Department Seminars & Colloquia
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Orthogonal polynomials are a family of polynomials which are orthogonal with respect to certain inner product. The n-th moment of orthogonal polynomials is an important quantity, which is given as an integral. In 1983 Viennot found a combinatorial expression for moments using lattice paths. In this talk we will compute the moments of several important orthogonal polynomials using Viennot's theory. We will also see their connections with continued fractions, matchings, set partitions, and permutations.
A naive definition of mathematics may be “the science of quantity and space”. And in the simpler explanation, they correspond to arithmetic and geometry. One of modern dictionaries, Encyclopedia Britannica, defines it as “the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects” But it is more difficult to understand. Probably it may not be easy to define unanimously the mathematics. But we all know that the mathematics is an indispensable tool for the science and the engineering in modern times.
The word engineering originated from the ingenium, may be defined as “the application of scientific, economic, social, and practical knowledge in order to invent, design, build, maintain, research, and improve structures, machines, devices, systems, materials and processes”. The engineering discipline is also extremely broad as the mathematics is. Therefore, the engineering area based on the classical mechanics will be dealt in this lecture. Since the early stage of mechanics was not much different from the physics and mathematics, the classical mechanics based areas like civil engineering, mechanical engineering, aerospace engineering, and etc., are now utilizing the mathematics most.
In this lecture, the intersection and the differences of three disciplines, mathematics, science and engineering, will be explained after defining the three disciplines. And then some famous and important contributors to the classical mechanics, who were then mathematician and/or physicist, will be introduced and their connection with mathematics will be illustrated along with some equations. And then, the cooperation and the convergence of mathematics and engineering will be emphasized by introducing some representative examples.
정확한 날씨예보 생산을 위한 예보관과 관측자료의 기여도가 각각 28%와 32%이나 수치예보모델이 미치는 영향은 40%라는 조사 결과가 있다. 기상청은 날씨예보 정확도 향상을 위하여 천리안 기상위성과 같은 관측 인프라 구축 뿐 아니라 수치예보모델 발전에 많은 투자를 하고 있다. 기상청 수치예보모델의 역사는 1989년 수치예보반을 설립하면서 처음 시작하였으며, 1995년 일본기상청의 전지구예보모델을 거쳐 영국의 통합모델을 도입 운영하고 있다. 기상청에서 운영하는 수치예보모델의 정확도는 세계 6위 또는 그 이상인 것으로 여러 비교 검증 결과가 보여주고 있으나 기상청은 이에 만족하지 않고 자체 선진 수치예보모델 기술 확보를 위하여 9년 동안 총 1000억원의 연구비를 투자하여 한국형수치예보모델 개발 사업을 추진 중에 있다. 수치예보모델의 문제 해결 및 발전을 위하여 응용수학과 같은 분야의 지식과 경험이 매우 필요하다. 이번 세미나에선 수치예보모델에 대한 응용수학 분야의 공동연구 과제 발굴에 필요한 기반 정보를 제공하기 위하여 기상청 수치예보모델의 현황과 문제점에 대해 발표할 예정이다.
Cluster algebras were discovered by Fomin and Zelevinsky in 2001. Since then, they have been shown to be related to diverse areas of mathematics and physics such as Total positivity, Quiver representations, String theory, Statistical physics, Non-commutative geometry, Teichmüller theory, Hyperbolic geometry, Tropical geometry, KP solitons, Integrable systems, Quantum mechanics, Lie theory, Algebraic combinatorics, Number theory and Poisson geometry.
We explain these connections between various fields in elementary languages.