Matrix-valued Monge-Kantorovich Optimal Mass Transport

We formulate an optimal transport problem for matrix-valued density functions. This is pertinent in the spectral analysis of multivariable time-series. The "mass" represents energy at various frequencies whereas, in addition to a usual transportation cost across frequencies, a cost of rotation is also taken into account. We show that it is natural to seek the transportation plan in the tensor product of the spaces for the two matrix-valued marginals. In contrast to the classical Monge-Kantorovich setting, the transportation plan is no longer supported on a thin zero-measure set.Comment: 11 page

Heavy-ion physics: freedom to do hot, dense, exciting QCD

In these two lectures I review the basics of heavy-ion collisions at relativistic energies and the physics we can do with them. I aim to cover the basics on the kinematics and observables in heavy-ion collider experiments, the basics on the phenomenology of the nuclear matter phase diagram, some of the model building and simulations currently used in the heavy-ion physics community and a selected list of amazing phenomenological discoveries and predictions.Comment: These lectures were given at the 2019 CERN Latin-American School of High-Energy Physics in Cordoba, Argentina, 13 - 26 March 2019 and the notes have been submitted to proceedings of CLASHEP 2019. These lecture notes are based on previous Heavy-Ion and extreme QCD lectures given at CLASHEP by A. Ayala (2017), E. Fraga (2015) and J. Takahashi (2013