The author covers smooth manifolds and

vector bundles, vector fields and differential equations, tensors, differential forms, the integration of manifolds, metric and symplectic structures, and a wide variety of other related subjects.

Together with my collaborators Oliver Braunling and Jesse Wolfson we have carefully studied one of the main tools of BBE, Tate

vector bundles, in an abstract context which allows to handle higher-dimensional situations.

Matsumura [2], here we note that the decomposition theorem also holds for direct images of canonical sheaves tensorized with Nakano semipositive

vector bundles.

He showed that MacP(d, n) plays the same role for matroid bundles as the ordinary Grassmannian plays for

vector bundles, and pointed out that the geometric realization of the order complex [parallel]MacP(d, n)[parallel] of MacP(d, n) is homeomorphic to the real Grassmannian Gr(d, n) if d equals i, 2,n - 2, [parallel]MacP(d, n)[parallel] of MacP(d, n) is homeomorphic to the real Grassmannian Gr(d, n) if d equals i, 2,n - 2, or n - 1.

Algebraic K-theory is a field of abstract algebra concerning projective modules over a ring and

vector bundles over schemes.

are linear connections on

vector bundles S(TM) and T[M.

The other 14 consider such topics as Picard groups of moduli spaces of torsion-free sheaves on curves,

vector bundles and the icosahedron, a quasi-complete homogeneous contact manifold associated to a cubic form, orthogonal bundles over curves in characteristic two, parabolic structures, and iterated destabilizing modifications for

vector bundles with connection.

Let E [right arrow] X and F [right arrow] X be smooth

vector bundles over an oriented manifold X, where E and F are either both complex or both real, and let

His major contributions include several work on conjectures, such as the Calabi conjecture, positive mass conjecture and existence of black holes, Smith conjecture, Hermitian Yang-Mills connection and stable

vector bundles, Frankel conjecture and Mirror conjecture, as well as new methods and concepts of gradient estimates and Harnack inequalities, uniformization of complex manifolds, harmonic maps and rigidity, minimal submanifolds, and also open problems in geometry, covering harmonic functions with controlled growth, rank rigidity of nonpositively curved manifolds, Kahler-Einstein metrics and stability of manifolds and Mirror symmetry.

Some probabilistic value distributions of the Riemann zeta function and its derivatives Junghun LEE, Tomokazu ONOZUKA and Ade Irma SURIAJAYA A remark on the decomposition theorem for direct images of canonical sheaves tensorized with semipositive

vector bundles Taro FUJISAWA Above two, communicated by Masaki KASHIWARA, M.

Also included is a developed theory of arithmetic Chern classes of integral automorphic

vector bundles with singular metrics, based on work by Burgos, Kramer, and KEhn, and material devoted to investigating volumes of orthogonal Shimura varieties.

Vector bundles on degenerations of elliptic curves and Yang-Baxter equations.