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One Dimensional Hyperbolic Systems of Conservation Laws - Bressan , Alberto
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 ut + f(u) x = 0. Here u = (u1, ... , un) is the vector of conserved quantities while the components of f = (f1, ... , fn) are called the fluxes. Integrating over the interval [a, b] one obtains d/dt ?ab u(t,x) dx = ?ab ut(t,x) dx = - ?ab f( u(t,x))x dx = f(u(t,a)) - f(u(t,b)) = [inflow at a ] - outflow at...
2.
Combinatorics, symmetric functions, and Hilbert schemes - Haiman , Mark
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 Sn 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.
3.
Periods of Limit Mixed Hodge Structures - Hain , Richard
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 Xt 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...
4.
Modular forms and aritmetic geometry - Kudla , Stephen S.
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...
5.
End Invariants and the Classification of Hyperbolic 3-Manifolds - Minsky , Yair N.
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...
6.
On the Cohomology of Locally Symmetric Spaces and of their Compactifications - Saper , Leslie
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...
7.
Cluster algebras: Noted for the CDM-03 conference - Fomin, Sergey; Zelevinsky, Andrei
This is an expanded version of the notes of our lectures given at the conference Current Developments in Mathematics 2003 held at Harvard University on November 21-22, 2003. We present an overview of the main definitions, results and applications of the theory of cluster algebras.
8.
Automorphic Number Theory - Iwaniec, Henryk
I shall emphasize the role of automorphic forms in harmonic analysis because they are indispensable in analytic number theory, which is my primary subject of interest. A lot has been presented to the general forum about modular forms in algebraic number theory, in particular after resolution of the Fermat last theorem [W], [BCDT]. Therefore I shall limit my venture to the topics which help to grasp the essence of modularity in number theory as a whole.
9.
On some problems of Kobayashi and Lang; algebraic approaches - Voisin, Claire
The goal of this survey paper is to present results on hyperbolicity of complex algebraic manifolds, which appeared after the papers [48] and [42], where a number of important and intriguing conjectures were proposed. Since the paper puts the accent on measure hyperbolicity on one hand, and on algebraic methods on the other hands, we hope it will not overlap too much with the beautiful paper [21] by Demailly, except for the basic definitions and starting points. The basic questions asked in [42], and in a different spirit in [48] concern the relationships between curvature properties of a given complex...
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