Study at TCU


Name NAKAI Hirofumi
Official Title Associate Professor
Affiliation Natural Sciences, Science and Engineering
Profile I am interested in the structure of the stable homotopy category of spectra, and I have been working on it using some gadgets like Adams or Adams-Novikov spectral sequence. Of course, the relevant computations are very hard as much as ``Tour de France''.
Recently, I have been trying to obtain information about the stable homotopy groups of spheres using cohomology theories that are not complex orientable. Typical examples of such theories are the topological modular forms and real Johonson-Wilson-theory, which is obtained by taking the homotopy fixed points of the spectrum representing the complex Johnson-Wilson theory. For example, it has already been shown abroad that this theory can be used to solve the problem of embedding projective spaces into Euclidean spaces. I believe that ER theory may be applied to other specific problems, such as determining the LS category of a space. In addition, I am currently working on a spectrum classification problem using quasi-equivalence related to ER theory.
Research Field(Keyword & Summary)
  1. (1) Stable homotopy theory

    Stable homotopy theory is a research area that studies the category of spectra. The main research in this field is to investigate the properties of spectra using stable homotopy groups and general cohomology theories.

  2. (2) Equivariant theory

    Equivariant theory is a branch of homotopy theory that studies the space with group actions. Of particular importance is the space of the group action of order 2. By considering the fixed point spectrum with respect to this action, the study of the 2-components of the stable homotopy group has recently been advanced.

Representative Papers
  1. (1) On β-elements in the Adams-Novikov spectral sequence. J. Topol. 2 (2009), no. 2, 295--320. (with D.C.Ravenel)
  2. (2) An algebraic generalization of image J. Homology Homotopy Appl. 10 (2008), no. 3, 321--333.
  3. (3) On the homotopy groups of E(n)-local spectra with unusual invariant ideals. Proceedings of the Nishida Fest (Kinosaki 2003), 319--332, Geom. Topol. Monogr., 10, Geom. Topol. Publ., Coventry, 2007. (with K.Shimomura)
  4. (4) The first cohomology group of the generalized Morava stabilizer algebra. Proc. Amer. Math. Soc. 131 (2003), no. 5, 1629--1639. (with D.C.Ravenel)
  5. (5) On the generalized Novikov first Ext group modulo a prime. Osaka J. Math. 39 (2002), no. 4, 843--865. (with D.C.Ravenel)
Grant-in-Aid for Scientific Research Support: Japan Society for Promotion of Science (JSPS)
Recruitment of research assistant(s) No
Affiliated academic society (Membership type) The Mathematical Society of Japan
Education Field (Undergraduate level) calculus, multivariable calculus, linear algebra, set theory, basic topology
Education Field (Graduate level) homotopy theory, algebraic topology