Proper actions on cohomology manifolds

Essential results about actions of compact Lie groups on connected manifolds are generalized to proper actions of arbitrary groups on connected cohomology manifolds. Slices are replaced by certain fiber bundle structures on orbit neighborhoods. The group dimension is shown to be effectively finite. The orbits of maximal dimension form a dense open connected subset. If some orbit has codimension at most , then the group is effectively a Lie group.

     

Correspondences Between 2-Brauer Characters of Solvable Groups

Let G be a finite solvable group and let p be a prime. Let P ∈ Syl _p (G) and N = N _G (P). We prove that there exists a natural bijection between the 2-Brauer irreducible characters of p′-degree of G and those of N _G (P).     

Corrigendum to “The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems” [J. Funct. Anal. 256 (9) (2009) 2967-3034]

In lines 8-11 of Lu (2009) [18, p. 2977] we wrote: “For integer m?3, if M is Cm-smooth and Cm−1-smooth L:R×TM→R satisfies the assumptions (L1)-(L3), then the functional Lτ is C²-smooth, bounded below, satisfies the Palais-Smale condition, and all critical points of it have finite Morse indexes and nullities (see [1, Prop. 4.1, 4.2] and [4])”. However, as proved in Abbondandolo and Schwarz (2009) [2] the claim that Lτ is C²-smooth is true if and only if for every (t,q) the function v?L(t,q,v) is a polynomial of degree at most 2. So the arguments in Lu (2009) [18] are only valid for the physical Hamiltonian in (1.2) and corresponding Lagrangian therein. In this note we shall correct our arguments in Lu (2009) [18] with a new splitting lemma obtained in Lu (2011) [20].     

Set Systems with L‐Intersections and k‐Wise L‐Intersecting Families

In this paper, by employing linear algebra methods we obtain the following main results:

• (i) Let and be two disjoint subsets of such that Suppose that is a family of subsets of such that for every pair and for every i. Then Furthermore, we extend this theorem to k‐wise L‐intersecting and obtain the corresponding result on two cross L‐intersecting families. These results show that Snevily's conjectures proposed by Snevily (2003) are true under some restricted conditions. This result also gets an improvement of a theorem of Liu and Hwang (2013).
• (ii) Let p be a prime and let and be two subsets of such that or and Suppose that is a family of subsets of [n] such that (1) for every pair (2) for every i. Then This result improves the existing upper bound substantially.

     

Nakai's conjecture for varieties smoothed by normalization

The notion of D-simplicity is used to give a short proof that varieties whose normalization is smooth satisfy Ishibashi's extension of Nakai's conjecture to arbitrary characteristic. This gives a new proof of Nakai's conjecture for curves and Stanley-Reisner rings.

     

关于低阶K群的若干问题

本文主要介绍了低阶Ｋ群Ｋ＿０，Ｋ＿１，Ｋ＿２及与它们相关的Ｂａｓｓ－Ｑｕｉｌｌｅｎ猜测、Ｂｉｒｃｈ－Ｔａｔｅ猜测研究的最新进展．Ｋ＿０与Ｋ＿１方面主要通过对Ｂａｓｓ～Ｑｕｉｌｌｅｎ猜测研究说明Ｋ＿０，Ｋ＿１的作用，同时也给出了一些多项式环上投射模自同构群的计算公式．在Ｋ＿２群方面，我们主要介绍了域上Ｋ＿２群，代数整数环上Ｋ＿２群以及著名的Ｂｉｒｃｈ－Ｔａｔｅ猜想．并且也给出了Ｋ＿２群一些计算方面的结果．     

Fermat最后定理—历史和证明简介

本文旨在介绍Ｆｅｒｍａｔ最后定量的历史和Ｗｉｌｅｓ最近所给的证明。首先简介了其在代数数论的发展过程中所起的作用，然后介绍椭圆曲线的基本概念，叙述Ｔａｎｉｙａｍａ－Ｗｅｉｌ猜想，即任一椭圆曲线都是模的。进而介绍Ｒｉｂｅｔ的工作。他证明了若Ｔａｎｉｙａｍａ－Ｗｅｉｌ猜想对半稳定的椭圆曲线成立则Ｆｅｒｍａｔ最后定理成立。最后介绍了ｌ－ａｄｉｃ Ｇａｌｏｉｓ表示的概念及Ｗｉｌｅｓ定理，即半稳定的椭圆曲线都     

The Novikov Conjecture for Low Degree Cohomology Classes

We outline a twisted analogue of the Mishchenko–Kasparov approach to prove the Novikov conjecture on the homotopy invariance of the higher signatures. Using our approach, we give a new and simple proof of the homotopy invariance of the higher signatures associated to all cohomology classes of the classifying space that belong to the subring of the cohomology ring of the classifying space that is generated by cohomology classes of degree less than or equal to 2, a result that was first established by Connes and Gromov and Moscovici using other methods. A key new ingredient is the construction of a tautological C* _r (, )-bundle and connection, which can be used to construct a C* _r (, )-index that lies in the Grothendieck group of C* _r (, ), where is a multiplier on the discrete group corresponding to a degree 2 cohomology class. We also utilise a main result of Hilsum and Skandalis to establish our theorem.