Hyperplane sections of determinantal varieties over finite fields and linear codes

We determine the number of ${\mathbb{F}}_q$-rational points of hyperplane sections of classical determinantal varieties defined by the vanishing of minors of a fixed size of a generic matrix, and identify the hyperplane sections giving the maximum number of ${\mathbb{F}}_q$-rational points. Further we consider similar questions for sections by linear subvarieties of a fixed codimension in the ambient projective space. This is closely related to the study of linear codes associated to determinantal varieties, and the determination of their weight distribution, minimum distance, and generalized Hamming weights. The previously known results about these are generalized and expanded significantly. Connections to eigenvalues of certain association schemes, distance regular graphs, and rank metric codes are also indicated.     

A construction of representations of affine Weyl groups

Let G be a complex, semisimple, simply connected algebraic group withLie algebra . We extend scalars to the power series field in one variable C(()), and consider the space of Iwahori subalgebras containing a fixed nil-elliptic element of C(()),i.e. fixed point varieties on the full affine flag manifold. We definerepresentations of the affine Weyl group in the homology of these varieties,generalizing Kazhdan and Lusztig's topological construction of Springer'srepresentations to the affine context.     

Maximal degeneracy points of GKZ systems

Motivated by mirror symmetry, we study certain integral representations of solutions to the Gel´fand-Kapranov-Zelevinsky (GKZ) hypergeometric system. Some of these solutions arise as period integrals for Calabi-Yau manifolds in mirror symmetry. We prove that for a suitable compactification of the parameter space, there exist certain special boundary points, which we called maximal degeneracy points, at which all solutions but one become singular.

     

A comparison theorem for l-adic cohomology

We show that, for certain types of rigid analytic varieties X and constructible l-adic sheaves (F_n)_n on, one has . As an application we obtain that, for an algebraic variety X and associated rigid analytic variety X^rig, the l-adic cohomology of X and X^rig agree.     

Every

It is known that congruence lattices of algebras in -permutable varieties satisfy non-trivial identities; however, the identities discovered so far are rather artificial and seem to have little intrinsic interest.

We show here that every -permutable variety satisfies the well-known and well-studied congruence identity . We also get a new condition equivalent to -permutability.

     

Harnack-Thom theorem for higher cycle groups and Picard varieties

We generalize the Harnack-Thom theorem to relate the ranks of the Lawson homology groups with -coefficients of a real quasiprojective variety with the ranks of its reduced real Lawson homology groups. In the case of zero-cycle group, we recover the classical Harnack-Thom theorem and generalize the classical version to include real quasiprojective varieties. We use Weil's construction of Picard varieties to construct reduced real Picard groups, and Milnor's construction of universal bundles to construct some weak models of classifying spaces of some cycle groups. These weak models are used to produce long exact sequences of homotopy groups which are the main tool in computing the homotopy groups of some cycle groups of divisors. We obtain some congruences involving the Picard number of a nonsingular real projective variety and the rank of its reduced real Lawson homology groups of divisors.

     

带有波前编码板的虹膜采集光学系统的设计

设计了一款采用波前编码技术的虹膜采集光学系统,以实现大景深、自然非约束状态下的成像.以传统的光学系统评价指标斯特利尔比和能量集中度作为波前编码系统的成像特征,利用ZEMAX软件对波前编码系统进行设计,权衡了离焦不敏感性和图像可恢复性.比较了带有立方相位编码板和带有修正项的立方相位编码板的虹膜采集系统以及传统系统的焦深延拓效果.发现采用立方相位编码板的虹膜采集光学系统的焦深延拓效果最佳,其焦深是传统系统的8倍.该系统增大了虹膜采集的距离,对于自然状态虹膜成像系统设计有一定的参考价值.