Combinatorial Algebra

早在世纪，著名的群的伯恩赛德问题就提出了这样的问题：是不是每个周期群都是有限的？ Kurosh-Levitzki问题是一个结合代数的类似问题：是不是每一个生成的零代数都是有限维的？ 反例最早是由Golod和Shafarevich在20世纪60年代提出的。 因此，很自然地，为了得到一个积极的解决办法，就需要用额外的假设重新表述这些问题。Zelmanov赢得了著名的菲尔兹奖于1994年，他的证明，每一个生成的剩余有限群是有限的，从而解决了所谓的限制伯恩赛德问题的群体。 我已经扩大Zelmanov的开创性工作组和李群理论，以一个单一的统一理论，有直接应用到其他分支的代数。 特别地，我证明了一个生成的诣零代数是有限维的，如果它是无穷小PI。 这是Zelmanov结果的一个重要推广。我还成功地将我的新理论应用于卡普兰斯基问题，该问题解决了增广理想是雅各布森根的群代数的结构。 Zelmanov提到Kaplansky问题作为下一个大的障碍，在群论后，限制伯恩赛德问题。

First posed early in the twentieth century, the famous Burnside Problem for groups asked: Is every finitely generated periodic group finite? The Kurosh-Levitzki Problem is an associative algebra analogue: Is every finitely generated nil algebra finite-dimensional? Counterexamples were first constructed by Golod and Shafarevich in the 1960's. It was natural, therefore, to reformulate these problems with additional hypotheses in order to obtain a positive solution. Zelmanov won the prestigious Fields Medal in 1994 for his proof that every finitely generated residually finite group is finite, thereby solving the so-called Restricted Burnside Problem for groups. I have extended Zelmanov's seminal work in group and Lie theory to a single unified theory that has direct applications to other branches of algebra. In particular, I have proved that a finitely generated nil algebra is finite-dimensional if it is infinitesimially PI. This is a significant generalisation Zelmanov's results. I have also had some success in applying my new theory to Kaplansky's Problem which addresses the structure of group algebras whose augmentational ideal is Jacobson radical. Zelmanov has referred to Kaplansky's Problem as the next big hurdle in group theory after the Restricted Burnside Problem.