• Speaker: Michael McNulty, MSU
• Title: Conditionally Stable Self-Similar Blowup for the Supercritical Quadratic Wave Equation Beyond the Light Cone
• Date: 09/21/2022
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Contact: Willie Wai-Yeung Wong (wongwil2@msu.edu)

Nonlinear wave equations of power-type serve as excellent toy models for geometric PDEs such as the Yang-Mills and wave maps equations. Of great interest in the energy supercritical setting is that of threshold phenomena. In this setting, unstable self-similar blowup solutions are believed to play an essential role in describing the threshold of singularity formation. We will discuss the stability of an explicitly known, unstable self-similar blowup solution of the energy supercritical quadratic wave equation in a region of spacetime which extends beyond the time of blowup. To overcome this instability, we introduce a new canonical method to investigate unstable self-similar solutions. This work represents the first steps toward an understanding of threshold phenomena in the energy supercritical setting.

• Speaker: Konstantin Matetski, MSU
• Title: Polynuclear growth and the Toda lattice
• Date: 10/05/2022
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Contact: Willie Wai-Yeung Wong (wongwil2@msu.edu)

Polynuclear growth is one of the basic models in the Kardar-Parisi-Zhang universality class, which describes a one-dimensional crystal growth. For a particular initial state, it describes the length of the longest increasing subsequence for uniformly random permutations (the problem first studied by S. Ulam). In my joint work with J. Quastel and D. Remenik we expressed the distribution functions of the polynuclear growth in terms of the solutions of the Toda lattice, one of the classical integrable systems. A suitable rescaling of the model yields a non-trivial continuous limit of the polynuclear growth (the KPZ fixed point) and the respective equations (Kadomtsev-Petviashvili).

• Speaker: Dimitris Vardakis, MSU
• Title: Buffon's needle problem for a random planar disk-like Cantor set
• Date: 11/09/2022
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Contact: Willie Wai-Yeung Wong (wongwil2@msu.edu)

The Favard length of the planar $1/4$-corner Cantor set is $0$. Estimates exists about the rate with which the Favard length of the previous steps goes to $0$, but the exact rate of decay is unknown. However, if one considers a random construction of the $1/4$-corner Cantor set, things might seem better. In fact, Peres and Solomyak showed that the rate of decay for the average Favard length for the random $1/4$-corner Cantor set is of order exactly $1/n$. We show that the rate of decay for a random disk-like analogue has again order $1/n$. This suggests that any ``reasonable'' random Cantor set of positive and finite length might decay at the same rate.

• Speaker: Leonardo Abbrescia, Vanderbilt University
• Title: TBA
• Date: 11/30/2022
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall
• Contact: Willie Wai-Yeung Wong (wongwil2@msu.edu)

TBA (likely virtual)