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.     

An Estimate for the Problem of Illumination of the Boundary of a Convex Body in E3

Let A be a convex body in Euclidean space E3. We denote by H(A) the smallest number of homothetic 'reduced copies of A by which it is possible to cover the whole of A. The conjecture of Hadwiger is H(A) 8. We prove that H(A) 16.     

Harmoniques Sphériques et Représentations Irréductibles de O ( ∞ )

We build an irreducible unitary representation of SO( ) from the usual one of SO( n ) in the space of harmonic homogeneous polynomials of degree m of ⁿ . We give a characterization of these new representations which extends in a natural way the finite dimensional characterization. In the particular case of SO( ), we thus get some results of Olshanskii (cf. [12]). This leads to a new proof of McKean conjecture about irreducible representations of ( infin ) (cf. [10]).     

有限子集系的Sperner系

1９８０年，Ｋｏ-Ｗｅｉ Ｌｉｈ提出如下猜想：如果Ｆ是由Ｂ^ｎ中固定秩的不同元素生成的序理想，那么Ｆ是Ｓｐｅｒｎｅｒ系．本文证实了当Ｆ是由Ｘ的子集Ｙ的所有相同秩的元素生成的序理想，猜想是正确的     

Strong mixing coefficients for non-commutative Gaussian processes

Bounds for non-commutative versions of two classical strong mixing coefficients for -Gaussian processes are found in terms of the angle between the underlying Hilbert spaces. As a consequence, we construct a -mixing -Gaussian stationary sequence with growth conditions on variances of partial sums. If classical processes with analogous properties were to exist, they would provide a counter-example to the Ibragimov conjecture.

     

The Brumer-Stark conjecture in some families of extensions of specified degree

As a starting point, an important link is established between Brumer's conjecture and the Brumer-Stark conjecture which allows one to translate recent progress on the former into new results on the latter. For example, if is an abelian extension of relative degree , an odd prime, we prove the -part of the Brumer-Stark conjecture for all odd primes with belonging to a wide class of base fields. In the same setting, we study the -part and -part of Brumer-Stark with no special restriction on and are left with only two well-defined specific classes of extensions that elude proof. Extensive computations were carried out within these two classes and a complete numerical proof of the Brumer-Stark conjecture was obtained in all cases.

     

Nonexistence of abelian difference sets: Lander's conjecture for prime power orders

In 1963 Ryser conjectured that there are no circulant Hadamard matrices of order 4$"> and no cyclic difference sets whose order is not coprime to the group order. These conjectures are special cases of Lander's conjecture which asserts that there is no abelian group with a cyclic Sylow -subgroup containing a difference set of order divisible by . We verify Lander's conjecture for all difference sets whose order is a power of a prime greater than 3.

     

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 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.     

Obstructions to nonnegative curvature and rational homotopy theory

We establish a link between rational homotopy theory and the problem which vector bundles admit a complete Riemannian metric of nonnegative sectional curvature. As an application, we show for a large class of simply-connected nonnegatively curved manifolds that, if lies in the class and is a torus of positive dimension, then ``most' vector bundles over admit no complete nonnegatively curved metrics.