Title: Behavior of the solution for a nonlinear hyperbolic equation with some boundary conditions
Abstract: In this presentation, I would like to introduce the PDE system under some mathematical modeling. It appears likely that significant nonlinear terms in the primary system and boundary conditions consist. First, I will show the existence and uniqueness of a solution for the primary system. Finally, I try to check the (energy) decay rates of solution for the system.
28. December 19th (Thur.), 2019
Time and Place: 4:30 pm - 5:30 pm (Room 1423)
Speaker: Dr. Eun-Kyung Cho (Hankuk University of Foreign Studies)
Title: Spacelike constant mean curvature and maximal surfaces in 3-dimensional de Sitter space via Iwasawa splitting (II)
Abstract: In this talk, we introduce the construction of maximal surfaces in 3-dimensional de Sitter space, by using Iwasawa splitting. If we have time, we also show some examples of maximal surfaces.
26. December 4th (Wed.), 2019
Time and Place: 4:30 pm - 5:30 pm (Room 1423)
Speaker: Dr. Yuta Ogata (Okinawa College)
Title: Spacelike constant mean curvature and maximal surfaces in 3-dimensional de Sitter space via Iwasawa splitting (I)
Abstract: In the first of talk, we introduce the construction of spacelike constant mean curvature (CMC) surfaces with mean curvature H >1 in 3-dimensional de Sitter space, by using Iwasawa splitting. We also study the relationships between Iwasawa splitting and singularities.
25. November 27th (Wed.), 2019
Time and Place: 4:00 pm - 5:00 pm (Room 1423)
Speaker: Dr. WAN Xueyuan (KIAS)
Title: Convexity of energy function associated to the harmonic maps between surfaces
Abstract: For a fixed smooth map u0u0 between two Riemann surfaces Σ and S with non-zero degree, we consider the energy function on Teichmuller space T of Σ that assigns to a complex structure t∈T on Σ the energy of the harmonic map ut:Σt:=(Σ,t)→S homotopic to u0. We prove that the energy function is convex at its critical points. If t0∈T is a critical point such that dut0dut0 is never zero, then the energy function is strictly convex at this point. As an application, in the case that u0 is a covering map, we prove that there exists a unique critical point t0∈T minimizing the energy function. Moreover, the energy density satisfies 12|du|2(t0)≡112|du|2(t0)≡1 and the Hessian of the energy function is positive definite at this point.
24. November 26th (Tue.), 2019
Time and Place: 4:00 pm - 5:00 pm (Room 1424)
Speaker: Dr. WU Peng (Shanghai Center of Math., Fudan)
Title: Complex structures on Einstein four-manifolds of positive scalar curvature
Abstract: The question that when a four-manifold with a complex structure admits a compatible Einstein metric of positive scalar curvature has been answered by Tian, LeBrun, respectively. In this talk we consider the inverse problem, that is, when a four-manifold with an Einstein metric of positive scalar curvature admits a compatible complex structure. We will show that if the determinant of the self-dual Weyl curvature is positive then the manifold admits a compatible complex structure.
23. November 22nd (Fri.), 2019
Time and Place: 4:00 pm - 5:30 pm (Room 7323)
Speaker: Dr. Kyunghwan Song (Ewha Womans University)
Title: New Orthogonality Criterion for Shortest Vector of Lattices and Its Extensions to Higher-Dimensional Spaces and other topics related to Isogeny-based Cryptosystem
Abstract: Complex theta functions carry much geometric data about their associated complex abelian varieties. We will recount some of this classic theory before discussing some of the challenges and features of studying p-adic (commutative) Lie groups. We will describe the structure of p-adic abelian varieties and how to interpret theta functions in a way that makes sense in this setting.
21. November 8th (Fri.), 2019
Time and Place: 4:30 pm - 5:30 pm (Room 7323)
Speaker: Seungjae Lee (Postech)
Title: On Hartogs type extension phenomena
Abstract: In this talk, I will introduce some generalizations of the Hartogs extension theorem. In 1906, Hartogs discovered that any holomorphic function which is defined on the complement of a compact set in a domain Ω ⊂ Cn , n ≥ 2 can be extended globally. This theorem, so-called the Hartogs extension the- orem, has explained one of the differences between one and multi-dimensional complex geometry. Recently, Nagata and Me gave a generalization for this theorem. They showed that any holomorphic function on the close of points whose have at least Levi rank (n − 2) on its boundary can be extended to the whole domain. In this presentation, I will give motivations and will explain our main results. If time is permitted, I will give the sketch of the proof of our theorem. This is joint work with Y. Nagata.
