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Wanke Yin, Wuhan University and Rutgers University
Let $M$ be a smooth real hypersurface in $\mathbb C^n$ with $n\geq 2$. For any $p\in M$ and any integer $s\in [1,n-1]$, Bloom in 1981 defined the following three kinds of integral invariants: invariant $a^{(s)}(M,p)$ defined in terms of contact order by complex submanifolds, invariant $t^{(s)}(M,p)$ defined by the iterated Lie bracket of vector fields and invariant $c^{(s)}(M,p)$ defined through the degeneracy of the trace of the Levi form. When $M$ is pseudoconvex, Bloom conjectured that these three invariants are equal. Bloom and Graham gave a complete solution of the conjecture for $s=n-1$. Bloom showed that the conjecture is true for $a^{(1)}(M,p)=c^{(1)}(M,p)$ when $n=3$. In this talk, I will present a recent joint work with Xiaojun Huang, in which we gave a solution of the conjecture for $s=n-2$. In particular, this gave a complete solution of the Bloom conjecture for $n=3$.
Todd Kemp, UCSD
Random Matrix Theory has become one of the hottest fields in probability and applied mathematics. With deep connections to analysis, combinatorics, and even number theory and representation theory, in the age of big data it is also finding its place at the heart of data science.
The field has largely focused on two kinds of generalizations of Gaussian random matrices, either preserving entry-wise independence or preserving rotational invariance. From another point of view, however, the classical Gaussian matrix ensembles can be viewed as Brownian motion on Lie algebras, and this Lie structure goes a long way in explaining some of their known fine structure. This suggests a third, geometric generalization of these ensembles to study: Brownian motion on the corresponding matrix Lie groups.
In this lecture, I will discuss the state of the art in our understanding of the behavior of eigenvalues of Brownian motion on Lie groups, focusing on unitary groups and general linear groups. No specialized background knowledge is required. There will be lots of pictures.
Katherine St. John
City University of New York & American Museum of Natural History
Trees are a canonical structure for representing evolutionary histories. Many popular criteria used to infer optimal trees are computationally hard, and the number of possible tree shapes grows super-exponentially in the number of taxa. The underlying structure of the spaces of trees yields rich insights that can improve the search for optimal trees, both in accuracy and running time, and the analysis and visualization of results. We review the past work on analyzing and comparing trees by their shape as well as recent work that incorporates trees with weighted branch lengths. This talk will highlight some of the elegant questions that arise from improving search and visualizing the results in this highly structured space. All are welcome.
There are no conferences next week.