In various cases, a law that holds in a group with high probability, must actually hold for all elements. For instance, a finite group in which the commutator law [x,y]=1 holds with probability at least 5/8, must be abelian. For infinite groups, one needs to work a bit harder to define the probability that a given law holds. One natural way is by sampling a random element uniformly from the r-ball in the Cayley graph and taking r to infinity; another way is by sampling elements using random walks. It was asked by Amir, Blachar, Gerasimova, and Kozma whether a law that holds with probability 1, must actually hold globally, for all elements. In a recent joint work with Be’eri Greenfeld, we give a negative answer to their question.

In this talk I will give an introduction to probabilistic group laws and present a finitely generated group that satisfies the law x^p=1 with probability 1, but yet admits no group law that holds for all elements.

Percolation clusters induce an intrinsic graph distance called the chemical distance. Besides its mathematical appeal, the chemical distance is connected to the study of random walks on critical percolation clusters. In this talk, I will begin with a brief survey on the chemical distance. Then, I will zoom in to the progress and challenges in the 2d critical regime. A portion of this talk is based on joint work with Philippe Sosoe.

We introduce an efficient and robust auto-tuning framework for hyperparameter selection in dimension reduction (DR) algorithms, focusing on large-scale datasets and arbitrary performance metrics. By leveraging Bayesian optimization with a surrogate model, our approach enables efficient hyperparameter selection with multi-objective trade-offs and allows us to perform data-driven sensitivity analysis. By incorporating normalization and subsampling, the proposed framework demonstrates versatility and efficiency, as shown in applications to visualization techniques such as t-SNE and UMAP. We evaluate our results on various synthetic and real-world datasets using multiple quality metrics, providing a robust and efficient solution for hyperparameter selection in DR algorithms.

I will talk about sharp isoperimetric inequalities on the hypercube refining several
results due to Kahn--Park and Talagrand.  In the second half of the talk I will present  Talagrand's isoperimetric inequalities
for functions taking values in a Banach space having finite cotype. In addition to that, several different proofs
 of recently resolved Talagrand's conjecture will be presented. 

This is joint with David Beltran and José Ramón Madrid Padilla.