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Optimal transport (OT) theory shares connections with geometry, analysis, probability theory, and other fields in mathematics. The recent resurgence of interests in OT stems from applied fields such as machine learning, image processing and statistics through the introduction of entropic regularization. In this talk, we will discuss the convergence of entropically regularized optimal transport. Our first result is about a large deviation principle of the associated optimizers in entropic OT and the second result is about the stability of the optimizers under weak convergence. To prove these results, we introduce a new notion called ‘cyclical invariance of measures. Lastly, we touch on a recent result which quantifies the stability by showing the rate of convergence of the entropic OT. This result is the first to find the rate of convergence of Sinkhorn algorithm for unbounded cost function.