Functional time series has become a recent focus of statistical research.
In this talk, we will discuss the applications of kernel methods in the analysis of nonparametric functional time series. In the first half, we propose the kernel estimates for the autoregressor in a nonparametric functional autoregression model. It consistency is proved and a valid bootstrap procedure is provided to construct the prediction regions. In the second half of the talk, we propose a class of estimators for the spectral density kernel, which is a key element encapsulates the second-order dynamics of a functional time series. The new class of estimators employs the inverse Fourier transform of a flat-top function as the weight function employed to smooth the periodogram. It is shown that using a flat-top kernel yields a bias reduction and results in a higher-order accuracy in terms of optimizing the integrated mean square error (IMSE).

In 1990 Lakshmibai and Sandyha proved a remarkable result which provides a purely combinatorial method of determining whether or not a Schubert variety is smooth. In this talk we will start by examining the combinatorial tools needed for this theorem, namely pattern containment and avoidance in permutations. We then move to the land of algebraic geometry, starting with a brief description of varieties and singular points on varieties. Finally we will construct Schubert varieties as special subsets of the complex full flag manifold and state without proof the Lakshmibai-Sandyha Theorem. In doing so we hope to show that the intersection of combinatorics and algebraic geometry is nonempty (although maybe it is only an epsilon neighborhood).

The stochastic inverse problem for determining parameter values in a physics model from observational data on the output of the model forms the core of scientific inference and engineering design. We describe a recently developed formulation and solution method for stochastic inverse problems that is based on measure theory and a generalization of a contour map. In addition to a complete analytic and numerical theory, advantages of this approach include avoiding the introduction of ad hoc statistics models, unverifiable assumptions, and alterations of the model like regularization. We present a high-dimensional application to determination of parameter fields in storm surge models. We conclude with recent work on defining a notion of condition for stochastic inverse problems and the use in designing sets of optimal observable quantities.

How does one measure area? As an example, how can one determine the area of a region on a map for the purpose of real estate appraisal? Wouldn't it be great if there were an instrument that would measure the area of a region by simply tracing its boundary? It turns out that there is such an instrument: it is called a planimeter. In this talk we will discuss a particular type of planimeter called the compensating polar planimeter. There will be a little bit of history and some analysis involving line integrals and Green's theorem. Finally, there will be a chance to see and touch actual examples of these fascinating instruments from the speaker's collection.

Free probability was introduced in the 1980’s by Voiculescu, with the aim of
studying von Neumann algebras by viewing them as non-commutative probability
spaces, and this analogy has proved quite powerful in operator algebra theory. In the
1990’s, Speicher was able to describe free independence using cumulants constructed
from the lattice of non-crossing partitions. In this talk we will give an introduction
to free probability and outline the role the non-crossing cumulants have played
in describing the theory. Time permitting we will also demonstrate some more
recent applications of combinatorics to free probability, such as in describing bi-free
probability, type B free probability, and boolean independence.

Massive data are often contaminated by outliers and heavy-tailed errors. In the presence of heavy-tailed data, finite sample properties of the least squares-based methods, typified by the sample mean, are suboptimal both theoretically and empirically. To address this challenge, we propose the adaptive Huber regression for robust estimation and inference. The key observation is that the robustification parameter should adapt to sample size, dimension and moments for optimal tradeoff between bias and robustness. For heavy-tailed data with bounded $(1+\delta)$-th moment for some $\delta>0$, we establish a sharp phase transition for robust estimation of regression parameters in both finite dimensional and high dimensional settings: when $\delta \geq 1$, the estimator achieves sub-Gaussian rate of convergence without sub-Gaussian assumptions, while only a slower rate is available in the regime $0<\delta <1$ and the transition is smooth and optimal.

In addition, non-asymptotic Bahadur representation and Wilks’ expansion for finite sample inference are derived when higher moments exist. Based on these results, we make a further step on developing uncertainty quantification methodologies, including the construction of confidence sets and large-scale simultaneous hypothesis testing. We demonstrate that the adaptive Huber regression, combined with the multiplier bootstrap procedure, provides a useful robust alternative to the method of least squares. Together, the theoretical and empirical results reveal the effectiveness of the proposed method, and highlight the importance of having statistical methods that are robust to violations of the assumptions underlying their use.

Models of information exchange that originate from economics provide interesting questions in probability. We will introduce some of these models, discuss open questions, and explain some recent results.
Joint with Wade Hann-Caruthers, Matan Harel, Vadim Martynov, Elchanan Mossel and Philipp Strack

We will discuss some geometric techniques used in proving ''arithmetic statistics'' results,
primarily using the case of Selmer groups for families of elliptic curves as a motivating example.

We consider the problem of computing a cluster of eigenvalues, and its associated eigenspace, of a (possibly unbounded) selfadjoint operator in a Hilbert space. A rational function of the operator is constructed such that the eigenspace of interest is its dominant eigenspace, and a subspace iteration procedure is used to approximate this eigenspace. The computed space is then used to obtain approximations of the eigenvalues of interest. An eigenvalue and eigenspace convergence analysis that considers both iteration error and discretization error is provided. A realization of the proposed approach for a model second-order elliptic operator is based on a discontinuous Petrov-Galerkin discretization of the resolvent, and a variety of numerical experiments illustrate its performance.

In recent years, there has been a great deal of excitement about 'big data' and about the new research problems posed by a world of vastly enlarged datasets.

In response, the field of Mathematical Statistics increasingly studies problems where the number of variables measured is comparable to or even larger than the number of observations. Numerous fascinating mathematical phenomena arise in this regime; and in particular theorists discovered that the traditional approach to covariance estimation needs to be completely rethought, by appropriately shrinking the eigenvalues of the empirical covariance matrix.

This talk briefly reviews advances by researchers in random matrix theory who in recent years solved completely the properties of eigenvalues and eigenvectors under the so-called spiked covariance model. By applying these results it is now possible to obtain the exact optimal nonlinear shrinkage of eigenvalues for certain specific measures of performance, as has been shown in the case of Frobenius loss by Nobel and Shabalin, and for many other performance measures by Donoho, Gavish, and Johnstone. We describe these results as well as results of the author and Behrooz Ghorbani on optimal shrinkage for Multi-User Covariance estimation and Multi-Task Discriminant Analysis.