFT. Gaining conceptual understanding is fundamental for progress towards deep results.

Aim 1: CFT Correlation Functions

Specifically, in Aim 1, I focus on CFT correlation functions and plan to reveal non-semisimple and logarithmic behavior, which is poorly understood even in the physics literature. For this, hidden symmetries from my earlier work will be exploited.

Aim 2: Probability Theory and Critical SM Models

Aim 2 combines this with probability theory: investigations of non-local quantities in critical SM models, relating to specific CFT correlation functions and to SLE.

Aim 3: Interplay of SLE, CFT, and Teichmüller Theory

In Aim 3, I investigate the interplay of SLE, CFT, and Teichmüller theory in terms of generalizations of so-called Loewner energy of curves. The main objective is to shed light on the hidden geometric interpretation of Loewner energy from the point of view of formulations of CFT in terms of Riemann surfaces, and eventually also to find its role within geometric quantization.

Aim 4: Isomonodromic Deformations

To elaborate the latter goal, Aim 4 combines these ideas with related structures in the theory of isomonodromic deformations. My starting point is the observation that Loewner energy minima and semiclassical limits of certain CFT correlations are both described by isomonodromic systems. I plan to make these connections explicit and implement them in order to discover intrinsic features of the interplay of the aforementioned structures.