m元n次方程或多项式的解或零点问题发展出代数几何理论。平行地，常微分方程解由李群解决，对应地，偏微分方程的解应该发展出代

[forecasting]

一元n次方程有无根式解的问题由群论判定，而m元n次方程或多项式的解或零点问题发展出代数几何理论。平行地，常微分方程解由李群解决，对应地，偏微分方程的解应该发展出代数微分几何理论？是啥？

[hbghlyj]

Sophus Lie and the role of Lie groups in mathematics.
有相关的介绍：

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.

[rgg]

Sophus Lie and the role of Lie groups in mathematics.
有相关的介绍：

多谢资料。
李群用来解微分方程要找每个方程单独的不变李群，这一步看文章的例子并不简单啊。

[forecasting]

Sophus Lie and the role of Lie groups in mathematics.
有相关的介绍：

谢谢回复和资料。我原贴中的问题是，微分方程对应于代数方程的代数几何的对等学科是什么？你提供的那文章我没仔细看，看到有数论里的自守函数，物理中的规范场论等等。微分方程可对应于代数几何的学科是什么？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？
不好意思，我不是特别耐心读书的人，想的比读的多。

[FoxMe]

伽罗瓦理论可以应用于微分方程，我知道但没学过。

偏微分方程感觉很难解，要么瞎猜，要么数值计算，不知道和代数怎么联系。

[wdds]

代数几何的就业方向是啥？

[fhnan]

代数几何的就业方向是啥？

只能当老师吧

[forecasting]

只能当老师吧

不愿意做老师，也可以做其他的啊，比如密码学，比如数值计算的各种应用领域，或者想赚钱去金融领域。

[forecasting]

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]

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

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]

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。

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。

[TheMatrix]

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。

嗯。我好像把你的意思给简单化了。