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\title{Titles and Abstracts}
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\subsection*{Yihong Du}
\noindent\textbf{Title: } Propagation dynamics of the Fisher--KPP nonlocal diffusion equation with free boundary
\noindent\textbf{Abstract: } Propagation has been modelled by reaction-diffusion equations since the pioneering works of Fisher and Kolmogorov--Peterovski--Piskunov (KPP).
Many new developments have been achieved in the past several decades on the modelling of propagation, with traveling wave and related solutions playing a central role. In this talk, I will report some recent results obtained with several collaborators on the Fisher--KPP equation with free boundary and ``nonlocal diffusion", where the diffusion operator is given by a convolution integral instead of the traditional Laplace operator. A key feature of this nonlocal equation is that the propagation may or may not be determined by traveling wave solutions. There is a threshold condition on the kernel function in the diffusion operator which determines whether the propagation rate is linear or superlinear in time, also known as accelerated spreading in the latter case, where the rate of spreading is not determined by traveling waves. For some typical kernel functions, sharp spreading rates will be presented.
\subsection*{Xuan Thinh Duong}
\noindent\textbf{Title: } Hardy spaces associated with operators and weak type estimates
\noindent\textbf{Abstract: } Let $(X, d, \mu )$ be a metric space with a metric $d$ and a doubling measure $\mu$.
Assume that $L$ is a non-negative self-adjoint operator on $L^2(X)$ whose heat kernel satisfies the Gaussian upper bound. In the last two decades, the Hardy space associated to the operator $L$, denote by $H^p_L(X)$ with $0 < p \le 1$, has had an important role in the study of boundedness of singular integrals with rough kernels which do not belong to the class of standard Calder\'on-Zygmund operators. In this talk, we show that if a sublinear operator $T$ is bounded from $L^2(X)$ to $L^2(X)$, and also bounded from $H^p_L(X)$ to $L^p(X)$ for some $0 < p < 1$, then $T$ is of weak type $(1,1)$. We then give some applications to estimates of singular integrals with rough kernels.
This is joint work with The Anh Bui.
\newpage
\subsection*{Daniel Hauer}
\noindent\textbf{Title: } Regularity and Separation for p-Laplace equations on Sub-Riemannian manifolds.
\noindent\textbf{Abstract: } We introduce a Sub-Riemannian framework and define Sobolev spaces in this framework. Under an additional assumption on the given metric, we show that those spaces are reflexive and separable. With this, we can realize the $p$-Laplace operator on such a manifold as the negative sub-differential operator of the $p$-Dirichlet energy. In the case, the given metric induces the Grushin space, we present a separation phenomenon showing that weak solutions of the elliptic and parabolic equation governed by the $p$-Laplace operator always admit an $L^q$-$L^r$-regularization effect for $1\le q0}
|S^\delta_t(\mathcal{L})f|$. As a result, we determine the sharp range of the summability indices $\delta$ for which almost everywhere convergence of $S^\delta_t(\mathcal{L})$ holds for every $f \in L^p(\mathrm{C}^d)$, when $p \geq 2$.
This talk is based on joint work with Sanghyuk Lee (Seoul National University) and Jaehyeon Ryu (KIAS).
\subsection*{Chongwei Liang}
\noindent\textbf{Title: } Muckenhoupt-type weights and the intrinsic structure in the Bessel setting
\noindent\textbf{Abstract: } We study the Muckenhoupt-type weights which reveal the intrinsic structure in the Bessel setting along the lines of Muckenhoupt--Stein and Andersen--Kerman. We introduce a new class of weights such that the Hardy--Littlewood maximal function is bounded on the weighted Lebesgue $L^p$ space if and only if the weight is in this new class. Moreover, along the lines of Coifman--Rochberg--Weiss, we investigate the commutator with the Bessel Riesz transform, and show that for the new Andersenâ€”Kerman type weight, the commutator is bounded on weighted $L^p$ if and only if the symbol is in the BMO space associated with Bessel operators.
\subsection*{Jiakun Liu}
\noindent\textbf{Title: } Monge--Amp\`ere equations and some applications
\noindent\textbf{Abstract: } In this talk we first give a brief introduction of Monge--Amp\`ere equation and Monge--Amp\`ere measure, together with the regularity theory under a doubling property. Then we give some interesting applications in geometry such as the isoperimetric inequality and the Brunn-Minkowski inequality, and applications in image recognition and a reconstruction problem in cosmology.
\newpage
\subsection*{Melissa Tacy}
\noindent\textbf{Title: } Filament structure in random plane waves
\noindent\textbf{Abstract: } Numerical studies of random plane waves, functions
\[
u=\sum_{j}c_{j}e^{\frac{i}{h}\langle x,\xi_{j}\rangle}
\]
where the coefficients $c_{j}$ are chosen ``at random'', have detected an apparent filament structure. The waves appear enhanced along straight lines. There has been significant difference of opinion as to whether this structure is indeed a failure to equidistribute, numerical artefact or an illusion created by the human desire to see patterns. In this talk I will present some recent results that go some way to answering the question. First we consider the behaviour of a random variable given by $F(x,\xi)=||u||_{L^{2}(\gamma_{(x,\xi)})}$ where $\gamma_{(x,\xi)}$ is a unit ray from the point $x$ in direction $\xi$. We will see that this random variable is uniformly equidistributed. That is, the probability that for any $(x,\xi)$, $F(x,\xi)$ differs from its equidistributed value is small (in fact exponentially small). This result rules out a strong scarring of random waves. However, when we look at the full phase space picture and study a random variable $G(x,\xi)=\Vert P_{(x,\xi)}u\Vert_{L^{2}}$ where $P_{(x,\xi)}$ is a semiclassical localiser at Planck scale around $(x,\xi)$ we do see a failure to equidistribute. This suggests that the observed filament structure is a configuration space reflection of the phase space concentrations.
\subsection*{Brett Wick}
\noindent\textbf{Title: } Wavelet Representation of Singular Integral Operators
Abstract:
In this talk, we'll discuss a novel approach to the representation of singular integral operators of Calder\'on-Zygmund type in terms of continuous model operators. The representation is realized as a finite sum of averages of wavelet projections of either cancellative or noncancellative type, which are themselves Calder\'on--Zygmund operators. Both properties are out of reach for the established dyadic-probabilistic technique. Unlike their dyadic counterparts, our representation reflects the additional kernel smoothness of the operator being analyzed. Our representation formulas lead naturally to a new family of $T1$ theorems on weighted Sobolev spaces whose smoothness index is naturally related to kernel smoothness. In the one parameter case, we obtain the Sobolev space analogue of the $A_2$ theorem; that is, sharp dependence of the Sobolev norm of T on the weight characteristic is obtained in the full range of exponents. As an additional application, it is possible to provide a proof of the commutator theorems of Calder\'on--Zygmund operators with BMO functions.
\subsection*{Po Lam Yung}
\noindent\textbf{Title: } Techniques for proving decoupling theorems
\noindent\textbf{Abstract: } Many techniques have been developed to prove decoupling theorems, drawing from areas such as PDE, geometry, and number theory. We will try to explain some of the ideas involved. Partly based on joint work with Shaoming Guo, Zane Kun Li and Pavel Zorin-Kranich. Time permitting we will also make a connection to the Hardy spaces for Fourier Integral Operators ($H^p_{FIO}$ for $p \geq 2$), introduced by Andrew Hassell, Pierre Portal and Jan Rozendaal.
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