20. October 30th (Wed.), 2019
Time and Place: 4:00 pm - 5:00 pm (Room 1424)
Speaker: Prof. Yuan Wei (Sun Yat-sen University)
Title: Volume comparison with respect to scalar curvature
Abstract: In Riemannian geometry, volume comparison theorem is one of the most fundamental results. The classic results concern volume comparison involving restrictions on Ricci curvature. In this talk, we will investigate the volume comparison with respect to scalar curvature. In particular, we show that one can only expect such results for small geodesic balls of metrics near V-static metrics. As for closed manifolds, we give a volume comparison theorem for metrics near stable Einstein metrics. In particular, it provides partially affirmative answers to both a conjecture of Schoen about hyperbolic manifolds and a conjecture proposed by Bray concerning the positive scalar curvature case respectively.
19. September 26th (Thur.), 2019
Time and Place: 10:30 am - 11:30 am (Room 1423)
Speaker: Prof. Hyun Suk Kang (GIST)
Title: Anisotropic curvature flow with the speed depending on position
Abstract: We consider anisotropic curvature flows where the curvature term is concave, inverse concave and of homogeneous degree one. Also the anisotropic factor depends on position not on the normal vector. The main ingredient is a pinching estimate which follows from Hamilton's maximum principle for tensors. We only allow small perturbation on the anisotropic factor to obtain the convergence of the flow and differential Harnack inequality. This is a joint work with Ki-Ahm Lee.
18. September 19rd (Thur.), 2019
Time and Place: 4:30 pm - 5:30 pm (Room 7323)
Speaker: Dr. Minh Hoang Nguyen (Vietnam National University)
Title: Construction of minimal annuli in $\widetilde{{\rm PSL}_2(\mathbb{R},\tau)}$ via variational method 2
Abstract: It is a classical result of Nirenberg and Pogorelov that a 2-sphere with positive Gauss curvature can be isometrically embedded in to Euclidean 3-space. In this talk, I will present a proof due to Jiaxing Hong of a disc with positive Gauss curvature and whose boundary has positive geodesic curvature can be isometrically embedded into Euclidean half 3-space. If time permits, I will include a discussion of my own recent work on isometric embeddings of 2-disc into hyperbolic space.
15. September 5rd (Thur.), 2019
Time and Place: 4:00 pm - 5:30 pm (Room 7323)
Speaker: Dr. Kyunghwan Song (Ewha Womans University)
Title: New Orthogonality Criterion for Shortest Vector of Lattices and Its Extensions to Higher-Dimensional Spaces
Speaker: Prof. Seungsu Hwang (Chung-Ang University)
Title: On the Besse conjeture
Abstract: On a compact n-dimensional manifold, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume is Einstein. It has been conjectured that a critical metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, will be Einstein. In this article we gives a short survey of the recent progress on this conjecture, proposed in 1987 by Besse.
Abstract: In this talk, I will first talk about the Yamabe problem and the Yamabe flow on Riemannian manifolds. Then I will explain what the Chern-Yamabe problem is, and talk about the Chern-Yamabe flow which is a geometric flow approach to solve the Chern-Yamabe problem.
12. August 9rd (Fri.), 2019
Time and Place: 4:00 pm - 5:00 pm (Room 7323)
Speaker: Prof. Sung-Jin Oh (KIAS)
Title: Bubbling for the hyperbolic Yang-Mills flow
Abstract: In this talk, I will provide an overview of the proof of the Threshold Conjecture for the (4+1)-dimensional hyperbolic Yang-Mills flow. An important step in the proof is a bubbling result for the hyperbolic Yang-Mills flow, which is analogous to those for elliptic PDEs in geometric analysis (harmonic maps, harmonic Yang-Mills connections on Riemannian manifolds). This is a joint work with D. Tataru (UC Berkeley).
