Georgia Institute of Technology College of Sciences

As the term "big data'' appears more and more frequently in our daily life and research activities, it changes our knowledge of how large the scale of the data can be and challenges the application of numerical analysis for performing statistical calculations. In this talk, I will focus on two basic statistics problems sampling a multivariate normal distribution and maximum likelihood estimation and illustrate the scalability issue that dense numerical linear algebra techniques are facing. The large-scale challenge motivates us to develop scalable methods for dense matrices, commonly seen in statistical analysis. I will present several recent developments on the computations of matrix functions and on the solution of a linear system of equations, where the matrices therein are large-scale, fully dense, but structured. The driving ideas of these developments are the exploration of the structures and the use of fast matrix-vector multiplications to reduce the general quadratic cost in storage and cubic cost in computation. "Big data'' offers a fresh opportunity for numerical analysts to develop algorithms with a central goal of scalability in mind. It also brings in a new stream of requests to high performance computing for highly parallel codes accompanied with the development of numerical algorithms. Scalable and parallelizable methods are key for convincing statisticians and practitioners to apply the powerful statistical theories on large-scale data that they currently feel uncomfortable to handle.

The Weak Structure Theorem of Robertson and Seymour is the cornerstone of many of the algorithmic applications of graph minors techniques. The theorem states that any graph which has both large tree-width and excludes a fixed size clique minor contains a large, nearly planar subgraph. In this talk, we will discuss a new proof of this result which is significantly simpler than the original proof of Robertson and Seymour. As a testament to the simplicity of the proof, one can extract explicit constants to the bounds given in the theorem. We will assume no previous knowledge about graph minors or tree-width. This is joint work with Ken Kawarabayashi and Robin Thomas

I'll introduce the Hilbert transform in a natural way justifying it as a canonical operation. In fact, it is such a basic operation, that it arises naturally in a range of settings, with the important complication that the measure spaces need not be Lebesge, but rather a pair of potentially exotic measures. Does the Hilbert transform map L^2 of one measure into L^2 of the other? The full characterization has only just been found. I'll illustrate the difficulties with a charming example using uniform measure on the standard 1/3 Cantor set.

We will discuss amenability of the topological full group of a minimal Cantor system. Together with the results of H. Matui this provides examples of finitely generated simple amenable groups. Joint with N. Monod.