When finite fields meet combinatorics
A finite field is a finite set that is a field. Due to finiteness, there are various interesting problems regarding finite fields in combinatorics. In this series of two talks, we describe two problems related to finite fields, which are derived from algebraic combinatorics and discrete geometry. In the first talk, we introduce an interesting counting formula related to q-binomial coefficients and see that counting combinatorial objects can be interesting in itself. In the second talk, we discuss the incidence problem and see how helpful spectral graph theory is for studying it. We mainly talk about the basic theory of quadratic forms and spectral graph theory. These talks are intended for general audiences in mathematics.
Toric varieties and their functions
A toric variety is roughly a space that contains a torus as a large part. As we have a good understanding of the torus, what matters for the description are points outside the torus. Such points can be characterized by looking at how functions defined on the torus extend, motivating the concept of a fan.
This talk is intended for audiences with very little knowledge of graduate-level mathematics. I will spend some time reviewing certain undergraduate mathematics including the definition of a torus. The final goal is to describe fiber bundles of toric varieties in terms of their fans.
The attainment of supremum of a continuous function on infinite dimensional space
It is well-known that every continuous functions defined on a compact metric space attains its supremum. Then, does the same phenomenon occur when the domain is not a compact space? We can easily construct a counterexample to this question by observing that a metric space is not necessarily bounded. Let us reconsider the same question for a subset of Banach space that is bounded but not compact. In this presentation, we will focus on the long-standing problem of whether a continuous linear operator defined on a Banach space attains its norm, a question that has been extensively studied by mathematicians such as Bishop, Phelps, and Lindenstrauss since the 1960s.
On toric Schubert varieties
A partial flag variety of type A is a smooth variety G/P, where G = GLn(ℂ) and P is a parabolic subgroup of G. Schubert varieties are special subvarieties of G/P, indexed by the corresponding Weyl group WP. We review known results on smooth toric Schubert varieties and consider further questions.
Brownian Motion and Stochastic Integration
Stochastic differential equations (SDEs) can be used to describe many problems in applied areas such as mathematical finance, fluid mechanics, and so on. On the other hand, Brownian motion is often used as the noise part of SDEs. Thus, to solve SDEs with Brownian motion, we need to study a stochastic integration with respect to Brownian motion (Itô calculus). In this talk, we introduce definitions and basic properties of Brownian motion and stochastic integration.
The Introduction of Erdős-Falconer problem over finite field
The Falconer distance problem asks a minimal Hausdorff dimension of compact subsets E ⊂ ℝd for which the distance set defined by Δ(E) := {|x-y| : x, y ∈ E } has a positive Lebesgue measure, where |⋅| is the Euclidean metric on ℝd. Recently, the Falconer distance problem has been extensively studied in the finite field setting. This problem is called Erdős-Falconer problem. In this talk, we review the Erdős-Falconer problem and introduce the variant of this problem. As a result, we obtain much stronger results on the Erdős-Falconer problem.
합성곱 신경망의 기초와 응용
인공 신경망 중에서도 광범위하게 사용되는 합성곱 신경망(Convolutional Neural Networks, CNN)은 이미지와 같은 고차원 데이터에서 효과적인 특징 추출과 학습을 가능하게 한다. CNN의 핵심 구성 요소인 합성곱(Convolution)과 풀링(Pooling) 연산은 데이터에서 유용한 특징을 추출하고, 데이터의 차원을 축소하여 처리 속도와 효율성을 높이는데 기여한다. CNN이 이미지를 처리하는 과정을 단계별로 설명하고, 이를 바탕으로 한 다양한 응용 분야를 소개한다. 특히, 얼굴 인식(Face Recognition) 기술 발전에 중요한 역할을 한 FaceNet과 그 구현 결과를 소개한다.
