Abstract:
we present a review of the coherent state calculation for a compact lie group g as a way to stablish a phase space and a hamiltonian dynamics (in the semiclassical limit) for a quantum system with the symmetry of g. the properties of these states are investigated in the general case as well as in the traditional examples of the harmonic oscillator and angular momentum. this material was part of the first summer mini-course on theoretical physics at the instituto de física de s？o carlos, universidade de s？o paulo.

Abstract:
ilya prigogine's work on the popularization of science is critically discussed. concerning his alleged theory that would find in chaotic dynamics the deterministic mechanism for an irreversible approach to equilibrium, i argue that his motivation is unsound, his contributions nonexistent and his conclusions exaggerated. the areas of physics to which he tries to draw attention are quite interesting, but he does a disservice to the public when he mystifies them.

Abstract:
a basic introduction to the su(1,1) algebra is presented, in which we discuss the relation with canonical transformations, the realization in terms of quantized radiation field modes and coherent states. instead of going into details of these topics, we rather emphasize the existing connections between them. we discuss two parametrizations of the coherent states manifold su(1,1)/u(1): as the poincaré disk in the complex plane and as the pseudosphere (a sphere in a minkowskian space), and show that it is a natural phase space for quantum systems with su(1,1) symmetry.

Abstract:
Apresentamos uma revis o do cálculo dos estados coerentes para um grupo de Lie G compacto como forma de se estabelecer um espa o de fase e uma dinamica Hamiltoniana (no limite semiclássico) para um sistema quantico com a simetria de G. As propriedades desses estados s o investigadas no caso geral e nos exemplos tradicionais do oscilador harm nico e do momento angular. O material apresentado foi parte do primeiro mini-curso de ver o em física teórica do Instituto de Física de S o Carlos, Universidade de S o Paulo.

Abstract:
The statistical properties of quantum transport through a chaotic cavity are encoded in the traces $\T={\rm Tr}(tt^\dag)^n$, where $t$ is the transmission matrix. Within the Random Matrix Theory approach, these traces are random variables whose probability distribution depends on the symmetries of the system. For the case of broken time-reversal symmetry, we present explicit closed expressions for the average value and for the variance of $\T$ for all $n$. In particular, this provides the charge cumulants $\Q$ of all orders. We also compute the moments $$ of the conductance $g=\mathcal{T}_1$. All the results obtained are exact, {\it i.e.} they are valid for arbitrary numbers of open channels.

Abstract:
Explicit formulas are obtained for all moments and for all cumulants of the electric current through a quantum chaotic cavity attached to two ideal leads, thus providing the full counting statistics for this type of system. The approach is based on random matrix theory, and is valid in the limit when both leads have many open channels. For an arbitrary number of open channels we present the third cumulant and an example of non-linear statistics.

Abstract:
We study the dynamical generation of entanglement for a very simple system: a pair of interacting spins, s1 and s2, in a constant magnetic field. Two different situations are considered:(a) s1 ->\infty, s2 = 1/2 and (b) s1 = s2 ->\infty, corresponding, respectively, to a quantum degree of freedom coupled to a semiclassical one (a qubit in contact with an environment) and a fully semiclassical system, which in this case displays chaotic behavior. Relations between quantum entanglement and classical dynamics are investigated.

Abstract:
We obtain explicit expressions for positive integer moments of the probability density of eigenvalues of the Jacobi and Laguerre random matrix ensembles, in the asymptotic regime of large dimension. These densities are closely related to the Selberg and Selberg-like multidimensional integrals. Our method of solution is combinatorial: it consists in the enumeration of certain classes of lattice paths associated to the solution of recurrence relations.

Abstract:
The statistics of quantum transport through chaotic cavities with two leads is encoded in transport moments $M_m={\rm Tr}[(t^\dag t)^m]$, where $t$ is the transmission matrix, which have a known universal expression for systems without time-reversal symmetry. We present a semiclassical derivation of this universality, based on action correlations that exist between sets of long scattering trajectories. Our semiclassical formula for $M_m$ holds for all values of $m$ and arbitrary number of open channels. This is achieved by mapping the problem into two independent combinatorial problems, one involving pairs of set partitions and the other involving factorizations in the symmetric group.

Abstract:
We show that the semiclassical approach to chaotic quantum transport in the presence of time-reversal symmetry can be described by a matrix model, i.e. a matrix integral whose perturbative expansion satisfies the semiclassical diagrammatic rules for the calculation of transport statistics. This approach leads very naturally to the semiclassical derivation of universal predictions from random matrix theory.