W dniu 12 września 2017 roku, w godzinach 10:00-12:00 w sali HS wygłoszą odczyty profesor Franz Lehnera z TU Graz (Austria):
Tropical Finite Free Probability,
oraz profesor Carlos Vargas Obieta z CIMAT (Meksyk):
Non-commutative distributions for simplicial comlexes.
Franz Lehner-Graz, "Tropical Finite Free Probability"
Recently A. Marcus and collaborators studied convolutions of polynomials and their relations to random matrices and free probability. In joint work with A. Peperko and A. Rosenmann we study analogs of these convolution in the tropical setting. More precisely, in Max-Plus algebra arithmetics take place over the semiring $(R,\oplus,\odot)$ where $a\oplus b=\max(a,b)$ and $a\odot b = a + b$. Most results of A. Marcus et al. have analogs in this setting, and the obtained formulas take a simpler form.
Carlos Vargas Obieta, "Non-commutative distributions for simplicial comlexes"
In non-commutative probability, it is useful to consider graphs as random variables via their adjacency matrices. From a simple operator-valued perspective, adjacency matrices may be essentially replaced by incidence or boundary matrices. The advantage is that boundary and incidence matrices can naturally be generalized to higher dimensions and hence a notion of non-conmutative distribution may be defined. For the case of the Boundary matrix J, the operator-valued distribution encodes topological information about the simplicial compex. In particular, the multivariate, discrete, analytic distribution of the positive element JJ*+J*J has the Betti numbers as the weights of the 0 eigenvalue.
This is relevant for topological data analysis (TDA) and the recent theory of homotopy probability spaces, started by Jae-Suk Park.