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### Analysis seminar meets Thursdays from 11:30-12:30 in SCEN322

Spring 2020 Schedule:

• January 16: JMM, no talk
• January 23: Alan Sola (Stockholm University)
• January 30: John Akeroyd
• February 3: OPEN
• February 13: Cody Stockdale (Washington University in St Louis)
• February 20: OPEN
• February 27: Caleb Parks
• March 5: Dallas Albritton (University of Minnesota)
• March 12: Ariel Barton
• March 19: Debraj Chakrabarti (Central Michigan University)
• March 26: Spring Break
• April 2: Andy Raich part 1
• April 9: Andy Raich part 2
• April 23: Meredith Sargent
• April 30: Tulin Kaman

## Analysis Seminar Titles and Abstracts

#### Ariel Barton (March 12): The Neumann problem for symmetric higher order elliptic differential equations

Abstract:

The second order differential equation $\nabla\cdot A\nabla u=0$ has been studied extensively. It is well known that, if the coefficients $A$ are real-valued, symmetric, and constant along the vertical coordinate (and merely bounded measurable in the horizontal coordinates), then the Dirichlet problem with boundary data in $L^q$ or $\dot W^{1,p}$, and the Neumann problem with boundary data in $L^p$, are well-posed in the half-space, provided $2-\varepsilon<q<\infty$ and $1<p<2+\varepsilon$.

It is also known that the Neumann problem for the biharmonic operator $\Delta^2$ in a Lipschitz domain in $\mathbb{R}^d$ is well posed for boundary data in $L^p$, $\max(1,p_d-\varepsilon)<p<2+\varepsilon$, where $p_d=\frac{2(d-1)}{d+1}$ depends on the ambient dimension~$d$.

In this talk we will establish well posedness of the $L^p$ Neumann problem, $p_d-\varepsilon<p<2+\varepsilon$, in the half-space for higher-order equations of the form $\nabla^m\cdot A\nabla^m u=0$, where the coefficients $A$ are real symmetric (or complex self-adjoint), and vertically constant.

#### Dallas Albritton (March 5): Weak-* stability and potential Navier-Stokes singularities

Abstract: A powerful technique, originally introduced by De Giorgi in the regularity theory of minimal surfaces, is to zoom in’ on a potential singularity and classify the blow-up limit’. We discuss applications of the above technique to the Navier-Stokes equations. In particular, when zooming in on a potential Navier-Stokes singularity, sequences of Navier-Stokes solutions whose initial data are converging only in a weak-* sense arise naturally. We identify a class of solutions satisfying the following stability property: weak-* convergence of the initial data in critical spaces implies strong convergence of the corresponding solutions. We apply the weak-* stability property to problems concerning blow-up criteria in critical spaces, minimal blow-up data, and forward self-similar solutions. Joint work with Tobias Barker (ENS).

#### Caleb Parks (February 27): A Generalized Voltera-Type Operator Between Hardy Spaces and Multiplication Operators Between Analytic Tent Spaces.

Abstract: We discuss a generalized Volterra-Type operator $V_g$ involving higher-order derivatives of $g$. Using tent space methods, we give an affirmative answer to the conjecture of Nikolaos Chalmoukis in 2019. This result completes the  characterization of $V_g$ mapping between different Hardy spaces. Along the way, we characterize the pointwise multipliers between the tent spaces. As a result, we recover the known characterizations of multipliers between Hardy and Bergman spaces.

#### Cody Stockdale (February 13): A Different Approach to Endpoint Weak-type Estimates for Calderón-Zygmund Operators

Abstract: The weak-type (1,1) estimate for Calderón-Zygmund operators is fundamental in harmonic analysis. We investigate weak-type inequalities for Calderón-Zygmund singular integral operators using the Calderón-Zygmund decomposition and ideas inspired by Nazarov, Treil, and Volberg. We discuss applications of these techniques in the Euclidean setting, in weighted settings, for multilinear operators, for operators with weakened smoothness assumptions, and in studying the dimensional dependence of the Riesz transforms.

#### John Akeryod (January 30): An invitation to extend Beurling’s Theorem.

Abstract: We review Beurling’s Theorem and its proof, and mention results that are related to it. There are some important open questions in this context that we carefully outline.

#### Alan Sola (January 23): Singularities of rational inner functions in dimension two and higher.

Abstract: I will present recent joint work with K. Bickel (Bucknell) and J.E. Pascoe (U Florida) concerning rational inner functions and their singularities. Our focus will be on integrability of derivatives as well as geometry of unimodular level surfaces. As we will see, the two-dimensional case is relatively well understood, while the situation in higher dimensions is much more complicated.

#### September 5: Chao Ding

Classical Maxwell Equations and a Generalized Maxwell Operator

#### September 19: Chao Ding

Special Bases for Solutions of the Generalized MaxwellEquation in 3-dimensional Space

#### October 10: Meredith Sargent

Old Results and New Observations for Optimal Approximants on the Bidisk

Abstract: In recent years, optimal polynomial approximants have been used to study cyclicity of functions in Dirichlet-type spaces on the complex unit disk, with particular interest being paid to the connection between the location of the zero sets of the optimal approximants and cyclicity, as well as a correspondence between optimal approximants and orthogonal polynomials. Some progress has been made to extend these ideas to the bidisk, where this question was also studied by electrical engineers for polynomials in the Hardy space. In this talk, the engineering results are reviewed, and new observations regarding zero sets and cyclic functions are discussed.

#### October 17: John Ryan

Towards a double layer potential operator for the Maxwell operator, 1

Abstract: We will first set up the Maxwell equation in n dimensional Euclidean space. We will establish that the equation is conformally invariant in degree one. We then use the conformal invariance of this Maxwell equation to introduction a Poisson kernel in upper half space and a double layer potential operator over a Lipschitz graph.
This is joint work with Chao Ding.

#### October 24: Minh Nguyen

U(n) Seiberg-Witten Equations

#### Special Visitor Seminar October 29, 1:00-1:50​ in SCEN 322:  Benjamin Steinhurst (McDaniel college)

Laplacians and Spectra on Laakso Spaces

Abstract: Laakso spaces are a family of fractal spaces that are constructed from a sequence of approximations via a projective (inverse) limit. This gives rise to a direct limit of the associated function spaces, ($L^{2}$, $Dom(\\\\Delta)$, $\\\\ldots$). The virtue of this is that the Laplacian is very well understood and has an explicitly computable spectrum and eigenspaces. We construct all of these objects and discuss how the spectrum changes in response to small changes in the geometry of the fractal.

#### October 31: Caleb Parks

Interpolation in Analytic Tent Spaces.

Abstract:Introduced by Coifman, Meyer, and Stein, the tent spaces have seen wide applications in Harmonic Analysis. Their analytic cousins have seen some applications involving the derivatives of Hardy space functions. Moreover, the tent spaces have been a recent focus of research. We introduce the concept of interpolating sequences for analytic tent spaces analogously to the same concepts for Bergman spaces. We then characterize such sequences in terms of Seip’s upper uniform density. We accomplish this by exploiting a kind of M\\”obius invariance for the tent spaces.

#### November 7: John Ryan

Towards a double layer potential operator for the Maxwell operator, 2

Abstract: we will continue to look at the Maxwell equation as an analog of the Laplace equations in n-dimensional Euclidean space. This leads us to some representation theory for the group SO(n) and to Rartitta-Schwinger/Stein-Weiss first order operators.