Automatic structures are algebraic structures, such as graphs, groups
and partial orders, that can be presented by automata. By varying the 
classes of automata (e.g. finite automata, tree automata, omega-automata) 
one varies the classes of automatic structures. The class of all automatic 
structures is robust in the sense that it is closed under many natural
algebraic and model-theoretic operations.  
In this talk, we give formal definitions to 
automatic structures, motivate the study, present many examples, and
explain several fundamental theorems.  Some results in the area
are deeply connected  with algebra, additive combinatorics, set theory, 
and complexity theory. 
We then motivate and pose several important  unresolved questions in the
area.

In this talk, I will talk about the existence of global weak solutions for the compressible Navier-Stokes equations, in particular, the viscosity coefficients depend on the density. Our main contribution is to further develop renormalized techniques so that the Mellet-Vasseur type inequality is not necessary for the compactness.  This provides existence of global solutions in time, for the barotropic compressible Navier-Stokes equations, for any $\gamma>1$, in three dimensional space, with large initial data, possibly vanishing on the vacuum. This is a joint work with D. Bresch, A. Vasseur.

We find a new duality for form factors of lightlike Wilson loops in planar N=4 super-Yang-Mills theory. The duality maps a form factor involving an n-sided lightlike polygonal super-Wilson loop together with m external on-shell states, to the same type of object but with the edges of the Wilson loop and the external states swapping roles. This relation can essentially be seen graphically in Lorentz harmonic chiral (LHC) superspace where it is equivalent to planar graph duality. However there are some crucial subtleties with the cancellation of spurious poles due to the gauge fixing. They are resolved by finding the correct formulation of the Wilson loop and by careful analytic continuation from Minkowski to Euclidean space. We illustrate all of these subtleties explicitly in the simplest non-trivial NMHV-like case.

In a robust decision, we are pessimistic toward our decision making when the probability measure is unknown. In particular, we optimise our decision under the worst case scenario (e.g. via value at risk or expected shortfall).  On the other hand, most theories in reinforcement learning (e.g. UCB or epsilon-greedy algorithm) tell us to be more optimistic in order to encourage learning. These two approaches produce an apparent contradict in decision making. This raises a natural question. How should we make decisions, given they will affect our short-term outcomes, and information available in the future?

In this talk, I will discuss this phenomenon through the classical multi-armed bandit problem which is known to be solved via Gittins' index theory under the setting of risk (i.e. when the probability measure is fixed). By extending this result to an uncertainty setting, we can show that it is possible to take into account both uncertainty and learning for a future benefit at the same time. This can be done by extending a consistent nonlinear expectation  (i.e. nonlinear expectation with tower property) through multiple filtrations.

At the end of the talk, I will present numerical results which illustrate how we can control our level of exploration and exploitation in our decision based on some parameters.