Title: Self-organized dynamics: aggregation and flocking
Abstract: Self-organized behaviors are commonly observed in nature and human societies, such as bird flocks, fish swarms and human crowds. In this talk, I will present some celebrated mathematical models, with simple small-scale interactions that lead to the emergence of global behaviors: aggregation and flocking. In particular, I will focus on the Euler-alignment system, and present some recent progress on the global wellposeness theory, large time behaviors, as well as interesting connections to classical equations in fluid mechanics.

Title: A variational approach to multi-dimensional scaling
Abstract: Dimension reduction is a crucial pre-processing step in many learning algorithms, and can be seen as a particular form of metric or graph embedding problem. Despite the widespread use of these algorithms, many of the non-linear versions of these algorithms lack rigorous theory. This talk will discuss recent work on multi-dimensional scaling, one of the most basic forms of non-linear dimension reduction. In particular, we pose this as a variant of an optimal transportation problem with a free marginal in the target space; in particular this can be seen as a Gromov-Wasserstein projection problem. We utilize a variational approach to derive regularity properties of the optimal embeddings selected by a simple version of multi-dimensional scaling, and provide new asymptotic consistency results for the algorithm.

Title: Positive weak solutions of nonlocal parabolic problems with logistic reaction term
Abstract: We study a parabolic reaction-diffusion equation with logistic reaction term and the fractional Laplacian as the diffusion operator. We discuss existence of a positive weak solution by constructing appropriate ordered sub- and supersolutions.

Title: Unique Maximum of Expected Score in Adaptive Algorithm- A New Approach for Secretary Problem
Abstract: Secretary problem appeared in the late 1950s and early 1960s. Starting from the idea of choosing the best secretary from a given number of candidates, the variants of this famous math-based decision-making have related to certain practical problems such as investment procedures and atomic bomb inspection programs. The Adaptive Algorithm suggested in Zhou et al. (2021) has drawn the main objective of Secretary Problem to maximize a numerical score for each candidate evaluation. The main goal of this project is to establish the existence and optimality of Adaptive Algorithm strategy by proving the existence of a unique maximum of expected score in this algorithm as well as limit of sequence of unique maximizer {xn} of each expected score function when n approaches infinity.