The contact connected sum is a well-understood operation for contact manifolds. I will discuss work in progress on how pseudo-holomorphic curves behave in the symplectization of the 3-dimensional contact connected sum, and as a result the connected sum formula of embedded contact homology.
Given a collection of functions in a Banach space, typically called a dictionary in machine learning, we study the approximation properties of its convex hull. Specifically, we develop techniques for bounding the metric entropy and n-widths, which are fundamental quantities in approximation theory that control the limits of linear and non-linear approximation. Our results generalize existing methods by taking the smoothness of the dictionary into account, and in particular give sharp estimates for shallow neural networks. Consequences of these results include: the optimal approximation rates which can be attained for shallow neural networks, that shallow neural networks dramatically outperform linear methods of approximation, and indeed that shallow neural networks outperform all continuous methods of approximation on the associated convex hull. Next, we discuss greedy algorithms for constructing approximations by non-linear dictionary expansions. Specifically, we give sharp rates for the orthogonal greedy algorithm for dictionaries with small metric entropy, and for the pure greedy algorithm. Finally, we give numerical examples showing that greedy algorithms can be used to solve PDEs with shallow neural networks.
Numerical algebraic geometry revolves around the study of solutions to polynomial systems via numerical method. Two of the fundamental tools in this field are the polyhedral homotopy of Huber and Sturmfels for computing isolated solutions and the concept of witness sets put forth by Sommese and Wampler as numerical representations for non-isolated solution components. In this talk, we will describe a stratified polyhedral homotopy method that will bridge the gap between these two largely independent area. Such a homotopy method will discover numerical representations of non-isolated solution components as by-products from the process of computing isolated solutions. We will also outline the pipeline of numerical algorithms necessary to implement this homotopy method on modern massively parallel computing architecture.
https://bluejeans.com/421317143/2787<br />
We consider the 2-dim capillary gravity water wave problem -- the free boundary problem of the Euler equation with gravity and surface tension -- of finite depth x2 \in (-h,0) linearized at a uniformly monotonic shear flow U(x2). Our main results consist of two aspects, eigenvalue distribution and inviscid damping. We first prove that in contrast to finite channel flow and gravity wave, the linearized capillary gravity wave has two unbounded branches of eigenvalues for high wave numbers. Under certain conditions, we provide a complete picture of the eigenvalue distribution. Assuming there are no singular modes, we obtain the linear inviscid damping. We also identify the leading asymptotic terms of velocity and obtain the stronger decay for the remainders.
The rank of a point $p$ with respect to a non-degenerate variety is the smallest number of the points in the variety that spans the point $p$. Studies about the ranks of points are interesting and important in various areas of applied mathematics and engineering in the sense that they are the shortest sizes of the decompositions of vectors into combinations of simple vectors.
In this talk, we focus on the ranks of points with respect to the rational normal curves, i.e. Waring ranks of binary forms. We introduce an algorithm that produces random points of given rank r. (Note that if we choose points randomly, we expect the rank of the points is just the generic rank.) Moreover, we check some known facts by Macaulay 2 computations. Lastly, we discuss the maximal and minimal rank of points in linear spaces.
Teams link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1649360107625?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%2206706002-23ff-4989-8721-b078835bae91%22%7d
Link: https://us06web.zoom.us/j/2782194473?pwd=L1Nnc0c1SXFFYkZqSkVGUGpEd2E4dz09
Consider two volume-preserving, smooth diffeomorphisms f and g of a compact manifold M. Define the random walk on M by selecting either f or g (i.i.d.) at each iterate. A number of questions arise in this setting:
Conjecturally, for a generic pair of f and g we should be able to answer the above. I will describe one sufficient criteria on f and g underwhich we can give some partial answers to the above questions. Such a criteria is expected to be generic amoung pairs of (volume-preserving) diffeomorphisms and should be able to be verified in a number of naturally occurring geometric settings where the above questions are not fully answered.
Link: https://bluejeans.com/520769740/3630
Parallel server systems have received a lot of attention since their introduction about 20 years ago. They are commonly used to model a situation where different type of jobs can be treated by servers with different specialties like data and call centers. Exact optimal policies are often not tractable for those systems. Instead, part of the literature was focused on finding policies that are asymptotically optimal as the load of the network approaches a value critical for stability (heavy traffic approximations). This is done by obtaining a weak convergence to a brownian control problem that is linked to a non-linear differential equation (Hamilton-Jacobi-Bellman). Asymptotically optimal policies have been analyzed for a long time under a restrictive assumption that is not natural for practical applications. This talk will present recent developments that allow for a more general asymptotic optimality result by focusing on the simplest non-trivial example.
In this talk, we focus on designing computational methods supported by theoretical properties for optimal motion planning and optimal transport (OT).
Over the past decades, motion planning has attracted large amount of attention in robotics applications. Given certain
configurations in the environment, the objective is to find trajectories which move the robot from one position to the other while satisfying given constraints. We introduce a new method to produce smooth and collision-free trajectories for motion planning task. The proposed model leads to short and smooth trajectories with advantages in numerical computation. We design an efficient algorithm which can be generalized to robotics applications with multiple robots.
The idea of optimal transport naturally arises from many application scenarios and provides powerful tools for comparing probability measures in various types. However, obtaining the optimal plan is generally a computationally-expensive task, sometimes even intractable. We start with the entropy transport problem as a relaxed version of original optimal transport problem with soft marginals, and propose an efficient algorithm to obtain the sample approximation for the optimal plan. We also study an inverse problem of OT and present the computational methods for learning the cost function from the given optimal transport plan.
We define a graph process based on a discrete branching process with deletions and mergers, which is inspired by the 4-cycle structure of the hypercube $\mathcal{Q}_d$ for large $d$. We prove survival and extinction under certain conditions on $p$ and $q$ that heuristically match the known expansions of the critical probabilities for bond percolation on the hypercube. Joint work with Laura Eslava and Sarah Penington. Based on https://arxiv.org/abs/2104.04407.
