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.

In this talk, we will talk about some optimization problems related to the Riemann zeta function and $L$-functions. In particular, we will talk about the distribution of the low-lying zeros of families of $L$-functions. We will see how we can use the one-level density theorems in the literature to estimate the proportion of non-vanishing of $L$-functions in a family at a low-lying height on the critical line. This is based on joint work with E. Carneiro and M. B. Milinovich. 

The notion of viscosity solution was introduced in 1980s by Evans and Crandall/Lions, which is one of the most important developments in the  theory of elliptic equations.  It provides a rigorous mathematical framework to describe the correct ``physical" solution of  first or second order PDEs when classical solutions might not exist. Important examples include first order Hamilton-Jacobi equations or second order degenerate elliptic equations (e.g mean curvature type equations) arising from control theory or front propagation problems in real applications.   In this talk, I will go over basic definitions, some important techniques, fundamental results and interesting examples.