To further enhance high-level mathematical foundational training and stimulate students' academic interest and exploratory spirit in geometry, the Geometry Seminar of the Climbing Plan at the School of Mathematics and Statistics has been held regularly in recent sessions. The seminar focuses on cutting-edge topics such as differential manifolds and Riemannian geometry, aiming to systematically integrate theoretical lectures with practical exercises to strengthen students' understanding and application of advanced geometric theories. Every Sunday morning, teachers and students gather together to explore the mysteries of abstract geometry, fostering a vibrant learning environment where teaching and learning mutually reinforce each other.
First Geometry Seminar
Hosted by Liu Chen, a 2023 undergraduate majoring in Mathematics and Applied Mathematics, with Professor Li Guanghan as the instructor, Liu Chen introduced the fundamental concepts of manifolds and coordinate chart transformations. Fan Huijun, Dean of the School of Mathematics and Statistics, attended the seminar to provide guidance, further energizing the discussion. Fan emphasized that theoretical learning should be combined with concrete examples, encouraging Liu Chen and the students to derive coordinate chart transformations on the sphere through intuitive calculations. This hands-on approach deepened everyone's understanding of the essence of manifolds and highlighted the importance of harmonizing rigorous abstraction with tangible examples, laying a solid foundation for learning basic geometric theories.
Second Geometry Seminar
Continuing as the host, Liu Chen was guided by Professor Xu Xu. This session focused on the rank theorem, a crucial method for constructing manifolds. The theorem states that if the Jacobian matrix of a mapping maintains constant rank, its level set naturally forms a manifold. Liu Chen demonstrated the practical applications of the rank theorem using classic manifolds like SO(n). Dean Fan Huijun once again participated in the seminar, offering specific guidance on the special case of linear mappings to help students connect abstract theorems with the solution sets of linear equations. Through systematic study and case analysis, students gained a deeper understanding of the rank theorem and its pivotal role in manifold construction.
Third Geometry Seminar
Liu Chen remained the host, delivering an in-depth lecture on quotient topology, covering the definitions and constructions of projective space RPn and Grassmann manifold G(n,k). From multiple perspectives, Liu Chen explained that projective space can be viewed as equivalence classes in Euclidean space, lines passing through the origin, or antipodal points on a sphere, among other forms. He also analyzed the topological properties induced by the corresponding equivalence relations. Using the diagonal method, Liu Chen cleverly proved the T2 and C2 conditions for RPn, thereby establishing its manifold structure. Additionally, he explored the construction and manifold properties of Grassmann manifolds as equivalence classes of specific matrices. The seminar's profound content and rigorous logic significantly expanded students' knowledge of topology and manifold theory, serving as an excellent example of integrating theoretical innovation with illustrative proofs.
Fourth Geometry Seminar
Hosted by Du Jinheng, a 2023 undergraduate majoring in Information and Computing Science, with Professor Li Guanghan as the instructor, the seminar focused on the nature and construction of tangent vectors on manifolds. Du Jinheng first introduced the dual space of vector spaces, laying the groundwork for defining cotangent spaces. Under Dean Fan Huijun's guidance, the seminar used polynomial examples to concretize abstract theories, further clarifying the practical significance of tangent vectors under coordinate charts. While the depth of the proofs discussed in this session leaves room for improvement—and related conclusions are detailed in standard literature—students actively engaged in calculations and teacher-student interactions, building a clear cognitive framework for the connections between manifolds, tangent vectors, and differential geometry.
Since its inception, the Geometry Seminar of the Climbing Plan has been meticulously organized, thematically focused, and substantively rich, fully demonstrating students' enthusiasm and potential for advanced mathematical research. The seminar not only strengthens theoretical foundations but also embodies an academic atmosphere that combines examples, emphasizes proofs, and pursues rigor. As the students have come to realize: abstract geometry requires not only profound logical thinking but also insight into typical examples and exploration. In the future, the Geometry Seminar will continue to deepen its discussions, enrich academic exchange formats, and provide solid support for students to climb the peak of geometric knowledge.
(Reporter: Fang Zijin, Hu Xuehong; Photographer: Liu Xiaohuan)
