Kangwon Geometry Seminar

Birational Equivalence and Compactifications 

Jaehyun Kim  (Ewha Womans University)

Classical objects in algebraic geometry are spaces defined as the common zero loci of polynomial equations. Such spaces, known as varieties, have been studied from various perspectives. In this talk, we focus on birational equivalence, which provides a framework for comparing varieties that are locally but not globally isomorphic. We also introduce the cylinder problem and briefly survey recent results in this area. 

Hyperbolic Geometry without Prerequisites: Distance, Curvature, and Model Equivalence 

Sanghoon Kwon (Catholic Kwandong University)

This lecture builds hyperbolic geometry using only calculus and linear algebra. After defining distance via curve length and tangent-vector speeds, we motivate negative curvature through familiar surfaces, culminating in an explicit treatment of the pseudosphere. We then introduce four equivalent models of the hyperbolic plane and prove the main isometry statements connecting them, highlighting Möbius transformations, the Cayley transform, and Lorentz symmetries. We end with cross-ratio computations and Klein’s transformation-group viewpoint, explaining geometrically why the Euclidean parallel postulate does not hold.

3:00pm- 3:40 pm :  학생 발표 및 토의

송기준:  Two perspectives on matrix : Linear Transformation and Change of basis

지난 1년동안 배운 선형대수학의 개념 중 행렬을 바라보는 두 가지의 관점에 대해서 발표할 것 입니다. 학기 선형대수에서 배운 선형변환과, 2학기에는 기저변환을 통합하여 행렬은 사실 이중적 의미로 볼 수 있음 고찰하고 이를 바탕으로 행렬을 재구성하여 닮음 관계를 이해할 수 있음을 보일 것 입니다.

홍영찬: Definition and examples of the Poisson process 

I will explain the definition and properties of the Poisson process and look at examples of the Poisson process 

김형주:  A Linear Algebraic Approach to Fourier Analysis: From String Vibration to Eigenfunction Diagonalization

I will introduce the wave equation derived from the vibration of a string and explore its solutions as a superposition of various modes. The primary objective of this presentation is to demonstrate how these complex wave functions can be decomposed into fundamental components using the concept of orthogonality from linear algebra. By interpreting Fourier series and integrals as a process of diagonalization through an orthogonal basis, I will show that solving complicated differential equations can be simplified into a series of independent algebraic calculations. Finally, I will present how this linear algebraic perspective provides a deeper insight into Fourier analysis.