Wednesday, February 1, 2023
02/01/2023 - 1:00pm
In this talk I will introduce an efficient method for solving 2nd order, linear, homogeneous ODEs whose solutions may vary between highly oscillatory and slowly changing over the solution interval.
The solver employs two methods: in regions where the solution varies slowly, it uses Chebyshev-grid based collocation with an adaptive stepsize, but in the highly oscillatory phase it automatically switches to constructing a local phase function. I propose a defect-correction iteration that gives an asymptotic series for such a phase function; this is numerically approximated on a Chebyshev grid. In the talk I will present how the method fits in the landscape of oscillatory solvers, details of the algorithm, results from numerical experiments, and a brief error analysis.
02/01/2023 - 4:15pm
Liouville quantum gravity (LQG) is aof surfaces that originated from string . Schramm Loewner evolution (SLE) is a family of curves describing scaling limits of many 2D models at their criticality. Before the rigorous study via LQG and SLE in probability, surfaces and scaling limits of models have been studied via another approach in theoretical physics called (CFT) since the 1980s. In this talk, I will demonstrate how a combination of ideas from LQG/SLE and CFT can be used to rigorously prove several long standing predictions in physics on surfaces and models, including the law of the modulus of the scaling limit of uniform triangulation of the annular topology, and the crossing formula for critical percolation on an annulus. I will then present some conjectures which further illustrate the deep and rich interaction between LQG/SLE and CFT. Based on joint works with Ang, Holden, Remy, Xu, and Zhuang.