Wednesdays at 4:15 PM in 384H.
This talk introduces tensor network skeletonization, which is a new coarse-graining algorithm for effective computing and transforming tensor networks, with applications in studying partition functions in statistical mechanics, Euclidean path integral of quantum many-body systems, and computational problems in graphical models. Building on top of recent work in tensor networks, this new algorithm introduces a novel skeletonization step for removing local short-range correlations. This key step allows for the bond dimension low so that the computation can be carried out efficiently without sacrificing the accuracy. Numerical examples are provided to demonstrate the effectiveness of the this approach.
In many applications, e.g. quantum systems governed by the Schroedinger equation and photonic systems governed by Maxwell’s equations, the novel dispersive properties of certain periodic structures can be used for stable and controlled localization of wave energy. We discuss the 2D Schroedinger equation for a large class of periodic potentials with the symmetry of a hexagonal tiling of the plane. The potentials we consider are superpositions of localized potential wells, centered on the vertices of a regular honeycomb structure, corresponding to the single electron model of graphene and its artificial analogues. We consider the regime of strong binding, where the depth of the potential wells is large. First, we show that for sufficiently deep potentials, the lowest two Floquet-Bloch dispersion surfaces, when appropriately rescaled, converge uniformly to those of the two-band tight-binding model, introduced by PR Wallace (1947) in his pioneering study of graphite. We then discuss corollaries, in the strong binding regime, on (a) the existence of spectral gaps for honeycomb potentials with PT symmetry-breaking perturbations, and (b) the existence of topologically protected edge states for honeycomb structures with "rational edges". This is joint work with C.L Fefferman and J.P Lee-Thorp. Extensions to the case of Maxwell’s equations (with Y. Zhu and J.P Lee-Thorp) are also discussed.
This talk explains how to formulate the now classical problem of optimal liquidation (or optimal trading) inside a Mean Field Game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader in front of a " background noise " (or " mean field "). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price. Here the trader faces the uncertainty of fair price changes too but not only. He has to deal with price changes generated by other similar market participants, impacting the prices permanently too, and acting strategically. Our MFG formulation of this problem belongs to the class of " extended MFG ", we hence provide generic results to address these " MFG of controls ", before solving the one generated by the cost function of optimal trading. We provide a closed form formula of its solution, and address the case of " heterogenous preferences " (when each participant has a different risk aversion). Last but not least we give conditions under which participants do not need to instantaneously know the state of the whole system, but can " learn " it day after day, observing others' behaviors.
Many interesting stochastic PDEs arising from statistical physics are ill-posed in the sense that they involve products between distributions, so the solutions to these equations are obtained after renormalisations, which typically change the original equation by a quantity that is infinity. I will use KPZ and Phi^4_3 as two examples to explain the meanings of these infinities. As a consequence, we will see how these two equations, interpreted after suitable renormalisations, arise naturally as (weakly) universal limits for two distinct classes of systems. Part of the talk based on joint works with Martin Hairer, Cyril Labbe and Hao Shen.
This talk is about the time dynamics of energy transfer in a turbulent flow. What is the precise mechanisms of energy transfer from large scales to small scales? This question is deeply coupled to whether there are singularities in the Euler equations. I will discuss the collision of vortex filaments, and argue that the critical events happen too quickly for resolution in either modern numerical simulations or experiments. I will present possible scenarios for what might happen, and describe a set of experiments (and numerical simulations of the experiments) designed to observe what happens in greater detail.
We study the motion of an incompressible, inviscid two-dimensional fluid in a rotating frame of reference. There the fluid experiences a Coriolis force, which we assume to be linearly dependent on one of the coordinates. This is a common approximation in geophysical fluid dynamics and is referred to as beta-plane. In vorticity formulation the model we consider is then given by the Euler equation with the addition of a linear anisotropic, non-degenerate, dispersive term. This allows us to treat the problem as a quasilinear dispersive equation whose linear solutions exhibit decay in time at a critical rate.
Our main result is the global stability and decay to equilibrium of sufficiently small and localized solutions. Key aspects of the proof are the exploitation of a “double null form” that annihilates interactions between waves with parallel frequencies and a Lemma for Fourier integral operators, which allows us to control a strong weighted norm and is based on a non-degeneracy property of the nonlinear phase function associated with the problem.
Joint work with Fabio Pusateri; prior work with Tarek Elgindi.
We provide analytic results for the optimal control problem faced by a market maker who can only obtain and dispose of inventory via a sequence of sealed-bid auctions. Under the assumption that the best competing response is exponentially distributed around a commonly discerned fair market price we examine properties of the market maker's optimal behavior. We show that simple adjustments to skew and width accommodate customer arrival imbalance. We derive a straightforward relationship between the market marker's fill probability and direct holding costs. A simple formula for optimal bidding in terms of inventory cost is presented. We present the results as a perturbation of an improvement to a "linear skew, constant width" (CWLS) market making heuristic.
Combining principles of compressed sensing with deep neural network-based generative image priors has recently been empirically shown to require 10X fewer measurements than traditional compressed sensing in certain scenarios. As deep generative priors (such as those obtained via generative adversarial training) improve, analogous improvements in the performance of CS and other inverse problems may be realized across the imaging sciences. In joint work with Paul Hand, we provide a theoretical framework for studying inverse problems subject to deep generative priors. In particular, we prove that with high probability, the non-convex empirical risk objective for enforcing random deep generative priors subject to compressive random linear observations of the last layer of the generator has no spurious local minima, and that for a fixed depth, these guarantees hold at order-optimal sample complexity.
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