Applications of Klee’s Dehn–Sommerville Relations

We use Klee’s Dehn–Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai’s conjecture providing lower bounds on the f-vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify Kühnel’s conjecture that gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k and the number of vertices, and (iii) partially prove Kühnel’s conjecture providing upper bounds on other Betti numbers of odd- and even-dimensional manifolds. For manifolds with boundary, we derive an extension of Klee’s Dehn–Sommerville relations and strengthen Kalai’s result on the number of their edges. I. Novik research partially supported by Alfred P. Sloan Research Fellowship and NSF grant DMS-0500748. E. Swartz research partially supported by NSF grant DMS-0600502.     

Kp,q-factorization of complete bipartite graphs

Let K _m,nbe a complete bipartite graph with two partite sets having m and n vertices, respectively. A K _p,q-factorization of K _m,n is a set of edge-disjoint K _p,q-factors of K _m,n which partition the set of edges of K _m,n. When p = 1 and q is a prime number, Wang, in his paper “On K _1,k -factorizations of a complete bipartite graph” (Discrete Math, 1994, 126: 359—364), investigated the K _1,q -factorization of K _m,nand gave a sufficient condition for such a factorization to exist. In the paper “K _1,k -factorizations of complete bipartite graphs” (Discrete Math, 2002, 259: 301—306), Du and Wang extended Wang’s result to the case that q is any positive integer. In this paper, we give a sufficient condition for K _m,n to have a K _p,q-factorization. As a special case, it is shown that the Martin’s BAC conjecture is true when p : q = k : (k+ 1) for any positive integer k.     

On the modularity of the GL<Subscript>2</Subscript>-twisted spinor L-function

In modern number theory there are famous theorems on the modularity of Dirichlet series attached to geometric or arithmetic objects. There is Hecke’s converse theorem, Wiles proof of the Taniyama-Shimura conjecture, and Fermat’s Last Theorem to name a few. In this article in the spirit of the Langlands philosophy we raise the question on the modularity of the GL₂-twisted spinor L-function Z _{G ⊗ h} (s) related to automorphic forms G,h on the symplectic group GSp₂ and GL₂. This leads to several promising results and finally culminates into a precise very general conjecture. This gives new insights into the Miyawaki conjecture on spinor L-functions of modular forms. We indicate how this topic is related to Ramakrishnan’s work on the modularity of the Rankin-Selberg L-series.     

Integral group ring of the Mathieu simple group M 12

We consider the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic groupM ₁₂. As a consequence, we confirm for this group the Kimmerle’s conjecture on prime graphs. The research was supported by OTKA grants No. T 43034, No.K61007 and Francqui Stichting (Belgium) grant ADSI107.     

The extension of the first main theorem for π-blocks

Let π be a set of primes and G a π-separable group. Isaacs defines the B _π characters, which can be viewed as the “π-modular” characters in G, such that the B _p′ characters form a set of canonical lifts for the p-modular characters. By using Isaacs’ work, Slattery has developed some Brauer’s ideals of p-blocks to the π-blocks of a finite π-separable group, generalizing Brauer’s three main theorems to the π-blocks. In this paper, depending on Isaacs’ and Slattery’s work, we will extend the first main theorem for π-blocks.     

Half-factorial-domains

LetR be a commutative domain with 1. We termR an HFD (Half-Factorial-Domain) provided the equality Π _i=1 ⁿ χ_i=Π{_f=1/^m}y _f impliesm=n, whenever thex’s and they’s are non-zero, non-unit and irreducible elements ofR. The purpose of this note is to study HFD’s, in particular, Krull domains that are HFD’s, and to provide examples of HFD’s, that contradict a conjecture of Narkiewicz.     

Cap representations of G2 and the spin L-function of PGSp6

We give a construction of a family of CAP representations of the exceptional group G ₂, whose existence is predicted by Arthur’s conjecture. These are constructed by lifting certain cuspidal representations of PGS _p6. To show that the lifting is non-zero, we establish a Rankin-Selberg integral for the degree 8 Spin L-function of these cuspidal representations of PGS _p6.     

Minami–Webb splittings and some exotic <Emphasis Type="Italic">p</Emphasis>-local finite groups

We give a new proof of the Minami–Webb formula for classifying spaces of finite groups by exploiting Symonds’s resolution of Webb’s conjecture. The methods are applicable to obtain a stable decomposition of Minami’s type for the classifying spaces of the three exotic p-local finite groups which were introduced by Ruiz and Viruel at the prime 7. We obtain a similar decomposition for the classifying spaces of a family of exotic p-local finite groups which were constructed by Broto, Levi and Oliver. The author was supported by the Nuffield Foundation Grant NAL/00735/G.     

<Emphasis Type="BoldItalic">π</Emphasis>-forms of Brauer’s <Emphasis Type="BoldItalic">k</Emphasis>(<Emphasis Type="BoldItalic">B</Emphasis>)—conjecture and Olsson’s Conjecture

Our main purpose of this paper is to give π-block forms of Brauer’s k(B) −conjecture and Olsson’s conjecture for finite π −separable groups.     

Splitting ρκλ into maximally many stationary sets

Letκ >ω be a regular cardinal and λ >κ a cardinal. Solovay’s classical result for κ[So] led Menas [Me] to conjecture that a stationary subset ofP _κλ would split into λ^<κ stationary set of size κ⁺ (see[BT]), the conjecture implies that the size is (κ⁺)^<κ as well. Part of this work was done during the author’s stay at Boston University as one of the Japanese Overseas Research Fellows. He gratefully acknowledge Professor Akihiro Kanamori’s hospitality. He also wishes to thank members of the set theory seminar at Waseda University for their interest at the early stage.     

