This pedagogical survey requires no prior knowledge of anything. Following Whaler et al, we extend the notion of Fuchsian differentials to non-commutative manifolds of odd dimension. We compute the cohomology of the standard C.O.D. operator, whose kernel is spanned by “null” functions analogous to the Fisher-Scrod germs. We investigate the remarkably surprising properties of these functions, such as: almost-sure permanence, quasi-conformal invariance and pseudo-holomorphicity in the lower-half plane. It is also found that their common zeros are distributed on Áz=-1/2 according to the Gaussian Unitary Ensemble of random matrices, as a consequence of the Yamasushi-Selberg trace formula. Applications to C* algebras as well as the famous Flounder nets of quantum cryptography will be described. To obtain a first principle derivation of the conformal field theory describing the edge excitations in a quantum Hall system has been a long standing problem. After reviewing the subject and the main challenges there, we present a microscopic derivation of the field theory of edge excitations (the so-called chiral Luttinger liquid theory) for the Laughlin states. Starting from the wave function describing an arbitrary incompressibly deformed Laughlin state, and using the description of the boundary of the quantum Hall droplet in terms of harmonic moments, we quantize these deformations by defining the explicit action of the creation (annihilation) operators of edge excitations on the many-electron wave-functions. In this way we obtain the low-energy projections of local operators, and derive the quantum field theory of edge excitations directly from quantum mechanics of electrons. We also comment that in the thermodynamic limit the incompressibly deformed Laughlin state is described by the dispersionless Toda hierarchy. Similar techniques have been recently used in the description of the Laplacian growth (Hele-Shaw) problem. We will discuss the predictions of the theory of parton saturation/Color Glass Condensate for particle production in p(d)A collisions. We concentrate on the nuclear modification factor R^pA for gluon production. We show that at moderately high energy/rapidity the nuclear modification factor R^pA exhibits Cronin enhancement. As the energy/rapidity increases, R^pA decreases. At sufficiently high energy/rapidity R^pA becomes less than 1 for all values of p_T indicating the onset of suppression of gluon production due to quantum small-x evolution effects. Our predictions are supported by the recently reported BRAHMS collaboration data on particle production at forward rapidity in dAu collisions at RHIC. .

Courant Institute of Mathematical Sciences, New-York