Georgia Institute of Technology College of Sciences

I will first give a brief review on simple and robust central-upwind schemes for hyperbolic conservation laws. I will then discuss their application to the Saint-Venant system of shallow water equations. This can be done in a straightforward manner, but then the resulting scheme may suffer from the lack of balance between the fluxes and (possibly singular) geometric source term, which may lead to a so-called numerical storm, and from appearance of negative values of the water height, which may destroy the entire computed solution. To circumvent these difficulties, we have developed a special technique, which guarantees that the designed second-order central-upwind scheme is both well-balanced and positivity preserving. Finally, I will show how the scheme can be extended to the two-layer shallow water equations and to the Savage-Hutter type model of submarine landslides and generated tsunami waves, which, in addition to the geometric source term, contain nonconservative interlayer exchange terms. It is well-known that such terms, which arise in many different multiphase models, are extremely sensitive to a particular choice their numerical discretization. To circumvent this difficulty, we rewrite the studied systems in a different way so that the nonconservative terms are multiplied by a quantity, which is, in all practically meaningful cases, very small. We then apply the central-upwind scheme to the rewritten system and demonstrate robustness and superb performance of the proposed method on a number numerical examples.

Let F = F_p for any fixed prime p >= 2. An affine-invariant property is a property of functions on F^n that is closed under taking affine transformations of the domain. We prove that all affine-invariant properties having local characterizations are testable. In fact, we show a proximity-oblivious test for any such property P, meaning that there is a test that, given an input function f, makes a constant number of queries to f, always accepts if f satisfies P, and rejects with positive probability if the distance between f and P is nonzero. More generally, we show that any affine-invariant property that is closed under taking restrictions to subspaces and has bounded complexity is testable. We also prove that any property that can be described as the property of decomposing into a known structure of low-degree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degree-d polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degree-structural properties and are therefore locally characterized. Our results depend on a new Gowers inverse theorem by Tao and Ziegler for low characteristic fields that decomposes any polynomial with large Gowers norm into a function of low-degree non-classical polynomials. We establish a new equidistribution result for high rank non-classical polynomials that drives the proofs of both the testability results and the local characterization of degree-structural properties. [Joint work with Eldar Fischer, Hamed Hatami, Pooya Hatami, and Shachar Lovett.]

The systematical study of model selection procedures, especially since the early nineties, has led to the design of penalties that often allow to achieve minimax rates of convergence and adaptivity for the selected model, in the general setting of risk minimization (Koltchinskii [3], Massart [4]). However, the proposed penalties often su.er form their dependencies on unknown or unrealistic constants. As a matter of fact, under-penalization has generally disastrous e.ects in terms of e¢ ciency. Indeed, the model selection procedure then looses any bias-variance trade-o. and so, tends to select one of the biggest models in the collection. Birgé and Massart ([2]) proposed quite recently a method that empirically adjusts the level of penalization in a linear Gaussian setting. This method of calibration is called "slope heuristics" by the authors, and is proved to be optimal in their setting. It is based on the existence of a minimal penalty, which is shown to be half the optimal one. Arlot and Massart ([1]) have then extended the slope heuristics to the more general framework of empirical risk minimization. They succeeded in proving the optimality of the method in heteroscedastic least-squares regression, a case where the ideal penalty is no longer linear in the dimension of the models, not even a function of it. However, they restricted their analysis to histograms for technical reasons. They conjectured a wide range of applicability for the method. We will present some results that prove the validity of the slope heuristics in heteroscedastic least-squares regression for more general linear models than histograms. The models considered here are equipped with a localized orthonormal basis, among other things. We show that some piecewise polynomials and Haar expansions satisfy the prescribed conditions. We will insist on the analysis when the model is .xed. In particular, we will focus on deviations bounds for the true and empirical excess risks of the estimator. Empirical process theory and concentration inequalities are central tools here, and the results at a .xed model may be of independent interest.

A well known result of Giroux tells us that isotopy classes of contact structures on a closed three manifold are in one to one correspondence with stabilization classes of open book decompositions of the manifold. We will introduce a stabilization-invariant property of open books which corresponds to tightness of the corresponding contact structure. We will mention applications to the classification of contact 3-folds, and also to the question of whether tightness is preserved under Legendrian surgery.