Calendar

Calabi-Yau metrics are central objects in K\”ahler geometry and also string theory. The existence of Calabi-Yau metrics on compact manifolds was answered by Yau in his solution of the Calabi conjecture, but the situation in the non-compact setting is much more delicate, and many questions related to the existence and uniqueness of non-compact Calabi-Yau metrics remain unanswered. I will give an introduction to this subject and discuss some ongoing joint work with T. Collins and S.-T. Yau, on a new relationship between complete Calabi-Yau metrics and a new Monge-Ampere equation.

Friday, Feb. 2nd at 12pm, with lunch, lounge at CMSA (20 Garden Street). Also by Zoom: https://harvard.zoom.us/j/92410768363

Billiards is one of the most-studied dynamical systems, modeling the behavior of a point particle bouncing around some space. If the space is a plane region bounded by an algebraic curve, then we may use techniques from algebraic geometry to study its billiards map. We explain how to view billiards as a complex algebraic correspondence, and we prove upper and lower bounds on the dynamical degree, the growth rate of the degrees of the iterates, in terms of the degree of the boundary curve. These degree growth rates are studied in mathematical physics, broadly speaking, as a way to identify integrable (exactly solvable) physical models. In our setting, this theory gives us an upper bound on the entropy, or chaos, of billiards in curves.

This semester we will go through the work of Burklund, Hahn, Levy and Schlank on the construction of counterexamples to the telescope conjecture.

Calabi-Yau metrics are central objects in K\”ahler geometry and also string theory. The existence of Calabi-Yau metrics on compact manifolds was answered by Yau in his solution of the Calabi conjecture, but the situation in the non-compact setting is much more delicate, and many questions related to the existence and uniqueness of non-compact Calabi-Yau metrics remain unanswered. I will give an introduction to this subject and discuss some ongoing joint work with T. Collins and S.-T. Yau, on a new relationship between complete Calabi-Yau metrics and a new Monge-Ampere equation.

Friday, Feb. 2nd at 12pm, with lunch, lounge at CMSA (20 Garden Street). Also by Zoom: https://harvard.zoom.us/j/92410768363

Stochastic partial differential equations (PDEs) find extensive applications across diverse domains such as physics, finance, biology, and engineering, serving as effective tools for modeling systems influenced by random factors. The analysis of the patterns in the peaks and valleys of stochastic PDEs is crucial for gaining deeper insights into the underlying physical phenomena.

One notable example is the KPZ equation, a fundamental stochastic PDE associated with significant models like random growth processes, Burgers turbulence, interacting particle systems, and random polymers. The study of the fractal structures inherent in the KPZ equation provides a quantitative characterization of the intermittent nature of its peaks, as well as those of the stochastic heat equation—a subject that has been extensively explored over the past few decades.

Conversely, the Parabolic Anderson model (PAM) serves as a prototypical framework for simulating the conduction of electrons in crystals containing defects. Investigating the intermittency of peaks in the PAM has been a prominent area of research, closely tied to the phenomenon of Anderson localization.

In this presentation, we delve into the fractal geometry of both the KPZ equation and the PAM, unveiling their multifractal nature. Specifically, we demonstrate that the spatial and spatio-temporal peaks of these equations exhibit infinitely many distinct values. Furthermore, we compute the macroscopic Hausdorff dimension (introduced by Barlow and Taylor) associated with these peaks.

The key findings presented here stem from a series of works that employ a diverse array of tools, ranging from random matrix theory and the Gibbs property of random curves to the utilization of regularity structures and paracontrolled calculus.