We present a new, recent doubling method for establishing a priori estimates, then classical solvability and regularity for solutions to fully nonlinear PDEs.  The method produces the missing estimates for the quadratic Hessian and prescribed hypersurface scalar curvature PDEs in dimension four.  It also gives new proofs of the estimates and regularity for Monge–Ampère and special Lagrangian equations and provides prospects for classical solvability of alternative Dirichlet problems.

In the first part of this talk, I will discuss sparsity in data science by looking at two applications. The first in machine learning, where I will briefly introduce supervised classification and neural networks. I will then show how sparsity can be used to prune and compress these networks while retaining or even improving accuracy. The second application involves image segmentation. Here, I will show the role of sparsity in capturing boundaries of salient objects in an image. In the second part of this talk, I will discuss my experience with mentoring undergraduates in research by looking at two case studies. One in segmentation, and the other, in data clustering. Finally, I'll talk a bit about some of my service contributions and how they tie into student affairs and teaching.

 I will take you on a journey through my teaching, teaching-related, and other scholarly activities. A special focus will be given to my redesign and implementation of Precalculus courses at Cal State LA, where I am the first point of contact of 3000 students every academic year.  The success of this implementation is historic as the pass rate of all the sections was over 87%, 40 percentage points higher than the one before the implementation. My model was borrowed by many other departments at Cal State LA and other campuses of the Cal State system. Two research papers about the best teaching, pedagogical, didactical, and assessment practices that led to this accomplishment, were already published. This implementation also had an important impact on the retention rates at Cal State LA.

The final destination of our journey is my research work on the new generation of Bose-Einstein condensates. With some collaborators, we recently obtained interesting results on a new class of PDEs and opened the door to other contributions in this new field.

Binary classification with linear classifiers is a fundamental problem in machine learning with broad applications. Traditionally, this problem focuses on using labeled data to learn a linear classifier, but data is often expensive to label or difficult to acquire. Instead of labeling all data points in a data set, we consider labeling a specially chosen subset of points and ask how well we can accomplish our learning task. To answer this question, we will reformulate this problem into the context of Gaussian spherical tessellations and study geometric properties of such tessellations. This work is joint with Rayan Saab and Eric Lybrand.

Topological and geometric data analysis (TGDA) is a powerful framework for quantitative description and simplification of datasets & shapes. It is especially suitable for modern biological data that are intrinsically complex and high-dimensional. Traditional topological data analysis considers the geometric features of a dataset, while in practice, there could be both geometric and non-geometric features. In this talk, I will introduce a persistent cohomology based method, enriched barcode to embed the non-geometric features in the topological invariants. I will then talk about a geometric method, unnormalized optimal transport for integrating heterogeneous datasets which is crucial for generating a comprehensive topological perspective for the system of interest. Scientific data often have limited size and high complexity, and a straightforward application of machine learning to raw data could result in suboptimal performances. To tackle this challenge, we integrate the TGDA method designed for biological data with deep learning. This topology-based deep learning strategy achieves top performance on standard benchmarks and D3R Grand Challenges, a worldwide competition series in computer-aided drug design. I will also show several applications of our geometric method to the analysis and integration of single-cell omics data. Finally, I will discuss future directions on data-driven modeling using topology, geometry, and machine learning-based approaches.