11. August 8th (Thu.), 2019
Time and Place: 4:00 pm - 5:00 pm (Room 7323)
Speaker: Prof. Homare Tadano (Tokyo University of Science)
Title: Some Bonnet--Myers Type Theorems for Transverse Ricci Solitons on Complete Sasaki Manifolds
Abstract: The aim of this talk is to discuss some generalization of Ricci solitons in Sasaki geometry. A Sasaki manifold is an odd dimensional Riemannian manifold such that the associated cone manifold is a Kahler manifold. This concept was introduced by S. Sasaki and Y. Hatakeyama in 1962 as a special kind of contact manifolds and studied as an odd dimensional counterpart of a Kahler manifold. Recently, a Sasaki--Einstein manifold has been an attractive object not only in mathematics but also in theoretical physics, since it plays an important role in AdS/CFT correspondence stemming from superstring theory. In this talk, after we review basic facts on Sasaki manifolds and some Bonnet--Myers type theorems for complete Ricci solitons, I would like to introduce some Bonnet--Myers type theorems for transverse Ricci solitons on complete Sasaki manifolds. Our results generalize previous Bonnet--Myers type theorems for complete Ricci solitons due to M. Fernandez-Lopez and E. Garcia-Rio (2008), M. Limoncu (2010, 2012), J.-Y. Wu (2018), and improve previous Bonnet--Myers type theorems for complete Sasaki manifolds due to I. Hasegawa and M. Seino (1981) and Y. Nitta (2009).
10. August 6th (Tue.), 2019
Time and Place: 4:00 pm - 5:00 pm (Room 7323)
Speaker: Prof. Homare Tadano (Tokyo University of Science)
Title: Geometry of Ricci Solitons
Abstract: The aim of this talk is to discuss the compactness of complete Ricci solitons. Ricci solitons were introduced by R. Hamilton in 1982 and are natural generalizations of Einstein manifolds. They correspond to self-similar solutions to the Ricci flow and often arise as singularity models of the flow. The importance of Ricci solitons was demonstrated by G. Perelman, where they played crucial roles in his affirmative resolution of the Poincare conjecture. In this talk, after we review basic facts on Ricci solitons, I would like to introduce some new Bonnet--Myers type theorems for complete Ricci solitons. Our results generalize previous Bonnet--Myers type theorems due to W. Ambrose (1957), J. Cheeger, M. Gromov, and M. Taylor (1982), M. Fernandez-Lopez and E. Garcia-Rio (2008), M. Limoncu (2010, 2012), Z. Qian (1997), Y. Soylu (2017), G. Wei and W. Wylie (2009), J.-Y. Wu (2018), and S. Zhang (2014).
9. August 2nd (Fri.), 2019
Time and Place: 3:30 pm - 5:30 pm (Room 1424)
Speaker: Prof. Joachim Koenig (KAIST)
Title: Pullback, specialization and lifting: arithmetic-geometric properties of Galois covers
Speaker: Prof. Seunghyeok Kim (Hanyang University)
Title: The positive mass theorem and its application to the boundary Yamabe problem II
Abstract: This talk is a continuation of the lecture delivered a month ago. In this time, I will first explain Schoen's argument on the compactness theorem for the classical Yamabe problem. I will also present several new features of the compactness theorem for the boundary Yamabe problem on general 4- and 5-manifolds.
[GS_M_APP] July 26th (Fri.), 2019
Time and Place: 4:00 pm - 5:30 pm (Room 1423)
Speaker: Prof. Hojoo Lee (Seoul National University)
Title: [GS_M_APP] 극소 곡면 그림 미술관
Abstract: 미국 수학자 Jesse Douglas에게 첫 번째 Fields 메달을 안겨 준 Joseph Plateau의 비누막 실험 문제부터, Henri Poincare의 위상 수학 숙제를 푼 업적으로 준다는 Fields 메달을 받으러 오지도 않았던 러시아 수학자 Grigori Perelman의 혁명적인 Ricci 곡률 흐름 이론에 이르기까지, 오랜 세월 동안 사랑받아 온 극소곡면들이 복소 해석학, 조화 함수론, 편미분 방정식론, 변분론, 기하학적 측도론, 미분 기하학, 재료 공학, 물리학, 화학, 생물학등 다양한 분야들을 연결하는 아름다움에 취해 보자.
