On the afternoon of June 25, the 13th lecture of Shenzhen University's "Master-to-Master" series was held in the multi-functional hall on the second floor of the Alumni Plaza at the Yuehai Campus. Professor Efim Zelmanov, Chair Professor at Southern University of Science and Technology, Director of the Shenzhen International Center for Mathematics (Zelmanov Mathematics Center), Member of the U.S. National Academy of Sciences, and Foreign Academician of the Chinese Academy of Sciences, delivered a lecture titled Exploring the Wonders of a New Mathematical World. Through an interdisciplinary lens, he analyzed the theoretical elegance and practical power of mathematics, offering the audience an academic feast rich in historical depth and innovative vision. Shenzhen University President Mao Junfa (Member of the Chinese Academy of Sciences), representatives from the Publicity Department of the Party Committee, Human Resources Department, School of Mathematical Sciences, and over 200 faculty and students attended the lecture.
President Mao Junfa presided over the event. In his opening remarks, he emphasized that the "Master-to-Master" series is a flagship academic initiative of Shenzhen University, providing students and faculty with direct access to scientific giants and fostering a culture of scientific spirit and humanistic sentiment on campus. President Mao presented Professor Zelmanov with a commemorative plaque, noting that Zelmanov was awarded the Fields Medal in 1994—the highest honor in mathematics—elected to the U.S. National Academy of Sciences in 2001, and honored as a Foreign Academician of the Chinese Academy of Sciences in 2021. Zelmanov revolutionized Jordan algebra theory by extending finite-dimensional Jordan algebra to infinite dimensions, resolving profound problems posed in 1938 by Jordan, von Neumann, and Wiener, as well as the Burnside conjecture. His work fundamentally reshaped Jordan algebra theory, establishing him as one of the greatest algebraists of the 20th century.
The Dual Nature of Mathematics: From Ancient Proofs to the Core of Modern Technology
Zelmanov highlighted mathematics as a discipline with a 2,000-year history, consistently serving as a driving force behind scientific progress. From early applications in land measurement and electromagnetism to the construction of abstract theoretical frameworks like geometry and partial differential equations, mathematics embodies both "practical tools" and "artistic thought." He contrasted mathematics with experimental sciences, noting that the former relies on a stable system of "proofs"—logics from Euclid’s era remain foundational standards.
Discussing the beauty of mathematics, Zelmanov used Fermat’s Last Theorem to illustrate its three aesthetic principles: profound proofs behind simple propositions, breakthrough cross-disciplinary applications (e.g., symmetry theory in physics), and universal expressions of concepts across contexts. He praised Galois theory as the "golden standard of mathematical beauty," noting how the young French mathematician’s group theory breakthrough in solving the quintic equation became a cornerstone for 20th-century particle physics research.
From Algebraic Equations to AI Algorithms: Cross-Temporal Applications of Mathematical Theory
Tracing the history of solving algebraic equations, Zelmanov revealed hidden links between mathematical theory and technological innovation. From quadratic to quartic equation solutions, through Ruffini and Abel’s explorations of higher-degree equations, to Galois’s unifying framework via group theory, he underscored mathematics’ evolving role.
In cybersecurity, he explained how Diffie-Hellman key exchange and RSA algorithms leverage the difficulty of discrete logarithms in finite fields. He stressed that these abstract mathematical structures form the bedrock of modern network security.
Mathematics and AI: The Indispensability of Fundamental Disciplines
Addressing debates over whether mathematics education should be reduced, Zelmanov emphasized that algebra, calculus, and probability are foundational to AI innovation. Using gradient descent algorithms in deep learning as an example, he demonstrated their reliance on calculus and linear algebra.
He noted that neural networks’ vector-matrix operations and Fourier transforms originate from core mathematical theories, asserting that breakthroughs in AI stem from the cumulative power of mathematical principles.
Nurturing the Roots of Mathematics, Cultivating Innovation
Zelmanov concluded with a metaphor: "Mathematics is like a plant," where pure and applied mathematics interdependently nourish its roots. Severing any part would endanger its vitality. From Galois’s group theory to modern cryptography, from complex analysis to AI algorithms, mathematics continues to reshape civilization through its "unreasonable effectiveness." He urged young scholars: "Mathematical research is not merely a pursuit of truth—it is a key force in shaping future technology."
Following the lecture, Professor Zelmanov engaged with faculty and students, addressing questions on choosing mathematical research directions and future planning. He emphasized that mathematics is a discipline of problem-solving, and foundational mathematics is indispensable for studying applied fields. "Do not fear difficult problems. Start with simple ones and gradually tackle harder ones—your level will rise accordingly," he advised. He encouraged students to identify research topics through persistent problem-solving, patience, and dedication: "Even the smallest progress demands extraordinary perseverance."
