In November 2023, Clarivate Plc announced that it had excluded the entire field of mathematics from the most recent edition of its influential list of authors of highly cited papers because of massive citation manipulation, which in return influences the so-called “Shanghai ranking” of top universities (or those claiming to be top). While most mathematicians would probably not care, the exclusion is in fact the tip of the iceberg of a parallel universe of predatory and mega-journals whose main purpose is to offer publishing opportunities for whoever is willing to pay the right price. I will explain how the system works, why we should care, and what measures we can all take against. In preparation, I invite you to think about the following questions: How often have you been contacted in the past months to attend a conference not in your field / submit a paper to  or edit a special issue in a journal you don’t know / review an article within 10 days or so? What do you know about the following journals: “Mathematics”, “Axioms” (published by MDPI), “Chaos, solitons, fractals” (Elsevier), “Journal of Difference Equations” (Springer)? Do you know the following mathematicians: Abdon Atangana, Dumitru Baleanu, Hari M. Srivastava? 

This talk is related to my work as  Chair of the Committee on Electronic Information and Communication (= publishing) of the International Mathematical Union.

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.