Given a formally integrable almost complex structure $X$ defined on the closure of a bounded domain $D \subset \mathbb C^n$, and provided that $X$ is sufficiently close to the standard complex structure, the global Newlander-Nirenberg problem asks whether there exists a global diffeomorphism defined on $\overline D$ that transforms $X$ into the standard complex structure, under certain geometric and regularity assumptions on $D$. In this talk I will present my recent result on the ½ estimate for global Newlander-Nirenberg problem on strongly pseudoconvex domains. The main ingredients in our proof are the construction of Moser-type smoothing operators on bounded Lipschitz domains using Littlewood-Paley theory and a convergence scheme of KAM type. 

In various cases, a law that holds in a group with high probability, must actually hold for all elements. For instance, a finite group in which the commutator law [x,y]=1 holds with probability at least 5/8, must be abelian. For infinite groups, one needs to work a bit harder to define the probability that a given law holds. One natural way is by sampling a random element uniformly from the r-ball in the Cayley graph and taking r to infinity; another way is by sampling elements using random walks. It was asked by Amir, Blachar, Gerasimova, and Kozma whether a law that holds with probability 1, must actually hold globally, for all elements. In a recent joint work with Be’eri Greenfeld, we give a negative answer to their question.

In this talk I will give an introduction to probabilistic group laws and present a finitely generated group that satisfies the law x^p=1 with probability 1, but yet admits no group law that holds for all elements.

https://sites.uci.edu/inverse/

Abstract: For over a decade, I have had the privilege to collaborate and learn from internationally acclaimed, Brooklyn-based choreographer Reggie Wilson and his Fist and Heel Performance Group https://www.fistandheelperformancegroup.org/ .  This engagement has given me new practices for teaching and mentoring, widened my sense of where and how mathematical knowledge is held, and surfaced unexplored areas near the heart of mainstream geometry and topology. In this short talk, I'll try to give a quick sense of how this came about, what it's taught me about being a mathematician, and why I think dance and bodies have more to teach us mathematically.