Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

Abstract: The Euclidean distance geometry problem arises in a wide variety of applications, from determining molecular conformations in computational chemistry to localization in sensor networks. Instead of directly reconstruct the incomplete distance matrix, we consider a low-rank matrix completion method to reconstruct the associated Gram matrix with respect to a suitable basis. Computationally, simple and fast algorithms are designed to solve the proposed problem. Theoretically, the well known restricted isometry property (RIP) can not be satisfied in the scenario. Instead, a dual basis approach is considered to theoretically analyze the reconstruction problem. Furthermore, by introducing a new condition on the basis called the correlation condition, our theoretical analysis can be also extended to a more general setting to handle low-rank matrix completion problems under any given non-orthogonal basis. This new condition is polynomial time checkable and holds for many cases of deterministic basis where RIP might not hold or is NP-hard to verify. If time permits, I will also discuss a combination of low-rank matrix completion with geometric PDEs on point clouds to understanding manifold-structured data represented as incomplete inter-point distance data.   This talk is based on:1. A. Tasissa, R. Lai, “Low-rank Matrix Completion in a General Non-orthogonal Basis”, arXiv:1812.05786 2018. 2. A. Tasissa, R. Lai, “Exact Reconstruction of Euclidean Distance Geometry Problem Using Low-rank Matrix Completion”, accepted, IEEE. Transaction on Information Theory, 2018. 3. R. Lai, J. Li, “Solving Partial Differential Equations on Manifolds From Incomplete Inter-Point Distance”, SIAM Journal on Scientific Computing, 39(5), pp. 2231-2256, 2017.

We will discuss the regularity of the conjugacy between an Anosov automorphism L of a torus and its small perturbation. We assume that L has no more than two eigenvalues of the same modulus and that L^4 is irreducible over rationals. We consider a volume-preserving C^1-small perturbation f of L. We show that if the Lyapunov exponents of f with respect to the volume are the same as the Lyapunov exponents of L, then f is C^1+ conjugate to L. Further, we establish a similar result for irreducible partially hyperbolic automorphisms with two-dimensional center bundle. This is joint work with Andrey Gogolev and Victoria Sadovskaya

The Georgia Scientific Computing Symposium is a forum for professors, postdocs, graduate students and other researchers in Georgia to meet in an informal setting, to exchange ideas, and to highlight local scientific computing research. The symposium has been held every year since 2009 and is open to the entire research community. This year, the symposium will be held on Saturday, February 16, 2019, at Georgia Institute of Technology. Please see <a href="http://gtmap.gatech.edu/events/2019-georgia-scientific-computing-symposium" title="http://gtmap.gatech.edu/events/2019-georgia-scientific-computing-symposium">http://gtmap.gatech.edu/events/2019-georgia-scientific-computing-symposium</a> for more information

I will present a proof of an abstract Nash-Moser Implicit Function Theorem. This theorem can cope with derivatives which are not boundly invertible from one space to itself. The main technique is to combine Newton steps - which loses derivatives with some smoothing that restores them.