The Chern Conjecture for Affinely Flat Manifolds Using Combinatorial Methods

An affine manifold is a manifold with a flat affine structure, i.e. a torsion-free flat affine connection. We slightly generalize the result of Hirsch and Thurston that if the holonomy of a closed affine manifold is isomorphic to amenable groups amalgamated or HNN-extended along finite groups, then the Euler characteristic of the manifold is zero confirming an old conjecture of Chern. The technique is from Kim and Lee's work using the combinatorial Gauss–Bonnet theorem and taking the means of the angles by amenability. We show that if an even-dimensional manifold is obtained from a connected sum operation from K(, 1)s with amenable fundamental groups, then the manifold does not admit an affine structure generalizing a result of Smillie.     

复环面情形的Suita猜想

对任意复环面的情形证明了推广的Suita猜想,即απK≥c~2(α∈R),其中c是修正后的对数容度,K是对角线上的Bergman核.还阐明了对任意亏格≥2的紧Riemann面情形的公开问题.文中结果的证明部分地依赖于椭圆函数理论.     

Proof of the Volume Conjecture for Whitehead Doubles of a Family of Torus Knots

A technique to compute the colored Jones polynomials of satellite knots,illus- trated by the Whitehead doubles of knots,is presented.Then the author proves the volume conjecture for Whitehead doubles of a family of torus knots and shows some interesting observations.     

The Two Hyperplane Conjecture

We introduce a conjecture that we call the Two Hyperplane Conjecture, saying that an isoperimetric surface that divides a convex body in half by volume is trapped between parallel hyperplanes. The conjecture is motivated by an approach we propose to the Hots Spots Conjecture of J. Rauch using deformation and Lipschitz bounds for level sets of eigenfunctions. We will relate this approach to quantitative connectivity properties of level sets of solutions to elliptic variational problems, including isoperimetric inequalities, Poincar′e inequalities, Harnack inequalities, and NTA(non-tangentially accessibility). This paper mostly asks questions rather than answering them, while recasting known results in a new light. Its main theme is that the level sets of least energy solutions to scalar variational problems should be as simple as possible.     

Steiner Ratio for Manifolds

The Steiner ratio characterizes the greatest possible deviation of the length of a minimal spanning tree from the length of the minimal Steiner tree. In this paper, estimates of the Steiner ratio on Riemannian manifolds are obtained. As a corollary, the Steiner ratio for flat tori, flat Klein bottles, and projective plane of constant positive curvature are computed.     

On a Criterion for Catalan's Conjecture

We give a new proof of a theorem of P. Mihailescu which states that the equation x ^p – y ^q = 1 is unsolvable with x, y integral and p, q odd primes, unless the congruences p ^q p (mod q ²) and q ^p q (mod p ²) hold.     

On Morse Conjecture for Flows on Closed Surfaces

Let M be a closed orientable surface and let ϕ be a C¹‐flow on M with set of singularities compact countable. In this paper, we prove the Morse conjecture for ϕ: if ϕ is topologically transitive then it is metrically transitive.     

Generic element inSK1 for simple algebras

We prove that the generic element inSK ₁(D) for a simple algebraD of the index divisible by 4 is nontrivial. It implies that the variety of an algebraic group SL(1,D) is not rational.     

The Kervaire-Laudenbach Conjecture and Presentations of Simple Groups

The statement “no non-Abelian simple group can be obtained from a non-simple one by adding one generator and one relator” first is equivalent to the Kervaire-Laudenbach conjecture, and second, becomes true under the additional assumption that an initial non-simple group is either finite or torsion free.Supported by RFBR grant No. 02-01-00170.__________Translated from Algebra i Logika, Vol. 44, No. 4, pp. 399–437, July–August, 2005.     

The Gold Partition Conjecture for 6-Thin Posets

We consider the Gold Partition Conjecture (GPC) that implies the 1/3–2/3 Conjecture. We prove the GPC in the case where every element of the poset is incomparable with at most six others. The proof involves the extensive use of computers. This paper contains results obtained using computer resources of the Interdisciplinary Centre for Mathematical and Computational Modelling (ICM), University of Warsaw.     

