From July 26-29, Prof. Zhizhang Wang from Fudan University was invited by Cuiying Honors College to teach a summer course “Differentiable Manifold Theory” to students in the Mathematics Cuiying class.

“Differentiable Manifold Theory” introduces the concepts and related applications of cluster, smooth (complex) manifold, smooth (holomorphic) map, complex toric torus, vector bundle and tangent bundle, etc. On this basis, de Rham cohomology theory is established and several specific topological spaces are studied.

In the four-day course, Prof. Zhizhang Wang taught the concepts of algebraic and analytic clusters in the Euclidean space, topological and differential manifolds, differential structures and complex structures on topological manifolds, verified examples of special linear groups and orthogonal transformation groups of real matrices, introduced smooth (holomorphic) maps, differentiation of smooth maps, as well as immersion and submersion, immersed and embedded submanifolds, complex structures on the torus induced by quotient. Four examples of constructing new vector series from existing vector series are given: Whitney sum, tensor product, outer product and dual bundle. Mathematical concepts such as moduli structures induced by vector bundles, differential forms and exterior differentiation on them, establishment, properties and applications of de Rham cohomology, and simple, singular homotopy theory and its relation to de Rham cohomology are explained.

"Theory of Differential Manifolds" was rich and compact. Prof. Zhizhang Wang introduced the core concepts of differential manifolds concisely and accurately in the limited teaching time, answered the students' questions in detail and enthusiasm, and explained their confusions in depth. Through this summer course, the students of the Mathematics Cuiying class understood the basic knowledge of differential manifolds, made up for the deficiencies in the geometric direction, broadened their academic horizons, felt the beautiful structure and profound ideas of modern mathematics, and appreciated the beauty of different mathematics.

（Translated by Che Zheng, proofread by Zhu Hanqing）