7. July 25th (Thu.), 2019
Time and Place: 4:00 pm - 5:30 pm (Room 7323)
Speaker: Dr. WonTae Hwang (KIAS)
Title: Elliptic Curves and Fermat's Last Theorem (Lecture note)
Abstract: In the history of number theory, the Fermat's Last Theorem, proven by Andrew Wiles and Richard Taylor, is one of the deep and beautiful results, in which, at least two seemingly unrelated mathematical areas were used in its proof. Roughly speaking, the theorem was proved by relating the existence of integer solutions of certain Diophantine equations to a special property, called modularity, of the corresponding elliptic curves. This kind of phenomenon occurs many times in various fields of math to reveal the beauty of mathematics. In this talk, we briefly review the story of the FLT (as non-experts) with the aid of the book: Rational Points on Elliptic Curves.
6. July 18th (Thu.), 2019
Time and Place: 4:00 pm - 5:30 pm (Room 7323)
Speaker: Dr. Jinwoo Shin (KIAS)
Title: Warped product Einstein manifolds (Lecture note)
Abstract: In this talk, we briefly describe the definition and geometric meanings of warped product Einstein manifolds.
5. April 29th (Mon.), 2019
Time and Place: 3:00 pm - 5:00 pm (Room 1423)
Speaker: Dr. Kyoung-Tark Kim (Pusan National University)
Title: The geometry of S(U(1) x U(1)) \ SU(1,1)
Abstract: In this talk, we introduce the left invariant Riemannian metric on the non-compact Lie group SU(1,1), and then consider about its left quotient space S(U(1) x U(1)) \ SU(1,1). By using the definition of Riemannian submersion, we can impose a Riemannian metric upon S(U(1) x U(1)) \ SU(1,1). It is diffeomorphically the upper half plane, but has a positive (non-constant) sectional curvature. We compute this fact using an Iwasawa decomposition of SU(1,1), and investigate the nature of the geometry of the left quotient space S(U(1) x U(1)) \ SU(1,1) (which is neither a homogeneous space nor a symmetric space). This is joint work with Professor Taechang Byun.
4. April 18th (Thu.), 2019
Time and Place: 2 pm - 3:30 pm (Room 7323)
Speaker: Dr. Daehwan Kim (KIAS)
Title: Solitons for the inverse mean curvature flow and their geometric properties
Abstract: Inverse mean curvature flow have been studied not only as a geometric flow but also for applications to prove geometric inequalities like Riemann-Penrose inequality, Minkowski type inequality, etc. Analyzing solitons of a geometric flow is a natural way to understand the flow. Examples of the homothetic and translating solitons of the inverse mean curvature flow in Euclidean space are provided. The incompleteness for the solitons are observed from several examples and then, the incompleteness of any translating soliton and homothetic solitons with restrict homothetic ratio can be proved by applying maximum principle. Their area growths are obtained.
Reference:
G. Drugan, H. Lee and G. Wheeler, Solitons for the inverse mean curvature flow. Pacific J. Math. 284 (2016), no. 2, 309-326.
D. Kim and J. Pyo, Translating solitons for the inverse mean curvature flow. Results Math. 74 (2019), no. 1, 74:64.
H. Omori, Isometric immersions of Riemannian manifolds. J. Math. Soc. Japan 19 (1967) 205-214.
3. April 5th (Fri.), 2019
Time and Place: 2:30 pm - 3:30 pm (Room 1424)
Speaker: Dr. Jeongmin Ha (Pusan National University)
Abstract: We will briefly review fundamentals of Complex and Kahler geometry: Almost complex manifold, Complex Structures, Kahler manifolds, and Kahler-Einstein metric, etc.
Abstract: SCV (Several Complex Variables) is a study of holomorphic functions of several variables, local theory for complex manifolds. We will briefly review fundamentals of SCV: Properties of holomorphic functions of several variables, Pseudo-convexity of domains, Levi Problem, and Bergman's kernel function, etc.
Reference:
복소해석기하학 (다변복소함수론), 김강태
Complex Analytic and Differential Geometry, Demailly