Writings

I post some of my writtings to record my growth. Some of them are drafts(not typed by latex), but I think it’s still useful (I may type them in latex if I have time).

• Physical picture of tensor products and direct sum: link Tensor products and direct sums of vector spaces are abstract concepts in linear algebra. In this article, we aim to elucidate these abstrct notions by developing into the physical manifestations of two systems: the 1/2- spin system and the two -particle system.

• Visualizasion for π1(SO(3)/D2) and rotation of eigenvectors:link In this article we will visualize SO(3)/D2 and π1(SO(3)/D2) to obtain a nice picture describing rotation of eigenframes.

• Notes on Higgs bundleslink Introduce definition of Higgs bundles and non abelian Hodge correspondence.

• Differential geometry of vector bundles(ch III in gtm65):link It introduces connections, curvatures and Chern classes in analysis version.

• Notes on fiber bundles:link It introduces fiber bundles, principal bundles, classification of principal bundles and characteristic classes including Chern classes, Stiefel -Whitney classes, and Pontrjagin classes. It’s the note for ch3,4 of this lecture notes.

• Notes on homotopy theory:link The main reference of this note is Hatcher’s algebraic topology and ch1,2 for this lecture notes.

• Notes on stable homotopy theory:link Introduce some basic defintion in stable homotopy theory such as spectrum, stable homology theory, stable homotopy category and so on, which also appear in many other topics. The main reference is link

Talks

• My Talks in Small workshop about several topics about geometry and topology, 24 summer:link It contains the following topics: sheaf theory, when do we have k-spectral bundles, how to tell a fiber bundle is trivial, introduction to stratified vector bundles, blow up and smoothness in algebraic geometry.

• Energy bands and Higgs bundles, 24/9/10 at Kunming Tianyuan Mathematics Research Center:slides In this talk I explain what energy bands are and show that Higgs bundles are useful to describe the topology of energy bands.

-A general introduction to intersection homology, 25/1/11 at Sustech 2nd Mathematics workshop:note In recent years, intersection homology has become an indispensable tool for studying the topology of singular spaces. While the main results of usual homology theories often fail for singular spaces, intersection homology effectively recovers these properties, bridging this critical gap. In this talk, I will present the foundational concepts of GM intersection homology (including simplicial intersection homology, PL intersection homology, and singular intersection homology) and some examples. Finally, I may explore the topic of non-GM (Goresky-MacPherson) intersection homology. Reference is here.