Optimal transport, concentration of measure and functional inequalities.

This talk is devoted to Talagrand's transport-entropy inequality and its deep connections to the concentration of measure phenomenon, large deviation theory and logarithmic Sobolev inequalities. After an introductive part on the field, I will present recent results obtained with P-M Samson and C. Roberto establishing the equivalence of Talagrand's inequality to a restricted version of the Log-Sobolev inequality. If time enables, I will also present some works in progress about transport inequalities in a discrete setting.