Abstracts
Abstracts
용권순
Title : Advanced Neural Network with Graph Structure: An Introduction to Graph Neural Network Learning and Its Recent Trends
Abstract
Deep neural networks have effectively solved problems that were previously challenging with traditional PDE (Partial Differential Equations) methods by learning non-linear functions, specifically through Multi-Layer Perceptrons (MLP). But, how do they fare with graph structures? It's possible to structure and represent all the world's data as graphs. The significant advantage of the structural properties of graphs is their capability to represent connections and topological information without the limitations of Euclidean space, preserving these relationships intact. However, despite these advantages, training such structures with traditional DNN architectures like MLPs, CNNs (Convolutional Neural Networks), and RNNs (Recurrent Neural Networks) presents challenges.[1]
In this section, I will briefly introduce how we can learn the connectivity of graphs without compromising topological information. I will focus on the essentials of Graph Convolutional Networks (GCN)[2] and Graph Attention Networks (GAT)[3]. Additionally, I will provide a quick overview of recent popular trends in the application of Graph Neural Networks, including generative models[4] and recommendation systems[5].
[1] William L. Hamilton, et.al, Representation learning on graphs methods and applications, IEEE Data Eng. Bull. 40(3): 52-74 (2017)
[2] Thomas N. Kipf, et.al, Semi-Supervised Classification with Graph Convolutional Networks , ICLR (Poster) 2017
[3] Petar Veličković, et.al, Graph Attention Networks, ICLR (Poster) 2018
[4] Clément Vignac, et.al, DiGress: Discrete Denoising diffusion for graph generation, ICLR 2023
[5] Xiangnan He, et.al, LightGCN: Simplifying and Powering Graph Convolution Network for Recommendation SIGIR 2020
박지수
Title : Secret Sharing
Abstract
Secret Sharing is a fundamental cryptographic tool utilized in various protocols. In a secret sharing scheme, a dealer possesses a secret, there exists a set of n parties, and an access structure A comprising subsets of parties. The dealer distributes shares among the parties, ensuring that: (1) any subset in A can reconstruct the secret with a specified threshold from its shares, and (2) any subset not in A cannot gain any partial information about the secret. Additionally, Homomorphic Secret Sharing (HSS) is introduced, facilitating compact local evaluations on its shares, enhancing the efficiency of cryptographic operations.
안준민
Title : Introduction to Coding Theory and Bounds of Codes
Abstract
Coding theory, as a part of information theory, emerged with the purpose of detecting and correcting errors occurring in any kind of noisy communication channel. In this talk, I briefly introduce the basic concepts of algebraic coding theory and some bounds of codes used to characterize how “good” a code is.
김예훈
Title : 주가의 변동성(volatility)
Abstract
주가(기초자산)의 변동성은 '가격이 얼마나 빨리 움직이는가'에 대한 시장의 변화 속도를 나타내는 값으로 금융시장에서 매우 중요한 parameter이다. 이 발표에서는 변동성을 수학적으로 표현하고, 변동성이 내포하는 의미, 변동성을 추정하는 방법, 확률적 변동성과 국소적 변동성 등에 대해서 이야기하려고 한다.
하구겸
Title : Divergence theorem in PDE
Abstract
When studying PDE at the early times of the graduate course, we inevitably get to encounter with some integrals in n-dimensional domain. It is quite tricky to do the rigorous calculations in n-variables. But the divergence theorem sometimes gives us useful informations which allows us to deal with such integrals more efficiently. I will talk about its applications in the field of PDE.
서민지
Title : Computational Methods in Transcranial Focused Ultrasound Therapy
Abstract
Transcranial focused ultrasound (tFUS) is an emerging technology capable of delivering focused ultrasound energy across the skull, offering a potential to treat various neurological conditions [1]. The fundamental of tFUS therapy lies in utilizing the ultrasound, a sound wave whose properties of propagation are determined by the medium which it travels. Accurately estimating intracranial wave propagation is essential for a safe and efficient treatment, yet the heteroge- neous structure of the skull poses a challenge to estimate it during the treatment process [2]. To address this, predicting the wave propagation within the skull with numerical simulation methods have been studied, and recently there have been efforts to apply deep learning techniques for tFUS simulation. This section will explore conventional numerial simulation methods for tFUS theraphy, delve into recent deep learning approaches for real-time tFUS simulation, and discuss the associated challenges.
[1] O. Naor, S. Krupa, and S. Shoham, “Ultrasonic neuromodulation,” Journal of neural engineering, vol. 13, p. 031003, 05 2016.
[2] W. Lee, D. Weisholtz, G. Strangman, and S.-S. Yoo, “Safety review and perspectives of transcranial focused ultrasound brain stimulation,” Brain & Neurorehabilitation, vol. 14, 03 2021.
김린화
Title : Basics of Coding Theory: An Introduction to Self-Dual Codes
Abstract
Coding theory has grown into a discipline intersecting mathematics and engineering with applications to almost every area of communication such as satellite and cellular telephone transmission, compact disc recording, and data storage. I will talk about the basic concepts of coding theory and self-dual codes.
송윤영
Title : Introduction to Subdivision Schemes
Abstract
Subdivision is an important tool for the generation of curves and surfaces in geometric modelling, having found its way into wide application in computer graphics and computer assisted geometric design(CAGD) in recent years. In this talk, a brief introduction to the basic ideas, examples and a general analysis will be given.
구도완
Title : From combinatorics to PDEs: existence of solutions via topological fixed point theorems
Abstract
In this talk, I will introduce Sperner's lemma, which is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem. Schaeffer's fixed point theorem, which is a infinite-dimensional version of Brouwer's theorem will be addressed and if time permits, I shall discuss how these topological results can be applied to obtain the existence of solutions to certain non-linear PDEs.
김원종
Title : Staggered DG methods on general meshes
Abstract
We introduce a lowest order staggered discontinuous Galerkin (SDG) method based on the first order system on polygonal meshes. The method is locally conservative, and flexible to rough grids such as distorted grids or polygonal meshes, especially for the case including hanging nodes. The optimal convergence is checked. The SDG method is desirable to the adaptive mesh refinements based on the a posteriori error estimators since it is flexible to general meshes. The numerical results demonstrate our theories for the method, and the singularity or the strong interior layer is treated well.
강민범
Title : Newton Polygons and Oscillatory Integral Operators
Abstract
Let $T^\lambda$ be the oscillatory integral operators of the form
T^\lambda(f)(x)=\int_{\mathbb{R}^n \ \text{or} \ \mathbb{R}^{n-1}} e^{i\lambda S(x,y)} \chi(x,y) f(y) dy \tag{1} \label{eq1}
with $S(x,y)$ being a real phase function and $\chi(x,y)$ being a smooth cutoff function supported near the origin.
In 70's, Arnold and Varchenko found an intriguing general conclusion on the decay rate of an oscillatory integral of higher dimensions and proved that it is determined by the Newton Polyhedron of its real analytic phase function. Phong and Stein were able to announce a general conclusion in 1997 that the oscillatory integral operator in \eqref{eq1} with $x,y \in \mathbb{R}$ and real analytic phase function has the same decay rate as Varchenko's conclusion for oscillatory integrals. In that paper, the space is partitioned according to the Newtown polygon of the phase Hessian $S^{\prime\prime}_{xy}(x,y)$.
In this talk, we introduce the Newton polygon for global domain and discuss the analogues of the result of Phong and Stein in global domain under some suitable assumptions.