New Developments ofAlgebtaic Topology

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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.