Department Seminars & Colloquia
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본 강연에서는 암호학의 여러분야를 소개하고 최근 연구되는 동형암호와 정수론의 근사계산이 어떻게 연결되는지 설명한다. 완전동형암호는 암호화된 상태에서 모든 계산을 지원하는 이상적인 암호로서 암호학계의 성배(holy grail)로 불리며 오랫동안 미해결 문제로 알려져 있었다. 2009년 Gentry에 의해 처음 만들어진 동형암호는 구현이 거의 불가능할 정도로 복잡하였으나 이후 많은 연구를 통해 실용 가능한 스킴들이 제시되고 있다. 완전동형암호는 클라우드 컴퓨팅에 꼭 필요한 암호기술로서 2011년 MIT Technical Review에서 10대 Emerging Technology로도 선정이 되는 등 IT전반에 큰 이슈가 되고 있다. 본 강연에서는 CVP, SVP등의 고차원 격자문제, AGCD, LWE등의 정수론 문제들의 근사 계산 이론을 소개하고 이를 바탕으로한 동형암호 설계와 이의 응용들을 살펴본다. 특히 데이터베이스에서의 암호화된 탐색과 연산, 개인정보유출이 없는 데이터 분석, 암호화된 DNA 분석등에서의 응용들을 소개한다.
The dynamical degree is the exponential rate of the volume growth. The dynamical degree is one of the essential quantity to study of rational surface mappings. For example, a birational mapping on $mathhbb{P}^2$ is birationally equivalent to a rational surface automorphism if and only if the dynamical degree is a Salem number. For any given birational mapping $f$ on $mathhbb{P}^2$, it is known that we can always construct a modification whose action on $H^{2,2}$ gives the dynamical degree of $f$.
We will discuss how to construct such modifications and how to compute the dynamical degree of a given rational map.
The revolution of molecular biology in the early 1980s has revealed complex network of non-linear and stochastic biochemical interactions underlying biological systems. To understand this complex system, mathematical modeling has been widely used.
In this talk, I will introduce the typical process of applying mathematical models to biological systems including mathematical representation of biological systems, model fitting to data, analysis and simulations, and experimental validation. I will also describe our efforts to develop and integrate mathematical tools across the different steps of the modeling process. Finally, I will discuss the shortcomings of our present approach and how they point to the parts of current toolbox of mathematical biology that need further mathematical development.
The Ramsey number of a graph G is the minimum integer n for which every edge coloring of the complete graph on n vertices with two colors admits a monochromatic copy of G. It was first introduced in 1930 by Ramsey, who proved that the Ramsey number of complete graphs are finite, and applied it to a problem of formal logic. This fundamental result gave birth to the subfield of Combinatorics referred to as Ramsey theory which informally can be described as the study of problems that can be grouped under the common theme that “Every large system contains a large well-organized subsystem.”
In this talk, I will review the history of Ramsey numbers of graphs and discuss recent developments.