Zoom link. https://umn.zoom.us/j/97659808242

Title. Discreteness of energy-minimizing measures.

Abstract. Click for pdf.

Zoom link. https://umn.zoom.us/j/93003755201

Title. Another counterexample to Zygmund's conjecture

Abstract. We present a simple dyadic construction that yields a new counterexample to Zygmund's conjecture. Our result recovers Soria's classical result in dimension three, through a different construction, and gives new ones in all other dimensions d>3.

Zoom link. https://umn.zoom.us/j/99146331309

Title. Rectifiability of Pointwise Doubling Measures in Hilbert Space

Abstract. Jones’ beta numbers measure the flatness of a set at various scales and windows. Since their introduction, beta numbers have served as an important tool to relate the geometric structure of sets and measures and to measure-theoretic quantities. We will extend results of Badger and Schul to show that an $L^2$ variant of the beta numbers can be used to characterize rectifiable pointwise doubling measures in Hilbert space. We will also discuss results for the related notions of graph rectifiability and fractional rectifiability.

Zoom link. https://umn.zoom.us/j/98202820964

Title. Random dyadic grids and what they can do for you.

Abstract. In the last decade dyadic analysis and probabilistic methods have been successfully used to obtain optimal weighted estimates. In this talk we introduce dyadic grids and their random shifted analogue. We discuss in an informal way some of the advantages of this tool and how it can be used to decompose your operator.

Zoom link. https://umn.zoom.us/j/97667237636

Title. Solvability of boundary value problems for the Schrodinger equation with nonnegative potentials

Abstract. Click for abstract

Zoom link. https://umn.zoom.us/j/92024464321

Title. Symplectic non-squeezing and Hamiltonian PDE

Abstract. In this talk we shall discuss infinite dimensional generalization of Gromov's sympelctic nonsqueezing result. As an application we will present mass subcritical and critical nonlinear Schrodinger equation. Nonsqueezing property of the said flows was already known, however the techniques used are based on finite dimensional Gromov's result, while ours presents a more natural way of looking at the Hamiltonian structure of the equations. Additionally, we shall remark on future projects in terms of application of the non-squeezing property. Joint work with Herbert Koch.

Zoom link. https://umn.zoom.us/j/93102186368

Title. Green's function for second order elliptic equations with singular lower order coefficients and applications

Abstract. We will discuss Green's function for second order elliptic operators of the form $\mathcal{L}u=-\text{div}(A\nabla u+bu)+c\nabla u+du$ in domains $\Omega\subseteq\mathbb R^n$, for $n\geq 3$. We will assume that $A$ is elliptic and bounded, and also that $d\geq\text{div}b$ or $d\geq\text{div}c$ in the sense of distributions.

In the setting of Lorentz spaces, we will explain why the assumption $b-c\in L^{n,1}(\Omega)$ is optimal in order to obtain a pointwise bound of the form $G(x,y)\leq C|x-y|^{2-n}$. Under the assumption $d\geq\text{div}b$, we will also discuss why this assumption is necessary to even have weak type bounds on Green's function. Finally, for the case $d\geq\text{div}c$, we will deduce a maximum principle and a Moser type estimate, showing again that the assumption $b-c\in L^{n,1}(\Omega)$ is optimal.

Our estimates will be scale invariant and no regularity on $\partial\Omega$ will be imposed. In addition, $\mathcal{L}$ will not

be assumed to be coercive, and there will be no smallness assumption on the lower order coefficients.

Zoom link. https://umn.zoom.us/j/91828133271

Title. Cones, rectifiability and singular integral operators

Abstract. Let K(x, V, s) be the open cone centred at x, with direction V, and aperture s. It is easy to see that if a set E satisfies for some V and s the condition:

"if x belongs to E, then E has an empty intersection with K(x, V, s)",

then E is a subset of a Lipschitz graph. To what extent can we weaken the condition above and still get meaningful information about the geometry of E? It depends on what we mean by "meaningful information'', of course. For example, one could ask for rectifiability of E, or if E contains big pieces of Lipschitz graphs, or if nice singular integral operators are bounded in L^2(E). In the talk I will discuss these three closely related questions.

