We construct natural Green forms for special cycles in orthogonal and unitary Shimura varieties, in all codimensions, and, for compact Shimura varieties of type O(p, 2) and U(p, 1), we show that the resulting local archimedean height pairings are related to special values of derivatives of Siegel Eisentein series. A conjecture put forward by Kudla relates these derivatives to arithmetic intersections of special cycles, and our results settle the part of his conjecture involving local archimedean heights.

These notes were written to be distributed to the audience of the first author's Takagi lectures delivered June 23, 2018. These are based on a work-in-progress that is part of a collaborative project that also involves Akshay Venkatesh. In this work-in-progress we give a new construction of some Eisenstein classes for GL N (Z) that were first considered by Nori [41] and Sczech [44]. The starting point of this construction is a theorem of Sullivan on the vanishing of the Euler class of SL N (Z) vector bundles and the explicit transgression of this Euler class by Bismut and Cheeger. Their proof indeed produces a universal form that can be thought of as a kernel for a regularized theta lift for the reductive dual pair (GL N , GL 1). This suggests looking to reductive dual pairs (GL N , GL k) with k ≥ 1 for possible generalizations of the Eisenstein cocycle. This leads to fascinating lifts that relate the geometry/topology world of real arithmetic locally symmetric spaces to the arithmetic world of modular forms. In these notes we don't deal with the most general cases and put a lot of emphasis on various examples that are often classical.

We use superconnections to define and study some natural differential forms on period domains D that parametrize polarized Hodge structures of given type on a rational quadratic vector space V . These forms depend on a choice of vectors v 1 , . . . , v r ∈ V and have a Gaussian shape that peaks on the locus where v 1 , . . . , v r become Hodge classes. We show that they can be rescaled so that one can form theta series by summing over a lattice L r ⊂ V r . These series define differential forms on arithmetic quotients Γ\D. We compute their cohomology class explicitly in terms of the cohomology classes of Hodge loci in Γ\D. When the period domain is a hermitian symmetric domain of type IV, we show that the components of our forms of appropriate degree recover the forms introduced by Kudla and Millson. In particular, our results provide another way to establish the main properties of these forms. 1

The Chixoy River in northern Guatemala exhibits a long‐term dependence of monthly streamflows. A broken line model is developed to simulate the long‐term dependence for the Chixoy River. The method for the determination of parameters for simulation is computationally easy. In addition, an algorithm is developed for preserving the memory of the historical streamflow record, which is operationally useful because if simulations of future streamflow are to be used for purposes of planning and design, such simulations should preserve a memory of the past, if long‐term persistence is assumed to exist. The broken line model is shown to be a useful operational alternative to an autoregressive model.

Abstract. We introduce a regularized theta lift for reductive dual pairs of the form (Sp 4 , O(V )) with V a quadratic vector space over a totally real number field F . The lift takes values in the space of (1, 1)-currents on the Shimura variety attached to GSpin(V ), and we prove that its values are cohomologous to currents given by integration on special divisors against automorphic Green functions. In the second part to this paper, we will show how to evaluate the regularized theta lift on differential forms obtained as usual (non-regularized) theta lifts.