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E6-1, 1401호 (오후 3시 - 5시)
KAIX LECTURE SERIES

시간 : 2019년 6월 24일 -28일 오후 3-5시

강연 내용 :

1. 평균 곡률 흐름의 소개 : 역사와 의의 소개, 특이점에 관한 주요 연구의 흐름 논의.기하학적 측도 이론/미분기하/조화 해석/함수 해석/편미분방정식/최소-최대 이론/동역학계 등이 어떻게 활용되는지 간략하게 소개.

2. 평균 곡률 흐름의 편미분방정식 : 평균 곡률 흐름이 편미분방정식 유도. 평균 곡률 흐름의 닫힌 자기동형해의 미분방정식 소개

3. 볼록하고 닫힌 평균 곡률 흐름의 특이점 : Huisken density 소개, 제 1형 특이점을 분석. Stampacchia(de Giorgi) iteration과 제 2형 특이점의 분석 소개.

4. 기울기 흐름으로서의 평균 곡률 흐름 : 평균 곡률 흐름이 Huisken density의 기울기 흐름임을 설명. 최소 곡면의 고립된 특이점과 기울기 흐름의 관계를 소개, 관련 특이점 연구들에 대해 논의. Huisken density를 이용한 엔트로피를 정의하고, 닫힌 곡면의 최소 엔트로피에 대해 논의.

5. 평균 곡률 흐름의 고대 해 : 고대 해, 접촉 해, 극한 해 등을 정의하고, 특이점 연구에의 응용을 소개. 닫힌 리찌 흐름과 평균 곡률 흐름의 붕괴를 정의하고, 붕괴하는 않는 해의 특이점에 대해 논의. 비붕괴성이나 작은 엔트로피가 극한 해의 점근 해를 결정함을 보이고, 기울기 흐름으로서 점근 해가 고정 됨을 논의. 결정된 점근 해 위에서 간단한 조화 해석학과 동역학계를 이용하여 극한 해를 분석하여 분류.

References

0. Xi-Ping Zhu, Lecture Notes on Mean Curvature Flows, AMS/IP

1. T. Colding and W. Minicozzi. Generic mean curvature flow I; generic singularities. Ann. of Math. (2), 175(2):755{833, 2012.

2. T. Colding and W. Minicozzi. Uniqueness of blowups and Lojasiewicz inequalities.

Ann. of Math. (2), 182(1):221{285, 2015.

3. J. Bernstein and L. Wang. A topological property of asymptotically conical

self-shrinkers of small entropy. Duke Math. J 166(3):403{435, 2017.

4. B. White. The size of the singular set in mean curvature flow of mean convex sets. J. Amer. Math. Soc. 13(3):665{695, 2000.

5. B. White. A local regularity theorem for mean curvature flow. Ann. of Math. (2), 161(3):1487-1519, 2005

6. X. Wang. Convex solutions to the mean curvature flow. Ann. of Math. (2), 173(3):1185{1239, 2011

7. S. Angenent, P. Daskalopoulos, and N. Sesum. Unique asymptotics of ancient convex mean curvature flow solutions. arXiv:1503.01178, 2015

8. S. Brendle and K. Choi. Uniqueness of convex ancient solutions to mean curvature flow in R3. arXiv:1711.00823v2, 2017

9. K. Choi, R. Haslhofer, and O. Hershkovits, Ancient low entropy flows, mean convex neighborhoods, and uniqueness, arXiv:1810.08467, 2018.

10. O. Chodosh and F. Schulze, Uniqueness of asymptotically conical tangent flows, arXiv:1901.06369, 2019.

11. J. Bernstein and L. Wang, A sharp lower bound for the entropy of closed hypersurfaces up to dimension 6, Invent. Math. 206

12. S. Brendle, Embedded self-similar shrinkers of genus 0, Ann. of Math. (2) 183 (2016), no. 2, 715–728.

13. L. Wang, Uniqueness of self-similar shrinkers with asymptotically conical ends, J. Amer. Math. Soc. 27 (2014), 613-638.

14. L. Wang, Asymptotic structure of self-shrinkers, https://arxiv.org/abs/1610.04904 (2016).

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자연과학동(E6) Room 1401
1월 16일(수)-1월 24일(목), Intensive Lectures on Algebraic Geometry by Prof. Y. Kawamata

Intensive Lecures on Semi-orthogonal decomposition of a derived category of an algebraic variety

Prof. Yujiro Kawamata (University of Tokyo)

16:00-17:15 in Jan. 16, 18, 22, 24, Room 1401

Abstarct.

I will start with the definition of a derived category of an algebraic variety. The derived category contains much essential information of the algebraic variety.

Then I consider a decomposition of the derived category into simpler components called a semi-orthogonal decomposition (SOD) and explain examples.

I will explain SOD’s related to the minimal model program in birational geometry. Some SOD’s are obtained by looking at so-called exceptional objects

or their deformed variants called relative exceptional objects. I will explain examples of SOD’s for some singular surfaces.

