Many equations that model fluid behavior are derived from systems that encompass multiple physical forces. When the equations are written in non dimensional form appropriate to the physics of the situation, the resulting PDEs often involve multiple non-dimensional parameters. Frequently some of these parameters are very small and they enter into the analysis in different ways. We will discuss one such system which has been proposed as a model for magnetostrophic turbulence and describe results that can be obtained in several different small parameter limits. In this talk we will concentrate on a forced drift-diffusion equation for the temperature where the fluid viscosity enters via the drift velocity. We examine the convergence of solutions in the limit as the viscosity goes to zero. We introduce a natural notion of ”vanishing viscosity” weak solutions and prove the existence of a compact global attractor for the critical drift-diffusion equation. This is joint work with Anthony Suen.

We revisit two research programs that were proposed in the 1960's, remained largely dormant for five decades, and then become hot areas of research in the last decade. The monograph ``Continuous Model Theory'' by Chang and Keisler, Annals of Mathematics Studies (1966), studied structures with truth values in [0,1], with formulas that had continuous functions as connectives, sup and inf as quantifiers, and equality. In 2008, Ben Yaacov, Bernstein, Henson, and Usvyatsev introduced the model theory of metric structures, where equality is replaced by a metric, and all functions and predicates are required to be uniformly continuous. This has led to an explosion of research with results that closely parallel first order model theory, with many applications to analysis. In my forthcoming paper ``Model Theory for Real-valued Structures'', the "Expansion Theorem" allows one to extend many model-theoretic results about metric structures to general [0,1]-valued structures--the structures in the 1966 monograph but without equality. My paper ``Ultrapowers Which are Not Saturated'', J. Symbolic Logic 32 (1967), 23-46, introduced a pre-ordering $\mathcal M\trianglelefteq\mathcal N$ on all first-order structures, that holds if every regular ultrafilter that saturates $\mathcal N$ saturates $\mathcal M$, and suggested using it to classify structures. In the last decade, in a remarkable series of papers, Malliaris and Shelah showed that that pre-ordering gives a rich classification of simple first-order structures. Here, we lay the groundwork for using the analogous pre-ordering to classify [0,1]-valued and metric structures.

We proposes a model-free and data-adaptive feature screening method for ultra-high dimensional data. The proposed method is based on the projection correlation which measures the dependence between two random vectors. This projection correlation based method does not require specifying a regression model, and applies to data in the presence of heavy tails and multivariate responses. It enjoys both sure screening and rank consistency properties under weak assumptions. A two-step approach, with the help of knockoff features, is advocated to specify the threshold for feature screening such that the false discovery rate (FDR) is controlled under a pre-specified level. The proposed two-step approach enjoys both sure screening and FDR control simultaneously if the pre-specified FDR level is greater or equal to 1/s, where s is the number of active features. The superior empirical performance of the proposed method is illustrated by simulation examples and real data applications.

A pointwise ergodic theorem for the action of a transformation $T$ on a probability space equates the global property of ergodicity of the transformation to its pointwise combinatorics. Our main result is a backward (in the direction of $T^{-1}$) ergodic theorem for countable-to-one probability measure preserving (pmp) transformations $T$. We discuss various examples of such transformations, including the shift map on Markov chains, which yields a new (forward) pointwise ergodic theorem for pmp actions of finitely generated countable groups, as well as one for the (non-pmp) actions of free groups on their boundary. This is joint work with Anush Tserunyan.

I will discuss an active project in computing the top weight cohomology of the moduli space $A_g$ of principally polarized abelian varieties of dimension $g$ for small values of $g$. This piece of the cohomology is controlled by the combinatorics of the boundary strata of a compactification of $A_g$. Thus, it can be computed combinatorially. This is joint work with Juliette Bruce, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.

A great number of multivariate statistical methods, such as principal component analysis, discriminant analysis, canonical correlation analysis and graphical lasso to name a few, require the estimate of covariance or correlation matrix of variables as one of the inputs. It is typical to use Pearson sample correlation matrix, which works well at capturing dependencies between normally distributed variables. In this work we consider the problem of estimating dependencies between zero-inflated measurements, which arise in miRNA data, microbiome data, physical activity data, etc. We propose truncated latent Gaussian copula to model the data with excess zeroes, which allows us to derive a rank-based estimator of latent correlation matrix without the estimation of marginal transformation functions. The new methodology is applied for the analysis of associations between gene expression and microRNA data of breast cancer patients, and for inferring the conditional independence graph in quantitate gut microbiome data.

TBA