Zoom link. https://umn.zoom.us/s/96104466915

Title. Transcendental Julia sets with Fractional Packing Dimension

Abstract. In this talk, we will define and compare different definitions of dimension (Hausdorff, Minkowski, and packing) used to analyze fractal sets. We will then define the basic objects in complex dynamics, and discuss some history of results about the dimension of the fractal Julia sets that famously show up in this area. No prior knowledge of complex dyanmics will be assumed. We will conclude by discussing my recent construction of a Julia set of a non-polynomial entire function with packing dimension strictly between one and two. We will see that Whitney decompositions, a foundational tool in harmonic analysis, play a vital role in the dimension calculation.

Zoom link. https://umn.zoom.us/j/99328467224

Title. Nonlinear stability of critical pulled fronts via resolvent expansions

Abstract. We consider invasion processes mediated by propagating fronts in spatially extended systems, in which a stable rest state invades an unstable rest state. We focus on the case of pulled fronts, for which the speed of propagation is the linear spreading speed, which marks the transition between pointwise decay and pointwise growth for the linearization about the unstable rest state in a co-moving frame. In a general setting of scalar parabolic equations on the real line of arbitrary order, we establish sharp decay rates and temporal asymptotics for perturbations to the front, under conceptual assumptions on the existence and spectral stability of fronts. Some of these results are known for the specific example of the Fisher-KPP equation, and so our work can be viewed as establishing universality of certain aspects of this classical model. Technically, our approach is based on a detailed study of the resolvent operator for the linearization about the critical front, near its essential spectrum.

Zoom link. https://umn.zoom.us/j/95439780402

Title. Existence, uniqueness and regularity of the minimizer of energy related to perimeter minus fractional perimeter

Abstract. We are interested in the asymptotic behaviors of the following energy functional $E(\Omega)=\sigma Per(\Omega)+\beta V_K(\Omega)$ defined for $|\Omega|=m$. Here the perimeter tries to keep the mass together in a ball, and $V_K$ is a non-local repulsive interaction energy trying to spread the mass around. We will then discuss the existence, uniqueness snd regularity properties of the minimizers of the energy especially in the regime where the energy $E(\Omega)$ converges to Perimeter minus fractional perimeter.

Zoom link. https://umn.zoom.us/j/95836995047

Title. Regularity criteria for weak solutions to the three-dimensional MHD system

Abstract. In this talk we will first review various known regularity criteria and partial regularity theory for 3D incompressible Navier-Stokes equations.

I will then present two generalizations of partial regularity theory of Caffarelli, Kohn and Nirenberg to the weak solutions of MHD equations. The first one is based on the framework of parabolic Morrey spaces. We will show parabolic Hölder regularity for the "suitable weak solutions" to the MHD system in small neighborhoods. This type of parabolic generalization using Morrey spaces appears to be crucial when studying the role of the pressure in the regularity theory and makes it possible to weaken the hypotheses on the pressure.

The second one is a regularity result relying on the notion of "dissipative solutions". By making use of the first result, we will show the regularity of the dissipative solutions to the MHD system with a weaker hypothesis on the pressure ($P \in \mathcal{D}'$).

This is a joint work with Diego Chamorro.

Zoom link. https://umn.zoom.us/j/91577100972

Title. Recent Progress on Fourier Uncertainty

Abstract. The classical Heisenberg Uncertainty Principle shows that a function and its Fourier transform cannot be too concentrated around a point simultaneously. In other words, if we force a function and its Fourier transform to vanish outside a small neighborhood of a point, then the function is zero. This classical principle has been generalized to many levels in the past, including results of Hardy, Beurling and many others. In this talk, we will recall old and new results about Fourier ncertainty, focusing more on the most recent developments on the field and its relationship to various topics, such as the sphere packing problem, interpolation formulae and many others.

