Recursos de colección
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 manifold
(or complex algebraic variety) on one...
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.
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.
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 of the proofs of three main theorems on L-modules
that...
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 led to the recent
solution by Brock, Canary...
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 curves over ?, expanding on the...
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 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...
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 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.
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
u_{t} + f(u) _{x} = 0.
¶
Here u = (u_{1}, ... , u_{n}) is the vector of
conserved quantities while the components of f = (f_{1}, ... , 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...