By Fanze Kong
Welcome back to a new academic year and a fresh quarter! As I write this on a beautiful autumn afternoon, I am delighted to share my experiences over the last year in the Applied Math Department at UW and to offer some of my thoughts on the breadth and depth of applied mathematics.
Growing up in a small town in China with few sources of entertainment, I discovered both joy and challenge in exploring the world of mathematics. As Albert Einstein writes in his landmark book The World as I See It, ``The most beautiful experience we can have is the mysterious." While saying this, he certainly means that there is beauty in remaining curious about the enigmatic and mysterious phenomena around us. This sense of curiosity has continually shaped my journey in mathematics, inspiring me to choose it as my major and ultimately motivating me to pursue research as a postdoctoral scholar at the University of Washington. In this journey, I have discovered the fascination of mathematics through its profound and far-reaching applications across the natural and social sciences, revealing patterns and insights that deepen our understanding of the world. This perspective was first sparked during the third year of my undergraduate studies, in the course Introduction to PDEs, where my instructor often remarked, ``Everything is a partial differential equation (PDE)." (Of course, data science was not so popular at that time.) Inspired by this guiding principle, I pursued research on biological PDE problems [1] under his mentorship—an experience that not only deepened my understanding of applied math, but also led me successfully to the completion of my Master’s degree.
Witnessing the intimate interplay between rigorous derivation and insightful mathematical interpretation, I firmly recognized the power of mathematics to illuminate diverse phenomena across physics, biology, and the social sciences. Such fascination with mathematics further motivated me to eagerly explore the broad landscape of applied mathematics, leading me to apply for a PhD program. Fortunately, I was accepted by two supervisors at the University of British Columbia and joined their research group. One of my supervisors, a preeminent scholar in the analysis of nonlinear PDEs, introduced me to a variety of mathematical tools in the analysis of reaction-diffusion equations (a representative class of nonlinear PDEs) and I eagerly immersed myself in these elegant techniques, surprisingly discovering that many of them originated from applied problems. This experience led me to view myself as an analyst, driven by the inherent challenges of applied mathematics rather than being confined to applications in any specific branch of the natural or social sciences. Focusing on the analysis of Keller-Segel models, which describe the collective cellular movement, I have acquired a range of mathematical techniques such as variational methods, bifurcation analysis, reduction approaches, etc [2]. I later employed these techniques to study mean-field game models, which serve as a paradigm for describing strategic interactions among a large number of players in the financial market [3]. At this stage, I became deeply absorbed in identifying commonalities between equations arising in diverse contexts, focusing particularly on the mathematical theories underlying the methods used to tackle problems in applied mathematics. I came to embrace the belief that much of the most elegant mathematical theories has grown out of attempts to understand real-world phenomena. A highlight in this journey is that I won the Cecil Graham Doctoral Dissertation Award and presented a lecture at the CAIMS/SIAM Joint Annual Meeting 2025 in Montreal.
The Department of Applied Mathematics at the University of Washington offered me the opportunity to continue my journey as a postdoctoral scholar (Pearson Fellow), where I have greatly enjoyed engaging in stimulating conversations with faculty members from diverse research areas within the department. During many profound conversations with Prof. Hong Qian, I began to view applied mathematics from a completely new perspective. It is too narrow to regard applied mathematics merely as a tool that serves other sciences, nor should it be seen as the construction of highly abstract mathematical theories without practical relevance. Rather, it lies in between -- a discipline that seeks to uncover the underlying structures connecting theory and real-world phenomena, and one that cultivates its own distinctive features and culture. Also, Prof. Hong Qian introduced me to the vast landscape of statistical physics and showed how physical principles can emerge from pure mathematical theory. He also believes that the mathematical foundations of data science should be developed in a similar way, reflecting deep connections with statistical physics. Inspired by this vision, I began exploring questions at the interface of data science and Fokker–Planck equations (See its application in large language models [4]) with the non-gradient drift, which resulted in the completion of several papers, e.g. [5]. This experience made me realize that applied mathematics is truly a discipline for perceiving the world. Many fundamental principles from physics can be derived within the framework of applied mathematics [6]. My journey in applied mathematics continues, and I am excited by how the field has evolved over the past centuries.
From my personal perspective (Of course taught by Prof. Hong Qian), the scope and focus of applied mathematics have evolved. It has gradually separated from pure mathematics to become a discipline aimed at explaining the world. Applied mathematics is neither merely a tool for addressing specific engineering problems nor an abstract mathematical theory completely independent of practical applications.
[1]. L. Chen, F. Kong, and Q. Wang, Global and exponential attractor of the repulsive Keller--Segel model with logarithmic sensitivity. Eur. J. Appl. Math., 32(4):599--617, 2021.
[2]. F. Kong, J. Wei, and L. Xu, Existence of multi-spikes in the Keller--Segel model with logistic growth. Math. Models Methods Appl. Sci., 33(11):2227--2270, 2023.
[3]. M. Cirant, F. Kong, J. Wei, X. Zeng, Critical mass phenomena and blow-up behaviors of ground states in stationary second order mean-field games systems with decreasing cost. J. Math. Pures Appl., 198: 103687, 2025.
[4]. J. Carson, A Stochastic Dynamical Theory of LLM Self-Adversariality: Modeling Severity Drift as a Critical Process. arXiv preprint arXiv:2501.16783, 2025.
[5]. F. Kong, C. Lai, and Y. Lu, Moment Estimate and Variational Approach for Learning Generalized Diffusion with Non-gradient Structures. arXiv preprint arXiv:2508.01854, 2025.
[6]. B. Miao, H. Qian, and Y. Wu, Emergence of Newtonian deterministic causality from stochastic motions in continuous space and time. arXiv preprint arXiv:2406.02405, 2024.