References

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E6-1, seminar room 4415
Mini-workshop on Variational Methods for Elliptic PDE.

Schedule

14:00-14:50 Vitaly Moroz (Swansea University)

Ground-states of a Schrödinger-Poisson-Slater type equation Abstract: Schrödinger-Poisson-Slater equation is a nonlinear modification of Schrödinger equation with a repulsive nonlocal Coulomb potential and a local nonlinearity. We develop a variational framework for a class of Schrödinger-Poisson-Slater type equations and discuss existence, positivity and radial symmetry of ground state solutions

15:00-15:50 Jongmin Han (Kyung Hee University)

On the self-dual Einstein-Maxwell-Higgs equation on compact surfaces

Abstract: In this talk, we discuss recent progress on the study of the self-dual Einstein-Maxwell-Higgs equation on compact surfaces. We give a brief background on the equation and known results on the plane. The main topics are two theorems on the existence of topological and nontopological type solutions as the coupling parameter tends to be zero.

16:00-16:50 Jinmyoung Seok (Kyonggi University)

Existence criterion for standing waves to pseudo-relativistic nonlinear Schrodinger equations

Abstract : The pseudo-relativistic Schrodinger operator is defined by the square root of the elliptic operator , that arises when we consider a relativistic versions of quantum mechanics. Especially, it is used for describing spin zero particles as is Klein-Gordon operator. In this talk, I will present an almost complete existence criterion for standing waves to the pseudo-relativistic nonlinear Schrodinger equations, which exhibit an intermediate features between nonrelativistic nonlinear Schrodinger equations and nonlinear half wave equations.

We will see that this criterion depends on the quantity , not only on n and nonlinear exponent p, where denotes the frequency of standing wave . Due to the supercriticality of the equation, the standard variational method does not seem to work although the equation enjoys a variational structure. Instead, we approach by adopting the contraction mapping principle. It turns out that a uniform L^p estimate for the pseudo-relativistic operator plays a crucial role for obtaining the existence of solutions in the full range of supercritical exponent p. This shall be obtained by the Hörmander-Mikhlin Theorem based on a symbol analysis of the pseudo-relativistic operator.

### Organizers

Sang-il Oum, Younjin Kim

## Invited Speakers

Jeong Han Kim (김정한), KIAS, SeoulMartin Balko, Charles University, Prague

Dániel Gerbner, Alfréd Rényi Institute of Mathematics, Budapest

Cory T. Palmer, University of Montana, Missoula

Boram Park (박보람), Ajou University

Dong Yeap Kang (강동엽), KAIST

## Schedule

### Feb. 11, 2019, Monday

1:30pm-2:20pm Jeong Han Kim: Entropy and sorting

2:20pm-3:10pm Cory T. Palmer: Generalized Turán problems – Berge hypergraphs

Coffee Break

4:00pm-4:50pm Martin Balko: Ramsey numbers of edge-ordered graphs

4:50pm-5:40pm Dong Yeap Kang: T.B.A.

Banquet

### Feb. 12, 2019, Tuesday

9:30am-10:20am Boram Park: Sum-free set problem on integers

Coffee Break

11:00am-11:50am Dániel Gerbner: Generalized Turán problems – counting subgraphs

Lunch

We plan to provide meals to all participants and provide a room at a near-by hotel for invited speakers and selected participants. Please register in the following form below by January 28, Monday; please register early if you want to receive the support for the accommodation.

*https://dimag.ibs.re.kr/event/2019-1-graph-theory/*

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자연과학동 2411호
Workshop on analysis of nonlinear waves

### Organizers

권순식

March 7

15:00 Conserved energies for NLS, KdV and Gross-

Pitaevskii (Herbert Koch)

16:10 TBA (Yonggeun Cho)

March 8

9:30 Conserved energies for NLS, KdV and Gross-

Pitaevskii (Herbert Koch)

11:00 Scattering of the defocusing generalized

Benjamin-Ono equations (Kiyhun Kim)

14:00 Conserved energies for NLS, KdV and Gross-

Pitaevskii. (Herbert Koch)

15:30 Ill-posedness of the Hall-MHD equations without

resistivity (In-Jee Jung)

16:40 Well-posedness of the Hall-MHD equations

without resistivity (Sung-Jin Oh)

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자연과학동(E6-1) 1층 공동강의실 (1501호)
수리과학 기반 인공지능 및 빅데이터 산학연 포럼

14:00 ~ 14:30 최우성 박사 (한국전력공사)

PHM(Prognostics and Health Management) 분야에서의 베이지안 및 딥러닝 활용 사례

14:45 ~ 15:15 김재덕 박사 (삼성전자)

Towards Practicality: Theorems should not limit practicality

15:15 ~ 16:00 자유토론 및 티타임

16:00 ~ 16:30 유용균 박사 (한국원자력연구원)

최적설계 분야의 딥러닝 응용

16:45 ~ 17:15 조현제 박사 (삼성전자)

기업 관점에서의 자동화된 데이터 분석 시스템

17:15 ~ 18:00 자유토론 및 티타임