Georgia Institute of Technology College of Sciences

The curve complex of Harvey allows combinatorial representation of a surface mappingclass group by describing its action on simple closed curves. Similar complexes of spheres,free factors, and free splittings allow combinatorial representation of the automorphisms ofa free group. We consider a Birman exact sequence for combinatorial models of mappingclass groups and free group automorphisms. We apply this and other extension techniquesto compute the automorphism groups of several simplicial complexes associated with map-ping class groups and automorphisms of free groups.

The first part, consists on a result in the area of commutators. The classic result by Coifman, Rochber and Weiss, stablishes a relation between a BMO function, and the commutator of such a function with the Hilbert transform. The result obtained for this thesis, is in the two parameters setting (with obvious generalizations to more than two parameters) in the case where the BMO function is matrix valued. The second part of the thesis corresponds to domination of operators by using a special class called sparse operators. These operators are positive and highly localized, and therefore, allows for a very efficient way of proving weighted and unweighted estimates. Three main results in this area will be presented: The first one, is a sparse version of the celebrated $T1$ theorem of David and Journé: under some conditions on the action of a Calderón-Zygmund operator $T$ over the indicator function of a cube, we have sparse control.. The second result, is an application of the sparse techniques to dominate a discrete oscillatory version of the Hilbert transform with a quadratic phase, for which the notion of sparse operator has to be extended to functions on the integers. The last resuilt, proves that the Bochner-Riesz multipliers satisfy a range of sparse bounds, we work with the ’single scale’ version of the Bochner-Riesz Conjecture directly, and use the ‘optimal’ unweighted estimates to derive the sparse bounds.

Constrained low rank approximation is a general framework for data analysis, which usually has the advantage of being simple, fast, scalable and domain general. One of the most known constrained low rank approximation method is nonnegative matrix factorization (NMF). This research studies the design and implementation of several variants of NMF for text, graph and hybrid data analytics. It will address challenges including solving new data analytics problems and improving the scalability of existing NMF algorithms. There are two major types of matrix representation of data: feature-data matrix and similarity matrix. Previous work showed successful application of standard NMF for feature-data matrix to areas such as text mining and image analysis, and Symmetric NMF (SymNMF) for similarity matrix to areas such as graph clustering and community detection. In this work, a divide-and-conquer strategy is applied to both methods to improve their time complexity from cubic growth with respect to the reduced low rank to linear growth, resulting in DC-NMF and HierSymNMF2 method. Extensive experiments on large scale real world data shows improved performance of these two methods.Furthermore, in this work NMF and SymNMF are combined into one formulation called JointNMF, to analyze hybrid data that contains both text content and connection structure information. Typical hybrid data where JointNMF can be applied includes paper/patent data where there are citation connections among content and email data where the sender/receipts relation is represented by a hypergraph and the email content is associated with hypergraph edges. An additional capability of the JointNMF is prediction of unknown network information which is illustrated using several real world problems such as citation recommendations of papers and activity/leader detection in organizations.The dissertation also includes brief discussions of relationship among different variants of NMF.

This dissertation concerns isoperimetric and functional inequalities in discrete spaces. The majority of the work concerns discrete notions of curvature. There isalso discussion of volume growth in graphs and of expansion in hypergraphs. [The dissertation committee consists of Profs. J. Romberg (ECE), P. Tetali (chair of the committee), W.T. Trotter, X. Yu and H. Zhou.]