09/12 2007, 星期三
所想
symmetry resulted in groups, another way of saying this is that group is automorphism.For finite groups and symmetric groups, I think it is obvious. Symmetric group is permutation group of N letters. Every element is automorphism. So I consider for fundamental group and homotopy groups. What kind of symmetric lead to these kinds of groups. Finally, I found that the homotopy group of a topological space X is the symmetry of the spaces of all fibers over X. （Kanex）That is the symmetry of the Universal cover of X. Then the fundamental group is the special case of them.Why to choose the group as invariant of Topological space? I am always considering this problem. Just because it is simple? I think the main reason in the symmetry. Say if we have a category C, we want to find some invariant of it in category D, then the functor is all functor from C to D modulo the automorphism of C(Kanex) 环： Ring can be also taken to be an invariant of Topological space. Actually because of more structure comparing to groups. After introducing Cup product,the cohomological group becomes ring. It carries more information. A natural generalization is K group. (Actually K ring). Rings plays important roles in Algebraic Geometry. 模： What is module. I think we should take it in two ways: 1. Module is an action of Ring on solution space. 2. Module ,The action of Ring on Aut(solution space). Considering some generalization of solution space. If we replace solution space by variety.(like elliptic curves,algebraic surface), due 2, we obtain :Some Ring--àAut(S)(Aut(E)). I think this is the better opinion to module. Module is solving the algebraic equation! Of course, why we consider module in Algebra? The reason is that we want to know ring and algebra. We can consider the module category over different rings. We can consider various Derived category of module category to build some connection between the Ring
数理学院的老师你熟吗？比如程曹宗老师
熟吧
程老师是数理学院副院长.
？
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太深奥了。我看不懂。
太深奥了。我看不懂。