5/9(Thu) 14:00 - 14:40 유재현(KIAS)

Title: Bochner-Riesz mean for the twisted Laplacian in $\mathbb R^2$

Abstract: We study the Bochner--Riesz problem for the twisted Laplacian on two-dimensional Euclidean space $\R^2$. It has been conjectured that the Bochner--Riesz means for the twisted Laplacian of order $\delta$ converges in $L^p$ if and only if  $\delta> \max(0, |(p-2)/p|-1/2)$. We prove the conjecture by obtaining uniform $L^p$ bounds on the Bochner--Riesz means up to the sharp summability indices. This talk is based on a joint work with Eunhee Jeong and Sanghyuk Lee. 

5/9(Thu) 14:40 - 15:20 고혜림(서울대)

Title: Maximal estimates for averages over $2$-dimensional surfaces in $\mathbb R^4$ 

Abstract: We explore the $L^p$-improving properties of the maximal estimates for $2$-dimensional surfaces in $\mathbb R^4$. We obtain the sharp $L^p$--$L^q$ estimates except for endpoints for nondegenerate surfaces satisfying specific rank condition. This is a joint work with Seheon Ham. 

5/9(Thu) 15:40 - 16:20 우관(UNIST)

Title: Initial value problem for sub-diffusion equations 

Abstract: Fractional calculus is a well-established and long standing field of research, and recent interest in Sobolev space theory for time-fractional equations has gained. 

Despite progress, certain aspects remain unclear, prompting further investigation.

This talk delves into the derivation of time-fractional parabolic equations via anomalous diffusion phenomena and identifies suitable solution and data spaces for initial value problems in anomalous diffusion.

We also discuss key properties such as trace and extension theorems and convergence to initial data.

5/9(Thu) 16:20 - 17:00 조성웅(KAIST)

Title: Learning Operators in Complex Function Spaces Using Hypernetworks with Limited Resources 

Abstract: Fast and accurate prediction of complex physical dynamics poses significant challenges across various fields, particularly on resource-constrained hardware where real-time processing is essential. The Deep Operator Network (DeepONet) offers a framework for learning nonlinear mappings between function spaces but suffers from high computational costs and extensive parameter requirements, especially for complex (discontinuous or non-smooth) target functions. This study introduces HyperDeepONet, utilizing the hypernetwork's capabilities to learn complex operators with fewer parameters effectively. We propose that DeepONet and its variants, which integrate input function information into the target function, represent specific cases of HyperDeepONet. Our analysis suggests that HyperDeepONet achieves the required learning accuracy with significantly reduced complexity. Furthermore, HyperDeepONet has demonstrated the ability to learn various operators more efficiently than existing benchmarks. 

5/9(Thu) 17:00 - 17:40 이명수(KAIST)

Title: A BGK-type model for relativistic gas mixtures 

Abstract:  In this talk, we briefly introduce the BGK model which is developed to approximate the Boltzmann equation with lower computational costs. Then, we present an approximation model of the relativistic Boltzmann equation for gas mixtures, based on the BGK-type approach. We see that our model satisfies important physical properties, such as the conservation laws and the H-theorem. 

5/10(Fri) 10:00 - 10:40 이주영(서울대)

Title: The critical weighted inequalities of the spherical maximal function 

Abstract: Weighted inequality on the Hardy-Littlewood maximal function is completely understood while it is not well understood for the spherical maximal function. For the power weight $|x|^{\alpha}$, it is known that the spherical maximal operator on $\R^d$ is bounded on $L^p(|x|^{\alpha})$ only if $1-d\leq \alpha<(d-1)(p-1)-1$ and under this condition, it is known to be bounded except $\alpha=1-d$. In this talk, we introduce the result of the case of the critical order, $\alpha=1-d$. 

5/10(Fri) 10:40 - 11:20 조주희(서울대)

Title: Almost everywhere convergence of sequence of Schrödinger means 

Abstract: In this talk, we consider pointwise convergence of the Schrödinger means along sequences tn that converge to zero. We discuss sufficient conditions for the convergence and explain the key observation, which is that bounds on the maximal function $sup_n |e^{i t_n ∆}f|$ can be deduced from those on $sup_{0 < t<=1} |e^{i t_n ∆}f|$ when $\{t_n\}$ is contained in the Lorentz space $\ell^\infty$. We will discuss sharp counterexamples for the related maximal estimates. 

5/10(Fri) 11:20 - 12:00 최재환(KAIST)

Title: Characterizations of weighted Besov and Triebel-Lizorkin spaces with variable smoothness 

Abstract: This presentation delves into the characterization of weighted Besov and Triebel-Lizorkin spaces with variable smoothness, expanding on the understanding of norm equivalences within these function spaces. While Besov and Triebel-Lizorkin spaces without weights have been thoroughly characterized, the characterizations involving weights remained unexplored. Our study confirms that norm equivalences persist even when variable smoothness and weights are incorporated. This achievement is made possible through the application of shifted maximal functions, Peetre’s maximal functions, and the reverse Hölder inequality, offering new insights into the structure and behavior of these complicated function spaces. 

5/10(Fri) 12:00 - 12:40 최미란(서강대)

Title: Stable one- and two-dimensional dispersion-managed solitons

Abstract: In this talk, we analyze the dynamics of one- and two-dimensional spatial solitons for the dispersion-managed nonlinear Schr¨odinger equation, as it arises in the context of fiber-optics communication. We establish threshold phenomena for the existence of these solitons, which are found as minimizers of nonlinear and nonlocal variational problems. This is based on joint works with Dirk Hundertmark and Young-Ran Lee.