The ABC Conjecture Implies Vojta's Height Inequality for Curves

Following Elkies (Internat. Math. Res. Notices7 (1991) 99-109) and Bombieri (Roth's theorem and the abc-conjecture, preprint, ETH Zürich, 1994), we show that the ABC conjecture implies the one-dimensional case of Vojta's height inequality. The main geometric tool is the construction of a Belyǐ function. We take care to make explicit the effectivity of the result: we show that an effective version of the ABC conjecture would imply an effective version of Roth's theorem, as well as giving an (in principle) explicit bound on the height of rational points on an algebraic curve of genus at least two.     

The Merrifield–Simmons Conjecture Holds for Bipartite Graphs

Let be a graph and the number of independent (vertex) sets of G. Then the Merrifield–Simmons conjecture states that the sign of the term only depends on the parity of the distance of the vertices in G. We prove that the conjecture holds for bipartite graphs by considering a generalization of the term, where vertex subsets instead of vertices are deleted.     

Broue's Conjecture for the Hall-Janko Group and its Double Cover

Broué's abelian defect conjecture suggests a deep linkbetween the module categories of a block of a group algebraand its Brauer correspondent, viz. that they should be derivedequivalent. We are able to verify Broué's conjecturefor the Hall–Janko group, even its double cover 2.J₂,as well as for U₃(4) and Sp₄(4). In fact we verify Rickard'srefinement to Broué's conjecture and show that the derivedequivalence can be chosen to be a splendid equivalence for theseexamples. 2000 Mathematical Subject Classification: 20C20, 20C34.     

The Bass Conjecture and Group von Neumann Algebras

The precise relationship between the Bass conjecture for the Hattori–Stallings trace for projective ZG-modules and the map from reduced K-theory of ZG to reduced K-theory of the von Neumann algebra, NG, of G, of G is determined. As a consequence it is shown this map is zero for all groups G. It is also shown that the map induced on K-theory from the inclusion of NG to the ring of closed, densely defined operators affiliated to NG is an isomorphism. Together with the above result, this gives some positive evidence for the validity of the Division Ring Conjecture for torsion free groups.     

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.     

The Tate Conjecture for Cubic Fourfolds over a Finite Field

We prove the Tate conjecture for codimension 2 cycles on an ordinary cubic fourfold over a finite field. The proof involves the construction of canonical coordinates on the formal deformation space via a crystalline period map.     

The -method, weak Lefschetz theorems, and the topology of Kähler manifolds

A new approach to Nori's weak Lefschetz theorem is described. The new approach, which involves the -method, avoids moving arguments and gives much stronger results. In particular, it is proved that if and are connected smooth projective varieties of positive dimension and is a holomorphic immersion with ample normal bundle, then the image of in is of finite index. This result is obtained as a consequence of a direct generalization of Nori's theorem. The second part concerns a new approach to the theorem of Burns which states that a quotient of the unit ball in () by a discrete group of automorphisms which has a strongly pseudoconvex boundary component has only finitely many ends. The following generalization is obtained. If a complete Hermitian manifold of dimension has a strongly pseudoconvex end and for some positive constant , then, away from , has finite volume.

     

The integral Novikov conjectures for S-arithmetic groups I

We prove the integral Novikov conjecture for torsion free S-arithmetic subgroups Γ of linear reductive algebraic groups G of rank 0 over a global field k. They form a natural class of groups and are in general not discrete subgroups of Lie groups with finitely many connected components. Since many natural S-arithmetic subgroups contain torsion elements, we also prove a generalized integral Novikov conjecture for S-arithmetic subgroups of such algebraic groups, which contain torsion elements. These S-arithmetic subgroups also provide a natural class of groups with cofinite universal spaces for proper actions. Partially Supported by NSF grants DMS 0405884 and 0604878.