台大校區機率演講公告Probability Seminars on NTU Campus

This is the third and last talk of this series. In the first talk, we discussed about the (1-dimensional) Selberg central limit theorem, and then we discussed its multi-dimensional variant in the second talk. In these works, the main probabilistic techniques are method of moments and Cramer-Wold devices, and therefore we lost track of its rate of convergence. In light of this issue, in the third talk, we will focus on the study of rate of convergence of the Selberg central limit theorem.

Still, we will emphasize the probabilistic aspect of this work and avoid non-probabilistic technologies. We will also review essential contents from previous talks; therefore, audiences who missed the previous talks are also welcome. Lastly, this talk is based on a joint work with Peng-Jie Wong: "On Rate of Convergence in Central Limit Theorem of Selberg and Bourgade" (https://doi.org/10.48550/arXiv.2601.02781).

Lecture 1: Taming Infinities.
Lecture 2: A gentle introduction to regularity structures.
Lecture 3: Constructing quantum gauge theories.

Continuing from the last talk, we will discuss the multivariate Selberg central limit theorem. The audiences who missed last talk is also welcome since we will review essential contents from last talk. Still, we will focus on probabilistic techniques used in this work and avoid non-probabilistic terminologies. This talk is based on the article: Log-correlation for shifted Dirichlet 𝐿-functions and Dedekind zeta functions of quadratic fields (https://doi.org/10.1515/forum-2023-0224).

In this talk, I will talk about the univariate case of Selberg central limit theorem in a bit more detail. We will mainly discuss the new method introduced by Radziwill and Soundararajan (DOI 10.4171/LEM/63-1/2-1) and will avoid non-probabilistic technical calculations.

The Mandelbrot Percolation serves one of the most intuitive examples of a fractal set that is the projection of a Galton-Watson tree, with good separation and hence dimensional properties.

Yet, attempts on understanding its properties finer than dimensions also yield substantial results. This talk highlights Steffen Winter's takes on controlling from below its critical probability for percolation via meticulous studies on two forms of nearest-neighbour relations.

Reference:

Klatt, M. A.; Winter, S.
Geometric functionals of fractal percolation
2020. Advances in applied probability, 52 (4), 1085–1126

Klatt, M. A.; Winter, S.
Almost sure convergence and second moments of geometric functionals of fractal percolation
2023. Advances in Applied Probability, 55 (4), 1–33