We show the existence of optimal transport maps in the case when the cost function is the distance induced by a crystalline norm in ℝⁿ, assuming that the initial distribution of mass is absolutely continuous with respect to $\mathcal{L}$ⁿ. The proof is based on a careful decomposition of the space in transport rays induced by a secondary variational problem having the Euclidean distance as cost function. Moreover, improving a construction by Larman, we show the existence of a Nikodym set in ℝ³ having full measure in the unit cube, intersecting each element of a family of pairwise disjoint open lines only in one point. This example can be used to show that the regularity of the decomposition in transport rays plays an essential role in Sudakov-type arguments for proving the existence of optimal transport maps.

Discrete moments of the Riemann zeta function were studied by Gonek [3], [4] and Hejhal [8] in the 1980s. They independently formulated a conjecture concerning the size of these moments. In 1999 Hughes, Keating, and O'Connell [9], by employing a random matrix model, made this conjecture more precise. Subject to the Riemann hypothesis, we establish upper and lower bounds of the correct order of magnitude in the case of the fourth moment.

The object of this paper is to describe the Jacquet module functor on Harish-Chandra modules via the localisation method of Beĭlinson and Bernstein.

Let $\mathfrak{g}$ be a simple Lie algebra. For a level κ (thought of as a symmetric $\mathfrak{g}$-invariant form of $\mathfrak{g}$, let $\hat{\mathfrak{g}}_\kappa$ be the corresponding affine Kac-Moody algebra. Let Gr_G be the affine Grassmannian of $\mathfrak{g}$, and let D_κ(Gr_G)-mod be the category of κ-twisted right D-modules on Gr_G. By taking global sections of a D-module we obtain a functor Γ:D_κ(Gr_G)-mod → $\mathfrak{g}$_κ-mod. It is known that this functor is exact and faithful when κ is less than critical or irrational. In this paper, we show that the functor Γ is also exact and faithful when κ is the critical level.

Let $\mathfrak{g}$ be a reductive Lie algebra over an algebraically closed field of characteristic zero, and let $\mathfrak{k}$ be a subalgebra reductive in $\mathfrak{g}$. We prove that $\mathfrak{g}$ admits an irreducible ($\mathfrak{g}$,$\mathfrak{k}$)-module M which has finite $\mathfrak{k}$-multiplicities and which is not a ($\mathfrak{g}$,$\mathfrak{k}$′)-module for any proper inclusion of reductive subalgebras $\mathfrak{k}$⊂$\mathfrak{k}$′⊂$\mathfrak{g}$ if and only if $\mathfrak{k}$ contains its centralizer in $\mathfrak{g}$. The main point of the proof is a geometric construction of ($\mathfrak{g}$,$\mathfrak{k}$)-modules which is analogous to cohomological induction. For $\mathfrak{g}=\mathfrak{g}\mathfrak{l}$(n) we show that whenever $\mathfrak{k}$ contains its centralizer, there is an irreducible ($\mathfrak{g}$,$\mathfrak{k}$)-module M of finite type over $\mathfrak{k}$ such that $\mathfrak{k}$ coincides with the subalgebra of all $g∈\mathfrak{g}$ which act locally finitely on M. Finally, for a root subalgebra $\mathfrak{k}⊂\mathfrak{g}\mathfrak{l}(n)$, we describe all possibilities for the subalgebra $\mathfrak{l}$⊃$\mathfrak{k}$ of all elements acting locally finitely on some M.

In this paper, we study the relations between the log term of the Szegö kernel of the unit circle bundle of the dual line bundle of an ample line bundle over a compact Kähler manifold. We prove a local rigidity theorem. The result is related to the classical Ramadanov conjecture.

We discuss ways that the ring of coefficients for a topological quantum field theory (TQFT) can be reduced if one restricts somewhat the allowed cobordisms. When we apply these methods to a TQFT associated to SO(3) at an odd prime p, we obtain a functor from a somewhat restricted cobordism category to the category of free finitely generated modules over a ring of cyclotomic integers: ℤ[ζ_2p] if p≡−1 (mod 4), and ℤ[ζ_4p] if p≡1 (mod 4), where ζ_k is a primitive kth root of unity. We study the quantum invariants of prime power order simple cyclic covers of 3-manifolds. We define new invariants arising from strong shift equivalence and integrality. Similar results are obtained for some other TQFTs, but the modules are guaranteed only to be projective.