We show that a Gorenstein subcanonical codimension 3 subscheme Z⊂X=ℙ^N, N≥4, can be realized as the locus along which two Lagrangian subbundles of a twisted orthogonal bundle meet degenerately and conversely. We extend this result to singular Z and all quasi-projective ambient schemes X under the necessary hypothesis that Z is strongly subcanonical in a sense defined below. A central point is that a pair of Lagrangian subbundles can be transformed locally into an alternating map. In the local case our structure theorem reduces to that of D. Buchsbaum and D. Eisenbud [6] and says that Z is Pfaffian.

We also prove codimension 1 symmetric and skew-symmetric analogues of our structure theorems.

We give an upper bound on the dimension of the Hilbert scheme of quotients of an arbitrary vector bundle on a smooth projective curve, depending on the minimal degree of such a quotient. The bound is used for deriving effective base point freeness statements for generalized theta linear series on moduli spaces of vector bundles.

We consider the initial and boundary value problem for the quasi-linear parabolic system

$\begin{alignat*}{2} \frac{\partial u}{\partial t}-\mathrm{div} \sigma\big(x,t,u(x,t),Du(x,t)\big) &=f &\text{on }&\Omega\times(0,T),\\ u(x,t)&=0&\text{on }&\partial\Omega\times(0,T),\\ u(x,0)&=u_0(x)\quad&\text{on }&\Omega \end{alignat*}$

for a function u : Ω×[0,T)→ℝ^m with T>0. Here, f∈L^p′(0,T;W^p′(Ω;ℝ^m)) for some p∈(2n/2n+2),∞, and u₀∈L²(Ω, ℝ^m). We prove existence of a weak solution under classical regularity, growth, and coercivity conditions for σ but with only very mild monotonicity assumptions.

We prove that manifolds admitting a Riemannian metric for which products of harmonic forms are harmonic satisfy strong topological restrictions, some of which are akin to properties of flat manifolds. Others are more subtle and are related to symplectic geometry and Seiberg-Witten theory.

We also prove that a manifold admits a metric with harmonic forms whose product is not harmonic if and only if it is not a rational homology sphere.

We classify all complete embedded minimal surfaces in ℝ³; with three ends of genus g and at least 2g+2 symmetries. The surfaces in this class are the Costa-Hoffman-Meeks surfaces that have 4g+4 symmetries in the case of a flat middle end. The proof consists of using the symmetry assumptions to deduce the possible Weierstrass data and then studying the period problems in all cases. To handle the 1-dimensional period problems, we develop a new general method to prove convexity results for period quotients. The 2-dimensional period problems are reduced to the 1-dimensional case by an extremal length argument.

Let G be a connected reductive group over ℂ, and let $\mathfrak {g}$^∨ be the Langlands dual Lie algebra. Crystals for $\mathfrak {g}$^∨ are combinatorial objects that were introduced by M. Kashiwara (cf., e.g., [6]) as certain "combinatorial skeletons" of finite-dimensional representations of $\mathfrak {g}$^∨. For every dominant weight λ of $\mathfrak {g}$^∨ Kashiwara constructed a crystal B(λ) by considering the corresponding finite-dimensional representation of the quantum group U_q($\mathfrak {g}$^∨) and then specializing it to q=0. Other (independent) constructions of B(λ) were given by G. Lusztig (cf. [9]) using the combinatorics of root systems and by P. Littelmann (cf. [7]) using the "Littelmann path model." It was also shown in [5] that the family of crystals B(λ) is unique if certain reasonable conditions are imposed (cf. Theorem 1.1).

The purpose of this paper is to give another (rather simple) construction of the crystals B(λ) using the geometry of the affine Grassmannian \mathscr {G}$_G=G($\mathscr{K}$)/G($\mathscr{O}$) of the group G, where $\mathscr{K}$=ℂ((t)) is the field of Laurent power series and $\mathscr{O}$=ℂ[[t]] is the ring of Taylor series. We then check that the family B(λ) satisfies the conditions of the uniqueness theorem from [5], which shows that our crystals coincide with those constructed in the references above. It would be interesting to find these isomorphisms directly (cf., however, [10]).

In 1918 G. Hardy and J. Littlewood proved an asymptotic estimate for the Second moment of the modulus of the Riemann zeta-function on the segment [1/2,1/2+iT] in the complex plane, as T tends to infinity. In 1926 Ingham proved an asymptotic estimate for the fourth moment. However, since Ingham's result, nobody has proved an asymptotic formula for any higher moment. Recently J. Conrey and A. Ghosh conjectured a formula for the sixth moment. We develop a new heuristic method to conjecture the asymptotic size of both the sixth and eighth moments. Our estimate for the sixth moment agrees with and strongly supports, in a sense made clear in the paper, the one conjectured by Conrey and Ghosh. Moreover, both our sixth and eighth moment estimates agree with those conjectured recently by J. Keating and N. Snaith based on modeling the zeta-function by characteristic polynomials of random matrices from the Gaussian unitary ensemble. Our method uses a conjectural form of the approximate functional equation for the zeta-function, a conjecture on the behavior of additive divisor sums, and D. Goldston and S. Gonek's mean value theorem for long Dirichlet polynomials. We also consider the question of the maximal order of the zeta-function on the critical line.