In the 1960s, Lynden-Bell, studying the dynamics of galaxies around steady states of the gravitational Vlasov-Poisson equation, described a phenomenon he called "violent relaxation," a convergence to equilibrium through phase mixing analogous in some respects to Landau damping in plasma physics. In this talk, I will discuss recent work on this gravitational Landau damping for the linearised Vlasov-Poisson equation and, in particular, the critical role of regularity of the steady states in distinguishing damping from oscillatory behaviour in the perturbations. This is based on joint work with Mahir Hadzic, Gerhard Rein, and Christopher Straub.

We consider degenerate fully nonlinear equations, whose degeneracy rate depends on the gradient of solutions. We work under a Dini-continuity condition on the degeneracy term and prove that solutions are continuously differentiable. Then we frame this class of equations in the context of a free transmission problem. Here, we discuss the existence of solutions and establish a result on interior regularity. We conclude the talk by discussing a boundary regularity estimate; of particular interest is the case of point-wise regularity at the intersection of the fixed and the free boundaries. This is based on joint work with David Stolnicki.

Extrinsic flows are evolution equations whose speeds are determined by the extrinsic curvature of submanifolds in ambient spaces.  Some of the well-known ones are mean curvature flow, Gauss curvature flow, and Lagrangian mean curvature flow.

We focus on the special case in which the speed of a flow is given by powers of mean curvature for smooth convex hypersurfaces of graph type, i.e., ones that can be represented as the graph of a function.  Convergence and long-time existence of such flow will be discussed. Furthermore, C^2 estimates which are independent of height of the graph will be derived to see that the boundary of the domain of the graph is also a smooth solution for the same flow as a submanifold with codimension two in the classical sense.  Some of the main ideas, notably a priori estimates via the maximum principle, come from the work of Huisken and Ecker on mean curvature evolution of entire graphs in 1989.  This is a joint work with Ki-ahm Lee and Taehun Lee.

Popular automated market makers (AMMs) use constant function markets (CFMs) to clear the demand and supply in the pool of liquidity. A key drawback in the implementation of CFMs is that liquidity providers (LPs) are currently providing liquidity at a loss, on average. In this paper, we propose two new designs for decentralised trading venues, the arithmetic liquidity pool (ALP) and the geometric liquidity pool (GLP). In both pools, LPs choose impact functions that determine how liquidity taking orders impact the marginal exchange rate of the pool, and set the price of liquidity in the form of quotes around the marginal rate. The impact functions and the quotes determine the dynamics of the marginal rate and the price of liquidity. We show that CFMs are a subset of ALP; specifically, given a trading function of a CFM, there are impact functions and  quotes in the ALP that replicate the marginal rate dynamics and the execution costs in the CFM. For the ALP and GLP, we propose an optimal liquidity provision strategy where the price of liquidity maximises the LP's expected profit and the strategy depends on the LP's (i) tolerance to inventory risk and (ii) views on the demand for liquidity. Our strategies admit closed-form solutions and are computationally efficient.  We show that the price of liquidity in CFMs is suboptimal in the ALP. Also, we give conditions on the impact functions and the liquidity provision strategy to prevent arbitrages from rountrip trades. Finally, we use transaction data from Binance and Uniswap v3 to show that liquidity provision is not a loss-leading activity in the ALP.