Additive Latin transversals and group rings

LetA={a ₁, …,a _k} and {b ₁, …,b _k} be two subsets of an abelian groupG, k≤|G|. Snevily conjectured that, when |G| is odd, there is a numbering of the elements ofB such thata _i+b _i,1≤i≤k are pairwise distinct. By using a polynomial method, Alon affirmed this conjecture for |G| prime, even whenA is a sequence ofk<|G| elements. With a new application of the polynomial method, Dasgupta, Károlyi, Serra and Szegedy extended Alon’s result to the groupsZ _p ^r andZ _p ^rin the casek<p and verified Snevily’s conjecture for every cyclic group. In this paper, by employing group rings as a tool, we prove that Alon’s result is true for any finite abelianp-group withk<√2p, and verify Snevily’s conjecture for every abelian group of odd order in the casek<√p, wherep is the smallest prime divisor of |G|. This work has been supported partly by NSFC grant number 19971058 and 10271080.     

Generalized Mukai conjecture for special Fano varieties

Let X be a Fano variety of dimension n, pseudoindex i _X and Picard number ρ_X. A generalization of a conjecture of Mukai says that ρ_X(i _X −1)≤n. We prove that the conjecture holds for a variety X of pseudoindex i _X ≥n+3/3 if X admits an unsplit covering family of rational curves; we also prove that this condition is satisfied if ρ_X> and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension five.     

On the uniqueness of promotion operators on tensor products of type <Emphasis Type="Italic">A</Emphasis> crystals

The affine Dynkin diagram of type A _n ⁽¹⁾ has a cyclic symmetry. The analogue of this Dynkin diagram automorphism on the level of crystals is called a promotion operator. In this paper we show that the only irreducible type A _n crystals which admit a promotion operator are the highest weight crystals indexed by rectangles. In addition we prove that on the tensor product of two type A _n crystals labeled by rectangles, there is a single connected promotion operator. We conjecture this to be true for an arbitrary number of tensor factors. Our results are in agreement with Kashiwara’s conjecture that all ‘good’ affine crystals are tensor products of Kirillov-Reshetikhin crystals.     

Thurston’s pullback map on the augmented Teichmüller space and applications

Let f be a postcritically finite branched self-cover of a 2-dimensional topological sphere. Such a map induces an analytic self-map σ _f of a finite-dimensional Teichmüller space. We prove that this map extends continuously to the augmented Teichmüller space and give an explicit construction for this extension. This allows us to characterize the dynamics of Thurston’s pullback map near invariant strata of the boundary of the augmented Teichmüller space. The resulting classification of invariant boundary strata is used to prove a conjecture by Pilgrim and to infer further properties of Thurston’s pullback map. Our approach also yields new proofs of Thurston’s theorem and Pilgrim’s Canonical Obstruction theorem.     

Vojta’s conjecture on blowups of \mathbbPn{\mathbb{P}^n}, greatest common divisors, and the abc conjecture

We will prove some cases of Vojta’s conjecture on blowups of \mathbbPⁿ{\mathbb{P}^n}, using Schmidt’s subspace theorem. The results can be stated as inequalities of greatest common divisors. Moreover, from Vojta’s conjecture on one further blowup at an infinitely near point, we derive a still-open special case of the abc-conjecture.     

Transcendence degree of zero-cycles and the structure of Chow motives

In the paper we introduce a transcendence degree of a zero-cycle on a smooth projective variety X and relate it to the structure of the motive of X. In particular, we show that in order to prove Bloch’s conjecture for a smooth projective complex surface X of general type with p _g = 0 it suffices to prove that one single point of a transcendence degree 2 in X(ℂ), over the minimal subfield of definition k ⊂ ℂ of X, is rationally equivalent to another single point of a transcendence degree zero over k. This can be of particular interest in the context of Bloch’s conjecture for those surfaces which admit a concrete presentation, such as Mumford’s fake surface, see [Mumford D., An algebraic surface with K ample, (K ²) = 9, p _g = q = 0, Amer. J. Math., 1979, 101(1), 233–244].     

Some conjectures on the zeros of approximates to the Riemann ≡-function and incomplete gamma functions

Riemann conjectured that all the zeros of the Riemann ≡-function are real, which is now known as the Riemann Hypothesis (RH). In this article we introduce the study of the zeros of the truncated sums ≡ _N (z) in Riemann’s uniformly convergent infinite series expansion of ≡(z) involving incomplete gamma functions. We conjecture that when the zeros of ≡ _N (z) in the first quadrant of the complex plane are listed by increasing real part, their imaginary parts are monotone nondecreasing. We show how this conjecture implies the RH, and discuss some computational evidence for this and other related conjectures.     

A Property of <Emphasis Type="Italic">g</Emphasis>-Expectation

This paper proves that, under the hypothesis g(t, 0, 0) ≡ 0 and some natural assumptions, the generator g of a backward stochastic differential equation can be uniquely determined by the corresponding g-expectations with all terminal conditions. The main result of this paper also confirms and extends Peng Shige‘s conjecture.     

Entire functions sharing polynomials with their first derivatives

In this paper we estimate the size of the ρ_n’s in the famous L. Zalcman’s Lemma. With it, we obtain a uniqueness theorem for entire functions and their first derivatives, which improves and generalizes the related results of Rubel and Yang and of Li and Yi. Some examples are provided to show the sharpness of our result. As an application, we prove that R. Brück’s conjecture is true for a class of functions. Received: 30 October 2008, Revised: 5 February 2009     

On Browkin’s conjecture about the elements of order five in <Emphasis Type="Italic">K</Emphasis><Subscript>2</Subscript>(ℚ)

Based on the work of Lenstra, a succinct proof of Browkin’s conjecture about the elements of order five in K ₂(ℚ) is given. This work was supported by the National Natural Science Foundation of China (Grant No. 10371061).