Camera Calibration with Gradient Descent
배럴 왜곡(Barrel distortion)은 이미지 가장자리가 원통형 곡선으로 휘어지는 현상으로, 핸드폰 카메라, CCTV, 자동차용 카메라 등 다양한 카메라에서 볼록렌즈로 인해 발생한다. 이러한 왜곡을 정교하게 보정하는 것은 정확한 이미지 해석을 위해 필요하다. 카메라 배럴 왜곡을 수식으로 설명하고, 경사 하강법(Gradient Descent)을 이용한 보정 방법을 소개한다.
수학의 본질과 미래: 무한한 가능성을 향한 여정
본 프레젠테이션은 "생각하는 방법에 대한 학문"으로서의 수학을 재정의하고, 그 본질적 아름다움과 현대 사회에서의 중요성을 탐구합니다. 대수기하학과 특이점 이론의 통찰을 제공하며, AI 시대 수학자의 고유한 역할을 강조합니다. 학사, 석사, 박사 학위별 진로 옵션과 취업률을 구체적 통계와 함께 제시하고, 다양한 분야에서의 수학 전공자 수요 증가 추세를 조명합니다. 이를 통해 학생들에게 수학 연구의 무한한 가능성과 그 안에서 자신의 역할을 발견할 수 있는 영감을 제공합니다.
Time series forecasting using machine learning
A time series is a series of data points indexed in time order, and time series forecasting has been widely studied due to its significant impacts on various decisions with social implications. With the advent of neural networks, beginning with the invention of the perceptron by Rosenblatt, time series prediction using machine learning has gained considerable attention due to its importance. In this talk, we will introduce machine learning methods for time series prediction, starting with the most basic neural network.
A Brief Introduction to the Kullback-Leibler Divergence
Claude Shannon, known as the father of information theory, had a great influence on artificial intelligence, modern cryptography. Shannon systematized information theory from a mathematical rather than a physical point of view, and defined the (Shannon) entropy that measures the expected amount of information. The Kullback–Leibler divergence is a concept that characterizes the relative (Shannon) entropy, and is a type of statistical distance. In other words, the Kullback–Leibler divergence measures how different one probability distribution is from the other. In this talk, we briefly introduce the Kullback–Leibler divergence to help us step into information theory.
Convex splitting method for the parabolic sine-Gordon equation
In this talk, we introduce a linear convex splitting method for the parabolic sine-Gordon equation and present an unconditionally energy stable scheme for any time step. Additionally, we prove the maximum principle for the proposed scheme and demonstrate through various numerical experiments that the evolution of the parabolic Sine-Gordon equation follows mean curvature flow.
M-ideals of compact operators and norm attaining operators
Given two Banach spaces X and Y, it is natural to consider constructing another Banach space through their direct sum X ⊕ Y. A notable example is the space equipped with an ℓp-norm for 1 ≤ p ≤ ∞, which we denote by X ⊕p Y. Moreover, for a positive real number q satisfying 1/p + 1/q = 1, it holds that (X ⊕p Y)* = X* ⊕q Y* = X* ⊕q X⊥, where X⊥ is the annihilator of X in X* ⊕q Y*. In this talk, we focus on the case where p = ∞ (and thus q = 1).
For a closed subspace J of a Banach space X, J is said to be an M-summand in X if X = J ⊕∞ J# for some complement J# of J. Additionally, J is said to be an M-ideal in X if X* = J* ⊕1 J⊥. It is clear that if J is an M-summand in X, then J is also an M-ideal in X. However, the converse does not generally hold. Especially, the space ℒ(X,Y) of bounded linear operators cannot generally be decomposed into an ℓ∞-sum with the space 𝒦(X,Y) of compact operators if 𝒦(X,Y) is proper. Nevertheless, for some pairs (X,Y) of Banach spaces, 𝒦(X,Y) can be an M-ideal in ℒ(X,Y), even if it is proper. In this talk, we introduce various known properties satisfied in these cases and present new results that contribute to the study of norm-attaining operators.