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A central problem in algebra at the end of the 18^th century was that of solving algebraic equations. While 2^nd, 3^rd and 4^th degree equations could be solved explicitly by radicals it was suspected, particularly through the work of Lagrange (1771), that the general 5^th degree equation could not be solved in this way. Ruffini, in 1813 , proposed a proof of this; however the proof was generally found to be unsatisfactory. Abel gave another proof in 1824 which after subsequent repairs has been considered complete. But to Galois belongs the far-reaching idea (around 1830) of attaching to the equation a certain finite permutation group (of the roots), now called the Galois group. A remarkable theorem in Galois theory states that the solvability of this group is equivalent to the solvability of the equation by radicals. The equation x⁵-x-1/3=0 has Galois group S₅, the symmetric group of five letters which is not solvable; thus the Ruffini-Abel result follows.

When Sylow gave a lecture on these matters at the University of Oslo 1863, a farmer's son, by the name of Sophus Lie (1842-1899), was in the audience. Although his interests were oriented more towards Geometry than Algebra, Galois' ideas made a great impression on him. After his friendly and productive collaboration with Klein 1970-71, Lie conceived the idea of developing an analog for differential equations to Galois theory for algebraic equations. I shall try to explain the foundations of this theory.

hbghlyj 写了： 2023年 8月 24日 02:46 Sophus Lie and the role of Lie groups in mathematics.
有相关的介绍：

hbghlyj 写了： 2023年 8月 24日 02:46 Sophus Lie and the role of Lie groups in mathematics.
有相关的介绍：

wdds 写了： 2023年 8月 27日 18:43 代数几何的就业方向是啥？

fhnan 写了： 2023年 8月 28日 00:46 只能当老师吧

forecasting 写了： 2023年 8月 28日 10:20 说远了，就没人知道或者思考过这个方向吗？
如果微分方程的解集上能定义Zariski拓扑，那么研究微分方程的解集的也是代数几何，不过不应该是全部的微分方程。微分方程除加和微分而外，还有乘这一运算，所以不大可能得出同一数学结构

TheMatrix 写了： 2023年 8月 28日 11:48 你这个类比需要更强的根据。一元的就是群论，多元的就是代数 - 这好像不是这么简单的类比。

forecasting 写了： 2023年 8月 28日 15:06 老兄，我啥时候这么说了？我的原话是：研究代数方程的解集的是代数几何，那么研究微分方程的解集的是啥呢？
前面那些话当然是启发式的。类比有根据就不是类比了，是证明两个数学结构同构或者同态。