---------------------------------- My semester has ended, my daughter has chosen a college, and I finally have some time to deal with Hopf. Sorry for the long delay. 7 new papers this time, from Blanc-Johnson-Turner, Broto-Moller-Oliver, Carlson-Chebolu-Minac, Neusel-Sezer, Serikbaev-Bitibaeva-Yerzhanov-Myrazukulov, Yagita (2). Mark Hovey New papers appearing on hopf between 3/3/07 and 5/15/08 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc-Johnson-Turner/lgss Title: Local-to-global spectral sequences for the cohomology of diagrams Authors: David Blanc, Mark W. Johnson, and James M. Turner Address: Department of Mathematics, University of Haifa, 31905 Haifa, Israel Department of Mathematics, Penn State Altoona, Altoona, PA 16601, USA Department of Mathematics, Calvin College, Grand Rapids, MI 49546, USA Abstract: The cohomology of diagrams arises in various areas of mathematics, such as deformation theory, classifying diagrams of groups, and in homotopy theory, in the context of the rectification of homotopy-commutative diagrams, and thus in the study of higher homotopy and cohomology operations. For this purpose we construct ``local-to-global'' spectral sequences for the cohomology of a diagram, which can be used to compute the cohomology of the full diagram in terms of smaller pieces. We also explain why such a local-to-global approach is relevant to higher operations. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Broto-Moller-Oliver/bmo1 Authors: C. Broto, J. M. M\o{}ller, and B. Oliver Title: Equivalences between fusion systems of finite groups of Lie type Subject class: Primary 20D06, Secondary 55R37, 20D20 keywords: groups of Lie type, fusion systems, classifying spaces, p-completion Abstract: We prove, for certain pairs $G,G'$ of finite groups of Lie type, that the $p$-fusion systems $F_p(G)$ and $F_p(G')$ are equivalent. In other words, there is an isomorphism between a Sylow $p$-subgroup of $G$ and one of $G'$ which preserves $p$-fusion. This occurs, for example, when $G=\Gamma(q)$ and $G'=\Gamma(q')$ for a simple Lie ``type'' $\Gamma$, and $q$ and $q'$ are prime powers, both prime to $p$, which generate the same closed subgroup of $p$-adic units. Our proof uses homotopy theoretic properties of the $p$-completed classifying spaces of $G$ and $G'$, and we know of no purely algebraic proof of this result. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Carlson-Chebolu-Minac/fgt Finite generation of Tate cohomology Jon F. Carlson Department of Mathematics University of Georgia Athens, GA 30602, USA Sunil K. Chebolu Department of Mathematics University of Western Ontario London, ON N6A 5B7, Canada Jan Minac Department of Mathematics University of Western Ontario London, ON N6A 5B7, Canada Abstract: Let G be a finite group and let k be a field of characteristic p. If M is a finitely generated indecomposable non-projective kG-module, we conjecture that the Tate cohomology of G with coefficients in M is finitely generated over the Tate cohomology ring of G if and only if the support variety V_G(M) of M is equal to the entire maximal ideal spectrum V_G(k). We prove various results all of which support this conjecture. It is also shown that all finitely generated kG-modules over a group G have finitely generated Tate cohomology if and only if G has periodic cohomology. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel-Sezer/separating TITLE: Characterizing Separating Invariants AUTHORS: Mara D.~Neusel and M\"uf\.it Sezer ABSTRACT: We study separating algebras for rings of invariants of finite groups. We give an algebraic characterization for these. Furthermore, we describe a particularly nice separating subalgebra for rings of invariants of p-groups in characteristic p. This leads to a characterization of subalgebras such that their p-root and integral closure is equal to the ring of invariants. Finally, we present separating sets for invariants rings of nonmodular representations of abelian groups whose size depends only on the degree of the representation. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Serikbaev-Bitibaeva-Yerzhanov-Myrazukulov/flow [Your moderator was in a quandary over this paper, which is clearly not remotely algebraic topology, but decided to err on the side of openness] Integrable isotropic geometrical flows and Heisenberg ferromagnets N.S.Serikbaev, Zh.M.Bitibaeva, K.K.Yerzhanov, R.Myrzakulov* Department of General and Theoretical Physics, Eurasian National University, Astana, 010008, Kazakhstan Abstract Geometrical Flows (GF) play an important role in modern mathematics and physics. In this letter we have considered some integrable isotropic GF Ricci Flows (RF) and mean curvature flows (MCF) ~ which are related with integrable Heisenberg ferromagnets. In 2+1 dimensions, these GF have a singularity at t = t0. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Yagita/motsplitG Title: Note on the mod p motivic cohomology of algebraic groups. Author: Nobuaki Yagita AMS classification numbers: 55P35, 57T25. Adress of Author: Faculty of Education, Ibaraki University, Ibaraki, Japan. Abstract: Let G_k be a split reductive group over a field k of ch(k)=0 corresponding to a compact Lie group G. In this paper, we show that the mod p motivic cohomology is isomorphic to the tensor product of the usual mod p cohomology H^*(G;Z/p) and the motivic cohomology H^{*,*'}(Spec(k);Z/p), when G=SO_n,G_2,F_4,E_6. We also give an example of nonsplit case (G=G_2,p=2,k=R) which does not hold the above isomorphism. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Yagita/realchow Title: Note on motivic cohomology of anisotropic real quadrics. Author: Nobuaki Yagita AMS classification numbers: 55P35, 57T25. Adress of Author: Faculty of Education, Ibaraki University, Ibaraki, Japan. Abstract: In this paper, we compute the mod 2 motivic cohomology H^{*,*'}(X;Z/2) for the anisotropic quadric X over R the field of real numbers. -----------