Tropical totally positive matrices

We investigate the tropical analogues of totally positive and totally nonnegative matrices. These arise when considering the images by the nonarchimedean valuation of the corresponding classes of matrices over a real nonarchimedean valued field, like the field of real Puiseux series. We show that the nonarchimedean valuation sends the totally positive matrices precisely to the Monge matrices. This leads to explicit polyhedral representations of the tropical analogues of totally positive and totally nonnegative matrices. We also show that tropical totally nonnegative matrices with a finite permanent can be factorized in terms of elementary matrices. We finally determine the eigenvalues of tropical totally nonnegative matrices, and relate them with the eigenvalues of totally nonnegative matrices over nonarchimedean fields.Comment: The first author has been partially supported by the PGMO Program of FMJH and EDF, and by the MALTHY Project of the ANR Program. The second author is sported by the French Chateaubriand grant and INRIA postdoctoral fellowshi

What is absolutely continuous spectrum?

This note is an expanded version of the author's contribution to the Proceedings of the ICMP Santiago, 2015, and is based on a talk given by the second author at the same Congress. It concerns a research program devoted to the characterization of the absolutely continuous spectrum of a self-adjoint operator H in terms of the transport properties of a suitable class of open quantum systems canonically associated to H

Cohomological Donaldson-Thomas theory

This review gives an introduction to cohomological Donaldson-Thomas theory: the study of a cohomology theory on moduli spaces of sheaves on Calabi-Yau threefolds, and of complexes in 3-Calabi-Yau categories, categorifying their numerical DT invariant. Local and global aspects of the theory are both covered, including representations of quivers with potential. We will discuss the construction of the DT sheaf, a nontrivial topological coefficient system on such a moduli space, along with some cohomology computations. The Cohomological Hall Algebra, an algebra structure on cohomological DT spaces, will also be introduced. The review closes with some recent appearances, and extensions, of the cohomological DT story in the theory of knot invariants, of cluster algebras, and elsewhere.Comment: 33 pages, some references adde

Random matrices: The Universality phenomenon for Wigner ensembles

In this paper, we survey some recent progress on rigorously establishing the universality of various spectral statistics of Wigner Hermitian random matrix ensembles, focusing on the Four Moment Theorem and its refinements and applications, including the universality of the sine kernel and the Central limit theorem of several spectral parameters. We also take the opportunity here to issue some errata for some of our previous papers in this area.Comment: 58 page

Motivations and Physical Aims of Algebraic QFT

We present illustrations which show the usefulness of algebraic QFT. In particular in low-dimensional QFT, when Lagrangian quantization does not exist or is useless (e.g. in chiral conformal theories), the algebraic method is beginning to reveal its strength.Comment: 40 pages of LateX, additional remarks resulting from conversations and mail contents, removal of typographical error

Positivity and lower bounds for the density of Wiener functionals

We consider a functional on the Wiener space which is smooth and not degenerated in Malliavin sense and we give a criterion of strict positivity of the density. We also give lower bounds for the density. These results are based on the representation of the density by means of the Riesz transform introduced by Malliavin and Thalmaier and on the estimates of the Riesz transform given Bally and Caramellino