Zoom link. https://umn.zoom.us/j/94208927523

Title. The Dirichlet problem for weakly harmonic maps with rough data.

Abstract. In this talk, we will discuss the well-posedness issues for weakly harmonic maps subject to Dirichlet boundary data assuming a minimal regularity. After a brief description of the problem we will present our techniques which partly rely on certain fundamental notions in harmonic analysis such as Carleson measures, its intrinsic connection to the John-Nirenberg space BMO and the Laplace operator... With an appropriate reformulation of our problem, various solvability results (existence, uniqueness and regularity) will then be reviewed. Our approach (nonvariational), as we will see, is suitable for the analysis of critical or endpoint elliptic boundary value problems and hence can unambiguously be applicable to similar type of equations or systems driven by classical operators. For this talk, we only mention a generalization of our results to second-order constant elliptic systems.

This is a joint work with Herbert Koch.

Zoom link. https://umn.zoom.us/j/95505467670

Title. Iterative methods for nonlinear optimisation problems - Prospects and applications

Abstract. An important class of extremal problems in nonlinear functional analysis is the nonlinear optimisation problem where some of the objective functions are nonlinear. In many cases where existence of solutions is guaranteed, these solutions are not usually affordable in a direct way. Iterative methods (or algorithms), therefore provide a convenient way of approximating these solutions. To a large extent, most of these iterative methods can be traced to the popular gradient descent algorithm. This talk presents the prospects that may result in using a different approach, and its usefulness even to equivalent reformulations of the nonlinear optimisation problem. Its applicability in some areas of science and technology is highlighted and a spectacular application in radiotherapy treatment planning where algorithmic efficiency is especially required is demonstrated.

This work is Joint with Charles Ejike Chidume .

Zoom link. https://umn.zoom.us/j/99152540720

Title. An Algebraic Brascamp-Lieb Inequality

Abstract. The Brascamp-Lieb inequalities are a natural generalisation of many familiar multilinear inequalities that arise in mathematical analysis, classical examples of which include Holder’s inequality, Young’s convolution inequality, and the Loomis-Whitney inequality. Each Brascamp-Lieb inequality is uniquely defined by a 'Brascamp-Lieb datum', which is a pair consisting of a set of linear surjections between euclidean spaces and a set of exponents corresponding to these maps. It is common in applications to encounter nonlinear variants, where the linear maps are replaced with nonlinear maps between manifolds. By incorporating a dampening factor that compensates for local degeneracies, we establish a global nonlinear Brascamp-Lieb inequality for a broad class of maps that exhibit a certain algebraic structure, with a constant that explicitly depends only on the associated 'degrees' of these maps.

Zoom link. https://umn.zoom.us/j/96413946414

Title. On the Liouville problem for the stationary Navier-Stokes equations

Abstract. Uniqueness of weak solutions of the 3D Navier-Stokes equations is a challenging open problem. In this talk, we will discuss some recent results of this problem for the 3D stationary Navier-Stokes equations. More precisely, within the framework of the Lebesgue, Lorentz and Morrey spaces, we will observe that the null solution of these equations is the unique one. This kind of results are also known as Liouville-type results.

Zoom link. https://umn.zoom.us/j/99894236187

Title. Orthogonal Systems of Spline Wavelets as Unconditional Bases in Sobolev Spaces

Abstract. We exhibit the necessary range for which functions in the Sobolev spaces $L^s_p$ can be represented as an unconditional sum of orthonormal spline wavelet systems, such as the Battle-Lemari\'e wavelets. We also consider the natural extensions to Triebel-Lizorkin spaces. This builds upon, and is a generalization of, previous work of Seeger and Ullrich, where analogous results were established for the Haar wavelet system.