We introduce a new class of non-compact symplectic manifolds called
Liouville sectors and show they have well-behaved, covariantly functorial
Fukaya categories.  Stein manifolds frequently admit coverings by Liouville
sectors, which can be used to understand the Fukaya category of the total
space (we will study this geometry in examples). Our first main result in
this setup is a local-to-global criterion for generating Fukaya categories.
Our eventual goal is to obtain a combinatorial presentation of the Fukaya
category of any Stein manifold. This is joint work (in progress) with John
Pardon and Vivek Shende.

Variance components (a.k.a. random/mixed effects) models are commonly used to determine genetic variance-covariance matrices of quantitative phenotypic traits in a population. The eigenvalue spectra of such matrices describe the evolutionary response to selection, but may be difficult to estimate from limited samples when the number of traits is large. In this talk, I will discuss the eigenvalues of classical MANOVA estimators of these matrices, including a characterization of the bulk empirical eigenvalue distribution, Tracy-Widom fluctuations at the spectral edges under a ``sphericity'' null hypothesis, the behavior of outlier eigenvalues under spiked alternatives, and a statistical procedure for estimating true population spike eigenvalues from the sample. These results are established using tools of random matrix theory and free probability.

In this presentation, we present a simple high order weighted essentially non- oscillatory (WENO) schemes to solve hyperbolic conservation laws. The main advantages of these schemes presented in the paper are their compactness, robustness and could maintain good convergence property for solving steady state problems. Comparing with the classical WENO schemes by {G.-S. Jiang and C.-W. Shu, J. Comput. Phys., 126 (1996), 202-228}, there are two major advantages of the new WENO schemes. The first, the associated optimal linear weights are inde- pendent on topological structure of meshes, can be any positive numbers with only requirement that their summation equals to one, and the second is that the new scheme is more compact and efficient than the scheme by Jiang and Shu. Extensive numerical results are provided to illustrate the good performance of these new WENO schemes. 

The general average distance problem, introduced by Buttazzo, Oudet, and Stepanov,  asks to find a good way to approximate a high-dimensional object, represented as a measure, by a one-dimensional object. We will discuss two variants of the problem: one where the one-dimensional object is a measure with connected one-dimensional support and one where it is an embedded curve. We will present examples that show that even if the data measure is smooth the nonlocality of the functional can cause the minimizers to have corners. Nevertheless the curvature of the minimizer can be considered as a measure. We will discuss a priori estimates on the total curvature and ways to obtain information on topological complexity of the minimizers. We will furthermore discuss functionals that take the transport along the network into account and model best ways to design transportation networks. (Based on joint works with Xin Yang Lu and Slav Kirov.)

Modern financial portfolio construction uses mean-variance optimisation that requiers the knowledge of a very large covariance matrix. Replacing the unknown covariance matrix by the sample covariance matrix (SCM) leads to disastrous out-of-sample results that can be explained by properties of large SCM understood since Marcenko and Pastur.  A better estimate of the true covariance can be built by studying the eigenvectors of SCM via the average matrix resolvent. This object can be computed using a matrix generalisation of Voiculescu’s addition and multiplication of free matrices.  The original result of Ledoit and Peche on SCM can be generalise to estimate any rotationally invariant matrix corrupted by additive or multiplicative noise. Note that the level of rigor of the seminar will be that of statistical physics.

This is a joint applied math/probability seminar.