Abstract:
We find new representations for Itzykson-Zuber like angular integrals for arbitrary beta, in particular for the orthogonal group O(n), the unitary group U(n) and the symplectic group Sp(2n). We rewrite the Haar measure integral, as a flat Lebesge measure integral, and we deduce some recursion formula on n. The same methods gives also the Shatashvili's type moments. Finally we prove that, in agreement with Brezin and Hikami's observation, the angular integrals are linear combinations of exponentials whose coefficients are polynomials in the reduced variables (x_i-x_j)(y_i-y_j).

Abstract:
We prove that the correlations functions, generated by the determinantal process of the Christoffel-Darboux kernel of an arbitrary order 2 ODE, do satisfy loop equations.

Abstract:
The goal of this article is to rederive the connection between the Painlev\'e $5$ integrable system and the universal eigenvalues correlation functions of double-scaled hermitian matrix models, through the topological recursion method. More specifically we prove, \textbf{to all orders}, that the WKB asymptotic expansions of the $\tau$-function as well as of determinantal formulas arising from the Painlev\'e $5$ Lax pair are identical to the large $N$ double scaling asymptotic expansions of the partition function and correlation functions of any hermitian matrix model around a regular point in the bulk. In other words, we rederive the "sine-law" universal bulk asymptotic of large random matrices and provide an alternative perturbative proof of universality in the bulk with only algebraic methods. Eventually we exhibit the first orders of the series expansion up to $O(N^{-5})$.

Abstract:
To any solution of a linear system of differential equations, we associate a kernel, correlators satisfying a set of loop equations, and in presence of isomonodromic parameters, a Tau function. We then study their semiclassical expansion (WKB type expansion in powers of the weight hbar per derivative) of these quantities. When this expansion is of topological type (TT), the coefficients of expansions are computed by the topological recursion with initial data given by the semiclassical spectral curve of the linear system. This provides an efficient algorithm to compute them at least when the semiclassical spectral curve is of genus 0. TT is a non trivial property, and it is an open problem to find a criterion which guarantees it is satisfied. We prove TT and illustrate our construction for the linear systems associated to the q-th reductions of KP - which contain the (p,q) models as a specialization.

Abstract:
We write the loop equations for the $\beta$ two-matrix model, and we propose a topological recursion algorithm to solve them, order by order in a small parameter. We find that to leading order, the spectral curve is a "quantum" spectral curve, i.e. it is given by a differential operator (instead of an algebraic equation for the hermitian case). Here, we study the case where that quantum spectral curve is completely degenerate, it satisfies a Bethe ansatz, and the spectral curve is the Baxter TQ relation.

Abstract:
African trypanosomiasis is a severe parasitic disease that affects both humans and livestock. Several different species may cause animal trypanosomosis and although Trypanosoma vivax (sub-genus Duttonella) is currently responsible for the vast majority of debilitating cases causing great economic hardship in West Africa and South America, little is known about its biology and interaction with its hosts. Relatively speaking, T. vivax has been more than neglected despite an urgent need to develop efficient control strategies. Some pioneering rodent models were developed to circumvent the difficulties of working with livestock, but disappointedly were for the most part discontinued decades ago. To gain more insight into the biology of T. vivax, its interactions with the host and consequently its pathogenesis, we have developed a number of reproducible murine models using a parasite isolate that is infectious for rodents. Firstly, we analyzed the parasitical characteristics of the infection using inbred and outbred mouse strains to compare the impact of host genetic background on the infection and on survival rates. Hematological studies showed that the infection gave rise to severe anemia, and histopathological investigations in various organs showed multifocal inflammatory infiltrates associated with extramedullary hematopoiesis in the liver, and cerebral edema. The models developed are consistent with field observations and pave the way for subsequent in-depth studies into the pathogenesis of T. vivax - trypanosomosis.

Abstract:
Trypanosoma vivax is the main species involved in trypanosomosis, but very little is known about the immunobiology of the infective process caused by this parasite. Recently we undertook to further characterize the main parasitological, haematological and pathological characteristics of mouse models of T. vivax infection and noted severe anemia and thrombocytopenia coincident with rising parasitemia. To gain more insight into the organism's immunobiology, we studied lymphocyte populations in central (bone marrow) and peripherical (spleen and blood) tissues following mouse infection with T. vivax and showed that the immune system apparatus is affected both quantitatively and qualitatively. More precisely, after an initial increase that primarily involves CD4+ T cells and macrophages, the number of splenic B cells decreases in a step-wise manner. Our results show that while infection triggers the activation and proliferation of Hematopoietic Stem Cells, Granulocyte-Monocyte, Common Myeloid and Megacaryocyte Erythrocyte progenitors decrease in number in the course of the infection. An in-depth analysis of B-cell progenitors also indicated that maturation of pro-B into pre-B precursors seems to be compromised. This interferes with the mature B cell dynamics and renewal in the periphery. Altogether, our results show that T. vivax induces profound immunological alterations in myeloid and lymphoid progenitors which may prevent adequate control of T. vivax trypanosomosis.

Abstract:
We improve Haldane's formula which gives the number of configurations for $N$ particles on $d$ states in a fractional statistic defined by the coupling $g=l/m$. Although nothing is changed in the thermodynamic limit, the new formula makes sense for finite $N=pm+r$ with $p$ integer and $0

Abstract:
We calculate the partition function for "composite particles". For any finite number of states d, and in the following two cases: 1)all states have the same energy, 2)the energy is linearly distributed over the states, we transform the partition function into a finite sum of terms which exhibit trivially their $% d $dependance. The infinite d thermodynamic limit is obtained as well as the finite d-size corrections, etc. In the second case, we obtain the finite d-size corrections to "the universal chiral partition function for exclusion statistics".

Abstract:
This publication is an exercise which extends to two variables the Christoffel's construction of orthogonal polynomials for potentials of one variable with external sources. We generalize the construction to biorthogonal polynomials. We also introduce generalized Schur polynomials as a set of orthogonal, symmetric, non homogeneous polynomials of several variables, attached to Young tableaux.