Georgia Institute of Technology College of Sciences

In this talk, I will refine the concept of the symmetry group of a geometric object through its symmetry groupoid, which incorporates both global and local symmetries in a common framework. The symmetry groupoid is related to the weighted differential invariant signature of a submanifold, that is introduced to capture its fine grain equivalence and symmetry properties. The groupoid/signature approach will be connected to recent developments in signature-based recognition and symmetry detection of objects in digital images, including jigsaw puzzle assembly.

Many real-world problems reduce to optimization problems that are solved by iterative methods. In this talk, I focus on recently developed efficient algorithms for solving large-scale optimization problems that arises in medical imaging and image processing. In the first part of my talk, I will introduce the Bregman Operator Splitting with Variable Stepsize (BOSVS) algorithm for solving nonsmooth inverse problems. The proposed algorithm is designed to handle applications where the matrix in the fidelity term is large, dense, and ill-conditioned. Numerical results are provided using test problems from parallel magnetic resonance imaging. In the second part, I will focus on the Euler's Elastica-based model which is non-smooth and non-convex, and involves high-order derivatives. I introduce two efficient alternating minimization methods based on operator splitting and alternating direction method of multipliers, where subproblems can be solved efficiently by Fourier transforms and shrinkage operators. I present the analytical properties of each algorithm, as well as several numerical experiments on image inpainting problems, including comparison with some existing state-of-art methods to show the efficiency and the effectiveness of the proposed methods.

In this talk I am going to present a way to study numerically the complex domains of invariant Tori for the dissipative standar map. The numerical approach is based on a Nash Moser method. This is work in progress jointly with R. Calleja.

The Jacobian (or sandpile group) of a graph is a well-studied group associated with the graph, known to biject with the set of spanning trees of the graph via a number of classical combinatorial mappings. The algebraic definition of a Jacobian extends to regular matroids, but without the notion of vertices, many such combinatorial bijections fail to generalize. In this talk, I will discuss how orientations provide a way to overcome such obstacle. We give a novel, effectively computable bijection scheme between the Jacobian and the set of bases of a regular matroid, in which polyhedral geometry plays an important role; along the way we also obtain new enumerative results related to the Tutte polynomial. This is joint work with Spencer Backman and Matt Baker.