Title: Free boundary problems
Description: In broad terms, free boundaries appear when the solution to a problem consists of a pair: a function (often satisfying a partial differential equation), and a set associated to this function. Two central issues in the study of free boundary problems are:
What is the optimal regularity of the solution?
How smooth is the free boundary?
We will learn tools, such as monotonicity formulas, that will allow us to investigate these questions.
Lecture notes from Mariana's mini course are available here.
Title: Quantitative unique continuation properties
Description: Unique continuation property is a fundamental property for harmonic functions, as well as solutions to a large class of elliptic and parabolic PDEs. It says that if a harmonic function vanishes at a point to infinite order, it must vanish everywhere. In the same spirit, we are interested in quantitative unique continuation problems, where we use the local growth rate of a harmonic function to deduce some global estimates, such as estimating the size of its zero set or critical set. In this mini-course, I will introduce some modern tools and recent developments in the study of quantitative unique continuation problems.
Abstract: In this talk, we will discuss the long-time behavior of large solutions to nonlinear wave equations in the energy-supercritical setting. We will review the concentration compactness and rigidity arguments through the seminal works of Kenig-Merle and Duyckaerts-Kenig-Merle for the energy-critical and energy-supercritical problems in three dimensions. We will then discuss the generalization of these methods to higher odd dimensions and even dimensions.
Abstract: Submanifold geometry studies the geometry of manifolds embedded in higher-dimensional ambient manifolds. One of the key equations in the theory of submanifolds is the Gauss formula for the Levi-Civita connection. The formula decomposes the covariant derivative into tangential and normal components, and it relates the intrinsic connection to the extrinsic connection. In this talk, we show how the formula can be extended to the different types of Laplacians for vector fields, and we discuss applications to fluid mechanics.
Title: Weak-type inequalities for sparse operators via Bellman functions
Abstract: I’ll discuss recent joint work with Guillermo Rey and Kristina Skreb, where we find the exact Bellman function governing a certain weak-type inequality for sparse operators. We work in the dyadic setting, where sparse operators have become a standard tool in the last few years - largely due to the sharp results one obtains from strong-type bounds. Several open problems involve weak-type bounds, which are much more difficult to sharpen. We explore this aspect through the Bellman function method.
Title: Φ-admissible weights and degenerate elliptic equations
Abstract: The notion of a p-admissible weight was first introduced by Fabes, Kenig, and Serapioni in 1982, and it captures the main properties of a weight needed for implementation of Moser’s iterative scheme. As a result one obtains regularity of weak solutions to degenerate elliptic equations, where the degeneracy is controlled by such a weight. One of the main defining properties of p-admissible weights is a (pσ,p) Sobolev inequality with σ > 1, which in particular implies that these weights are doubling.
In this talk I will introduce a generalization of 2-admissible weights, the class of Φ-admissible weights. A weight is Φ-admissible if a (Φ,2) weighted Orlicz-Sobolev inequality holds. Such a weight might not be 2-admissible, and even non-doubling, therefore classical iterative schemes of Moser and DeGiorgi become unavailable. However, a modified DeGiorgi iteration can be performed and meaningful regularity theory developed for degeneracies controlled by certain Φ-admissible weights, thus making it an interesting class to explore and investigate further.
Abstract: We define a measurable subset of the real line as "thick" if the measure of the intersection of this set with any interval of length one is bounded from below. The classical theorem of Logvinenko and Sereda states that if the Fourier transform of a function is supported in some ball, then the function itself can be sampled from a thick set. In this talk, we will consider Schrödinger operators with increasing potentials and functions with bounded spectrum in the corresponding space. The spectral inequalities provide estimates for sampling functions with bounded spectrum from (relatively) thick sets. We will give an overview of some resent results and describe an application of the spectral inequalities to controllability theory. The talk is based on a joint work with Jiuyi Zhu.
Title: The Structure of Measure for which Symmetrization is Perimeter Non-increasing
Abstract: For a symmetrization procedure to be useful, the key property is that the measure of the perimeter of a set be non-increasing under the given symmetrization. Roughly speaking, given a symmetrization procedure with this property, one can use it to study associated geometric inequalities (e.g., Isoperimetric Inequalities, Sobolev Inequalities, etc.), sharp bounds on eigenvalues for elliptic problems (e.g. Faber-Krahn inequalities), and sharp a priori estimates on the Dirichlet problem for elliptic operators (Talenti-type inequalities). This gives rise to the natural question: "what geometric properties must a measure have in order to support a useful symmetrization procedure?" Despite intense interest in symmetrization procedures over the last 140 years, few general answers to this problem exist within the literature. This talk will focus upon new results answering this question in higher dimensions, joint with Kuan-Ting Yeh (Purdue University). There will be many pictures. If time permits, we will also discuss several related open problems.
Title: A general prescribed projection theorem for transversal projections
Abstract: A remarkable theorem of Davies says that if one is given a set A_θ for each angle θ, one can construct a set E which has projection A_θ in direction θ up to an error of measure zero. In joint work with Chang and Taylor, this result was generalized to a particular class of non-linear projections motivated by the problem of covering sets efficiently by translates of a given plane curve. In this talk, I will discuss that result as well as ongoing work (joint with Chang, Fraser, Mayer, and Taylor) to generalize this further to projections satisfying a simple transversality condition, as well as some geometric implications of this generalization.
Abstract: The existence of global solutions for the nonlinear Schrödinger equation (NLS) has been extensively studied. The problem becomes more complicated as we consider a general power-type nonlinearity of degree p. So far, global well-posedness for small data is known for p strictly greater than the Strauss exponent, as the dispersive effect becomes weaker for smaller p. In dimension 3, this Strauss exponent is 2, making 3D NLS with quadratic nonlinearity particularly interesting.
In this talk, I will present a result that shows the global existence and scattering for systems of quadratic NLS for small, localized initial data. We will review the space-time resonance method developed in previous works, which have all required 0-dimensional space-time resonance set. In applying the space-time resonance method to a general system of quadratic NLS, there are two cases when the space-time resonance sets are 3-dimensional, which cannot be handled by existing method. I will explain how we fully resolved one case and tackled the other case, which arises from the u¯u-type nonlinearity, by introducing an “ϵ”regularization to the lower frequency part of the nonlinear term.
Abstract: A vibrant and classic area of research is that of relating the size of a set to the finite point configurations that it contains. Here, size may refer to cardinality, dimension, or measure. It is a consequence of the Lebesgue density theorem, for instance, that sets of positive measure in R^n contain a similar copy (and all sufficiently small scalings) of any given finite point configuration. In another direction, a seminal result of Szemer´edi demonstrates the existence of arithmetic progressions in subsets of N with positive upper density.
In the fractal setting, there is a rich literature on finite point configurations, and some notions of size are more appropriate for certain settings. For example, it is known that full Hausdorff dimension is not enough to guarantee the existence of a 3-term arithmetic progression in subsets of R^n. Even full Hausdorff dimension and maximal Fourier dimension need not suffice. We will see, however, that sets of sufficient Newhouse thickness are guaranteed to contain arithmetic progressions. In another more topological direction, we also see that any bounded countable point set, such as an arithmetic progression of arbitrary length, can be realized in any second category Baire set. Connections will be drawn to a topological variant of the Erdös
similarity conjecture.
Abstract: In this talk, we will discuss Falconer's distance set conjecture in the case of polyhedral and continuously differentiable norms. Using the analogy of distances to projections, one can obtain sharp lower bounds for distance sets for general norms. This is joint work with Iqra Altaf and Ryan Bushling.