The growth of an infinite-dimensional algebra is a fundamental tool to measure its "size." The growth of noncommutative algebras plays an important role in noncommutative geometry, representation theory, differential algebraic geometry, symbolic dynamics, homological stability results, and more. 

We analyze the space of growth functions of algebras, answering a question of Zelmanov on the existence of certain 'holes' in this space, and provide evidence for the ampleness of the possible growth rates of algebras with prescribed properties; we conclude a strong quantitative solution of the Kurosh Problem on algebraic algebras.

Utilizing new layers of the interplay between noncommutative algebra and symbolic dynamics, we exhibit surprising pathologies in the prime spectrum and tensor product structure of algebras with polynomial growth, thereby providing counterexamples to questions of Bergman, Krause, Lenagan, and others; applying our methods to algebras of faster growth types, we resolve a conjecture of Bartholdi on amenable representations in exponential growth.

Finally, the largest objects (groups, algebras, Lie algebras) are, in many contexts, those containing free substructures. We discuss the coexistence of this phenomenon with finiteness properties -- in particular, "almost algebraicity" of algebras and "almost periodicity" of groups -- from algebraic, geometric, and probabilistic perspectives.

This talk is partially based on joint works with Bell, Goffer, and Zelmanov.

In topology, the difference between the category of smooth manifolds and the category of topological manifolds has always been a delicate and intriguing problem, called the "exotic phenomena". The recent work of Watanabe (2018) uses the tool "Kontsevich's invariants" to show that the group of diffeomorphisms of the 4-dimensional ball, as a topological group, has non-trivial homotopy type. In contrast, the group of homeomorphisms of the 4-dimensional ball is contractible. Kontsevich's invariants, defined by Kontsevich in the early 1990s from perturbative Chern-Simons theory, are invariants of (certain) 3-manifolds / fiber bundles / knots and links (it is the same argument in different settings). Watanabe's work implies that these invariants detect exotic phenomena, and, since then, they have become an important tool in studying the topology of diffeomorphism groups. It is thus natural to ask: how to understand the role smooth structure plays in Kontsevich's invariants? My recent work provides a perspective on this question: the real blow-up operation essentially depends on the smooth structure, therefore, given a manifold / fiber bundle X, the topology of some manifolds / bundles obtained by doing some real blow-ups on X can be different for different smooth structures on X

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.