司靈得 (Daniel Spector)

Fall 2026 Nonlinear Analysis Seminar Series

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Date: Tuesday, 15 Sep. 2026, 15:00-16:00 TST (GMT+8) , in Room M212 in NTNU Gongguan Campus Mathematics Building
Speaker: Prashant Athavale, Clarkson University
Title: Star-Norm Bounds and Total Variation Flow, with Applications to Crystallographic Imaging
Abstract:
The dual of the total-variation seminorm, sometimes called the star-norm in image processing, arises naturally in variational and multiscale image decomposition. One way to understand this norm is through the equation $f = \mathrm{div}\, p$. A bounded vector field $p$ satisfying this equation immediately provides an upper bound for the star-norm of $f$. Hierarchical constructions developed by Eitan Tadmor in connection with the Bourgain-Brezis problem give a constructive way of producing such bounded solutions. However, obtaining sharp and computationally useful estimates of the star-norm of a general image remains an interesting open problem.

I will use this question as a starting point for a related line of work based on multiscale total-variation methods. Hierarchical $(BV, L^2)$ decompositions lead, in a continuous-scale limit, to total-variation flow. For weighted TV flow, the accumulated flow parameter gives an explicit upper bound on the weighted star-norm of the residual and, after finite-time extinction, on the star-norm of the image itself. Introducing spatially varying weights also makes it possible to suppress noise while reducing diffusion across significant edges.

I will then discuss how these ideas arise in electron backscatter diffraction (EBSD), where images encode crystallographic orientations of polycrystalline materials. In this setting, denoising and restoration must account for crystallographic symmetry while preserving grain boundaries, whose geometry carries important physical information. I will describe our use of total-variation and weighted total-variation methods for restoring EBSD orientation maps and extracting their geometric structure.

I will conclude with several open questions, including sharper and more computable estimates of the star-norm, connections between hierarchical bounded solutions and TV-based multiscale representations, and variational methods that better preserve the geometry of crystallographic images. These problems suggest several directions in which techniques from nonlinear analysis, geometric analysis, and image processing may interact.

Date: Tuesday, 22 Sep. 2026, 15:00-16:00 TST (GMT+8) , in Room M212 in NTNU Gongguan Campus Mathematics Building
Speaker: Chun Ho Lau, National Taiwan Normal University
Title: When (does) a singular integral operator maps atoms to molecules
Abstract:
In real Hardy spaces $H^p(\mathbb{R}^n)$ for $0 < p \le 1$, to check whether a singular integral operator $T$ is bounded on $H^p(\mathbb{R}^n)$, it is common to show that $T(a)$ is a molecule with the assumption that $T^*(1) = 0$. In this talk, we consider a similar situation in the local Hardy spaces $h^p(\mathbb{R}^n)$, with a broader class of molecules defined by a weaker cancellation condition. Moreover, we will discuss the properties of the singular integral operator if it maps atoms to molecules. This talk is based on the papers with Galia Dafni, Tiago Picon, and Claudio Vasconcelos.

Date: Tuesday, 29 Sep. 2026, 15:00-16:00 TST (GMT+8), in Room M212 in NTNU Gongguan Campus Mathematics Building
Speaker: Guy Foghem, Brandenburg University of Technology Cottbus-Senftenberg
Title: Robust interpolation inequalities via Chebyshev-type integral inequalities
Abstract:
We establish robust log-convex interpolation inequalities within the scale of Gagliardo seminorms. We achieve this by deriving some Chebyshev-type integral inequalities for general non-synchronous functions. An application for establishing these robust interpolation inequalities stems from the study of the asymptotic nonlocal-to-local optimal stability of weak solutions to the boundary Dirichlet problem associated with the regional fractional $p$-Laplacian. More precisely, if $u_s \in W^{s,p}(\Omega)$ weakly satisfies $(-\Delta)_{p,\Omega}^s u_s = f_s$ in $\Omega$ and $\gamma_0^s(u_s) = g_s$ on $\partial\Omega$, with $\frac{1}{p} < s \le 1$ and $\Omega \subset \mathbb{R}^d$ is bounded Lipschitz, then, under appropriate convergence of the data $f_s$ and $g_s$ as $s \to 1^-$, we establish that $\|u_s - u_1\|_{W^{s,p}(\Omega)} \xrightarrow{s \to 1^-} 0$.

Date: Tuesday, 3 Nov. 2026, 09:00-10:00 TST (GMT+8) , in Room M212 in NTNU Gongguan Campus Mathematics Building
Speaker: Ramiro Lafuente, University of Queensland
Title: TBA
Abstract: TBA

Date: Tuesday, 1 Dec. 2026, 09:00-10:00 TST (GMT+8) , in Room M212 in NTNU Gongguan Campus Mathematics Building
Speaker: Jesse Gell-Redman, University of Melbourne
Title: TBA
Abstract: TBA