Simplicial complex theory and simplicial set theory.
The notion of fibre bundle first arose out of questions posed in the 1930s on the topology and geometry of manifolds. By the year 1950, the definition of fibre bundle had been clearly formulated, the homotopy classification of fibre bundles achieved, and the theory of characteristic classes of fibre bundles developed by several mathematicians: Chern, Pontrjagin, Stiefel, and Whitney. Fiber bundle theory is not only important in topology and differential geometry, but also is widely used in other branches of mathematics and physics. This course mainly introduces some basic concepts of fiber bundles and the application of fiber bundles in homology theory in comparison with the geometric level and the simple level.
1. Dale Husemoler, Fibre Bundles.
2. Simplicial Objects and Homotopy Groups, Lecture notes of Jie Wu.
3. Barratt, M. G., Gugenheim, V. K. A. M., and Moore, J. C., On semisimplicial fifibre-bundles, Am. J. Math., 81(3), 1959, 639-657.
4. Curtis, E. B., Simplicial homotopy theory, Adv. Math., 6(2), 1971, 107-209.
Generalities on Bundles, Vector Bundles, General Fibre Bundles, Principal Bundle and the Universal Principal Bundle, Grassmann Manifolds and the Classical Groups, Stability of the Homotopy Groups of the Classical Groups, Universal Bundles and Classifying Spaces for the Classical Groups，Simplicial fibre bundle theory.
Assistant Reserch fellow Jingyan Li received a PhD degree from the Department of Mathematics of Hebei Normal University in 2007. Before joining BIMSA in September 2021, she has taught in the Department of Mathematics and Physics of Shijiazhuang Railway University and the School of Mathematical Sciences of Hebei Normal University as an associate professor. Her research interests include topology data analysis and simplicial homology and homotopy.
Lecturer Email: firstname.lastname@example.org
TA: Dr. Zhuo Liu, email@example.com