My Journey with Mathematics
It was the summer of 2014. Holding my college entrance exam results in hand, I walked up to my high school homeroom teacher and math instructor, Mr. Lei Yunhai, and said, "Mr. Lei, I want to study mathematics." A look of joy crossed his face, but after a brief pause, he replied, "You might want to consider statistics." Back then, big data and artificial intelligence hadn't yet become the buzzwords they are today, and statistics wasn't a trendy discipline. Yet, in my youthful ignorance, I filled out my college application form with the words: Applied Statistics.
Planting the Seeds of Mathematics
In college, I met a remarkable professor, Mr. Wang Shutang, who taught us topology. His lectures were full of students who admire him and he was always peppered with his personal reflections on academic inquiry: "Science requires a spirit of dedication. Mathematics is about distilling the essence of ideas from the works of our predecessors-separating the wheat from the chaff, eliminating the false to retain the true, abstracting and generalizing. It's crucial for science to cultivate keen intuition and a vision that transcends mere logic..."
Up until then, my perception of mathematics was tinged with a faint sense of mystique, shaped by the glowing tributes of Newton and Gauss. Professor Wang's words lit a fire in me. His philosophy resonated deeply, and the seed of a passion for research took root in my heart. Even today, I often wake from dreams recalling his wisdom. Looking back, I realize that my life's trajectory began intertwining with mathematics from that moment onward.
Seven Years of Numbers
Four years later, I was admitted to graduate school to continue studying statistics. Seven years passed with a blink of eyes. Along the way, I made lifelong friends and developed a daily rhythm of mathematical research, spending my days immersed in calculating and problem-solving.
To me, there is no boundary between applied mathematics and pure mathematics. Applied mathematics provides rich examples, while pure mathematics advances theoretical understanding, creating a virtuous cycle. My work in statistics has always swung between tackling real-world challenges and revisiting theoretical foundations. Occasionally, I've fantasized about finding a single, all-encompassing solution to every problem, but I've learned that such shortcuts are illusory.
I often lost myself in my studies, spending entire days in libraries and study rooms, sometimes forgetting to eat. My first paper was on the Karhunen-Lo'eve theorem, where I classified bivariate random fields and established four necessary and sufficient conditions for the non-uniform convergence of random orthogonal series. At the age of 24, I tasted the satisfaction of producing original research, gaining confidence in my ability to make progress in mathematics.
My continued engagement with real-world problems led me to observe that a bivariate random field corresponds naturally to a 2-tensor, or a random matrix. I proposed definitions for four classes of matrix normal distributions and submitted the findings to Annals of Statistics. My growing fascination with tensors led me to extend these results to general matrix rings, reducing the complexity of matrix multiplication to (O(n^{5/2})) which surpasses the classical (O(n^3)). These three papers marked my entry into mathematical research, but I know I owe much of my progress to Professor Wang's inspiring words.
Doctoral Research and Beyond
My doctoral thesis was titled Generalized Additive Processes, a topic steeped in abstraction. In essence, it sought to uncover qualitative shifts arising from the accumulation of quantitative similarities among examples. I observed that additive processes, including Markov processes, share a common feature: they all have a predictable compensator. This observation led to the definition of a generalized additive process as the sum of a predictable process and a martingale, naturally extending the work of Doob and Meyer.
Once the definition was in place, Br'emaud's 1972 conjecture followed almost effortlessly. This result, included in my thesis, yielded further intriguing corollaries, such as the existence of a cluster process that is not infinitely divisible. Some may wonder why a seemingly natural definition requires so much groundwork, but I believe that mathematics derives its strengths from such cumulative efforts. These scattered results coalesce into a cohesive whole, as mathematics relies not on itself to advance but on the synthesis of diverse ideas and perspectives.
Recently, I've turned my attention to harmonic functions and representation theory, exploring their intricacies through practical examples. Unlike Professor Wang, whose philosophy laid my basis, I now pursue my own line of inquiry. Perhaps this is due to my encounters with Buddhism, which has taught me to be devout but not dogmatic. I balance my academic pursuits with hobbies like singing, running ten kilometers, swimming, and catching up with friends. These activities, along with my research, form the tapestry of my life.
A Vision for the Future
Next year, I will graduate from the doctoral school. At that time, I hope to find a position that allows me to continue my research and translate books that perhaps inspire me. Above all, I aspire to learn from the people I admire. Some have passed on, but their spirit burns brightly, like a torch passed from one generation to the next. I don't seek to become a blazing fire myself; I'm content to be a blade of grass-weathering the winds and rains, but growing with each drop of nourishment.