These notes are meant to provide a survey of some recent results and techniques in the theory of conservation laws. In one space dimension, a system of conservation laws can be written as u_t + f(u) _x = 0.

Here u = (u₁, ... , u_n) is the vector of conserved quantities while the components of f = (f₁, ... , f_n) are called the fluxes. Integrating over the interval [a, b] one obtains

d/dt ?_a^b u(t,x) dx = ?_a^b u_t(t,x) dx = - ?_a^b f( u(t,x))_x dx = f(u(t,a)) - f(u(t,b)) = [inflow at a ] - outflow at b].

In other words, each component of the vector u represents a quantity which is neither created nor destroyed: its total amount inside any given interval [a, b] can change only because of the flow across boundary points.

We survey the proof of a series of conjectures in combinatorics using new results on the geometry of Hilbert schemes. The combinatorial results include the positivity conjecture for Macdonald's symmetric functions, and the "n!" and "(n+1)^n-1" conjectures relating Macdonald polynomials to the characters of doubly-graded S_n modules. To make the treatment self-contained, we include background material from combinatorics, symmetric function theory, representation theory and geometry. At the end we discuss future directions, new conjectures and related work of Ginzburg, Kumar and Thomsen, Gordon, and Haglund and Loehr.

The first goal of this paper is to explain some important results of Wilfred Schmid from his fundamental paper [30] in which he proves very general results which govern the behaviour of the periods of a of smooth projective variety X_t as it degenerates to a singular variety. As has been known since classical times, the periods of a smooth projective variety sometimes contain significant information about the geometry of the variety, such as in the case of curves where the periods determine the curve. Likewise, information about the asymptotic behaviour of the periods of a variety as it degenerates sometimes contain significant information about the degeneration and the singular fiber. For example, the Hodge norm estimates, which are established in [30] and [3], describe the asymptotics of the Hodge norm of a cohomology class as the variety degenerates in terms of its monodromy. They are an essential ingredient in the study of the L₂ cohomology of smooth varieties with coefficients in a variation of Hodge structure [37, 4].

A second goal is to give some idea of how geometric and arithmetic information can be extracted from the limit periods, both in the geometric case and in the case of the limits of the mixed Hodge structures on fundamental groups of curves.

The aim of these notes is to describe some examples of modular forms whose Fourier coefficients involve quantities from arithmeticla algebraic geometry. Ath the moment, no general theory of such forms exists, but the examples suggest that they should be viewed as a kind of arithmetic analogue of theta series and that there should be an arithmetic Siegel-Weil formula relating suitable averages of them to special values of derivatives of Eisenstein series. We will concentrate on the case for which the most complete picture is available, the case of generating series for cycles on the arithmetic surfaces associated to Shimura curves over ?, expanding on the treatment in [40]. A more speculative overview can be found in [41].

These notes are a biased guide to some recent developments in the deformation theory of hyperbolic 3-manifolds and Kleinian groups. This field has its roots in the work of Poincaré and Klein, and connects to topology via Thurston's geometrization program, to analysis via the Ahlfors-Bers quasiconformal theory, and to complex dynamics via the work of Thurston, Sullivan and others. It encompasses many techniques and ideas and may be too big a subject for a single account. We will focus on the geometric study of ends of hyperbolic 3-manifolds and boundaries of deformation spaces, and in particular on the techniques that led to the recent solution by Brock, Canary and the author [82, 23] of the incompressible-boundary case of Thurston's "Ending Lamination Conjecture".

This expository article gives an introduction to the (generalized) conjecture of Rapoport and Goresky-MacPherson which identifies the intersection cohomology of a real equal-rank Satake compactification of a locally symmetric space with that of the reductive Borel-Serre compactification. We motivate the conjecture with examples and then give an introduction to the various topics that are involved: intersection cohomology, the derived category, and compactifications of a locally symmetric space, particularly those above. We then give an overview of the theory of L-modules and micro-support which was developed to solve the conjecture but has other important applications as well. We end with sketches of the proofs of three main theorems on L-modules that lead to the resolution of the conjecture. The text is enriched with many examples, illustrations, and references to the literature.