Quantile functions and distribution functions are mathematically equivalent ways to define a probability distribution. However, quantile functions have several advantages over distribution functions. In this talk, we present basic concepts of quantile functions and discuss various applications of these concepts in statistical modelling. In particular, we introduce new quantile functions and provide their applications in lifetime data analysis.

We consider a Dirichlet boundary value problem driven by the $(p,q)$-Laplace operator $-\Delta_p u-\mu\Delta_q u$, coupled with a reaction term that combines a critical Sobolev nonlinearity $\lambda |u|^{p^*-2}u$ with a singular nonlinearity $\alpha |u|^{-\gamma-1}u$, where $0<\gamma<1$.

The presence of the singular term destroys the $C^1$-differentiability of the natural energy functional on the whole Sobolev space, while the critical exponent destroys compactness of the associated Sobolev embedding; together, these two features place the problem outside the reach of classical variational methods.

We describe how both obstructions are overcome. The singular nonlinearity is first regularized by a smooth truncation, producing a genuinely $C^1$ approximating functional $I_\varepsilon$ to which an abstract critical point theorem, built on the Alexander--Spanier cohomological index, can be applied. This yields $m$ distinct pairs of nontrivial critical points once a Palais--Smale condition is verified below an explicit energy threshold $c^*$ determined by the best Sobolev constant.

A key technical point is established carefully: almost-everywhere convergence of gradients along Palais--Smale sequences must be obtained through the Boccardo--Murat theorem before the Brézis--Lieb lemma can be legitimately applied to the gradient terms.

Letting the regularization parameter vanish recovers solutions of the original singular problem, and a further limit as the singular parameter $\alpha\to 0^+$ shows that these solutions persist continuously into solutions of the corresponding nonsingular problem. Positivity of all solutions produced is obtained via a comparison argument with an explicit subsolution, combined with Lieberman's boundary regularity theory and the strong maximum principle.

The talk is intended to convey both the main multiplicity and stability results and the recurring technical pitfalls---circular use of compactness lemmas and loss of differentiability from singular terms---that arise in this class of variational problems, along with the methods used to resolve them.

We live in an era of rapidly growing, high-dimensional data generated by sources such as the web, social media, IoT devices, and online platforms. Extracting meaningful insights from such data is the fundamental subroutine of many modern technologies, including search engines, recommendation systems, artificial intelligence, and signal processing. However, the scale and dimensionality of these datasets often make direct computation prohibitively expensive due to the "curse of dimensionality". Dimensionality reduction provides an elegant solution by compressing high-dimensional data into a lower-dimensional representation while preserving the geometric structure of original data, ensuring that the result of the algorithm on the reduced data closely approximates to that of the original dataset.

In this talk, we will explore one of the most fundamental and powerful techniques in this area - JL Lemma (a.k.a. random projection). We will develop intuition for its effectiveness, discuss its theoretical guarantees, and highlight its extensions. We will also examine how this simple yet powerful idea enables faster and more efficient large-scale data analysis in practice.

Note: I plan to present a simpler proof of JL Lemma [1, 2]. The algorithm is based on a simple (random) matrix-vector multiplication approach, and the material should be accessible to anyone with an undergraduate background in linear algebra and probability.

[1] William B Johnson and Joram Lindenstrauss. "Extensions of Lipschitz mappings into a Hilbert space". In: Proceedings of the 1982 Conference in Modern Analysis and Probability. Vol. 26. Contemporary Mathematics. AMS, 1984, pp. 189-206. doi: 10.1090/conm/026/737400. [2] Dasgupta, S. and Gupta, A. An elementary proof of a theorem of Johnson and Lindenstrauss. Random Structures & Algorithms, 22(1):60-65, 2003. https://doi.org/10.1002/rsa.10073.