New Developments ofAlgebtaic Topology

在首席研究员土屋（Tsuchiya）的强烈要求下，Minami开始对Keller、Toen、Tabuada的d -范畴理论以及Toen- vezzosi和Lurie的推导代数几何进行了密集的复试。虽然在这些学科中尚未发现突出的结果，但在研究稳定同伦理论中最基本的问题之一霍普金斯色分裂猜想方面已经取得了一些成果。此外，对于代数几何母题理论中的基本Tate猜想，首席研究员Minami指出了某人对Tate猜想的著名解中的一个致命错误，并实现了一个可能有前途的新方法。研究者Tsuchiya在与满足C2条件的顶点算子代数相关的Riemman曲面上的共形场论方面取得了重大进展。研究者Furuta和Kametani在当前的代数拓扑水平下，也许得到了当hi为正时Bauer-Furuta-Seiberg-Witten不变量的决定性结果。研究者Shimakawa从标记位形空间和连续函子的角度对紧李群等变广义上同调理论进行了全新的研究。最近，研究者小林充分利用佩雷尔曼的方法，获得了一系列突出的结果，包括对勒布朗猜想的肯定回答。这是一项非常重要的成就，远远超出了首席调查员最初的预期。

Following some strong demand of the investigator Tsuchiya, the head investigator, Minami embarked upon intensive rematch on the dg-category theory of Keller, Toen, Tabuada, and on the derived algebraic geometry of Toen-Vezzosi and Lurie. Although prominent results are yet to be discovered in these disciplines, some dividends have been obtained to study the Hopkins Chromatic Splitting Conjecture, one of the most fundamental problems in the stable homotopy theory. Also, concerning the fundamental Tate conjecture in the motif theory of algebraic geometry, the head investigator Minami pointed out a fatal mistake in someone's acclaimed solution of the Tate conjecture, and realized a possibly new promising approach to the Tate conjecture.The investigator Tsuchiya made a big progress in the conformal field theory on the Riemman surface associated with the vertex operator algebra satisfying the C2 condition. The investigators Furuta and Kametani obtained a perhaps, under the current level of algebraic topology, the decisive result on the Bauer-Furuta-Seiberg-Witten invariants when hi is positive. The invesitagator Shimakawa opened up a completely new approach to the compact Lie group equivariant generalized cohomology theories, from the view point of the labeled configuration spaces and continuous functors.Very recently, the investigator Kobayashi, making full use of Perelman's method, has obtained a series of prominent results, including an affirmative andwer to Lebrun's conjecture. This is a very important achievement which is well beyong the head investigator's initial expectations.