세미나 일정

 2024년 5월 28일  

연사:  Kalachand Shuin (서울대) 4시 

발표 제목: l^2 decoupling for certain surfaces of finite type in R^3

초록: In this talk, we shall discuss the paper of Z. R. Li and J. Q. Zheng. We will see how to use  Bourgain-Demeter’s   Decoupling inequality of paraboloid to prove decoupling inequality for some degenerate hypersurfaces in R^3.

 2024년 5월 21일  

연사: 오세욱 (KIAS) 4시 

발표 제목: Estimates for maximal functions associated to hypersurfaces in R3

초록: We review the paper "Estimates for maximal functions associated to hypersurfaces in R3 with height h<2: part 2" by Buschenhenke, Ikromov, and Muller. The authors proved Lp estimates for maximal averages when 1<p<2. The result is sharp for "good" hypersurfaces. In this talk, the speaker explains their main results and introduces a new quantity determining the Lp boundedness of maximal averages.

 2024년 5월 14일  

연사: 유재현 (KIAS) 2시 

발표 제목: Local smoothing for the Hermite wave equation

초록: We review the paper "Local smoothing for the Hermite wave equation" by Robert Schippa.  We study the local smoothing conjecture for the Hermite wave equation proposed by the author in the article and how the author could obtain partial results on the conjecture. We also discuss the relationship between the local smoothing estimate and the $L^p$ boundedness for the Hermite Bochner-Riesz means.

 2024년 4월 30일  

연사: 이진봉 (서울대) 4시 

발표 제목: Maximal functions estimates of Pramanik-Yang-Zahl

초록: This talk is based on the paper "A Furstenberg-type problem for circles, and a Kaufman-type restricted projection theorem in R3" by M. Pramanik, T. Yang, and J. Zahl.  Their main tool is so-called maximal functions estimates, which is Theorem 1.7. The speaker explains contents of Theorem 1.7 and present a sketch of the proof in which the authors used a result from topological graph theory.

 2024년 4월 23일  

연사: Melissa Tacy (University of Auckland) 5시 

발표 제목: A quasimode approach to spectral multipliers

초록

 2024년 4월 16일  

연사: 이주영 박사 (서울대) 4시 

발표 제목: A weighted decoupling inequality and its application to the maximal Bochner-Riesz problem

초록: We review the recent paper "A weighted decoupling inequality and its application to the maximal Bochner-Riesz problem" by Shengwen Gan and Shukun Wu. They obtained a weighted $l^p$-decoupling inequality when $p=2n/(n-1)$. Precisely, they restricted the domain in a smaller set and obtained a smaller decoupling constant. The authors obtained a sharp estimate when $n=2$, and some partial results when $n\geq 3$. As an application, we will see how this refined decoupling inequality gives an improvement on the maximal Bochner-Riesz problem.

 2024년 4월 2일  

연사: 문경태 (포항공대) 4시 

발표 제목: A restriction estimate with a log-concavity assumption

초록  

 2024년 3월 26일  

연사: Luz Roncal (Basque Center for Applied Mathematics - BCAM, Spain) 4시 

발표 제목: Landis-type results for discrete equations

초록: We prove Landis-type uniqueness results for both the semidiscrete heat and the stationary discrete Schrödinger equations. To establish a nomenclature, we refer to Landis-type results when we are interested in the maximum vanishing rate of solutions to equations with potentials. The results are obtained through quantitative estimates within a spatial lattice which manifest an interpolation phenomenon between continuum and discrete scales. In the case of the elliptic equation, these quantitative estimates exhibit a rate decay which, in the range close to continuum, coincides with the same exponent as in the classical results of the Landis conjecture in the Euclidean setting. 

Joint work with Aingeru Fern\'andez-Bertolin and Diana Stan.

 2024년 3월 19일  

연사: 강현배 교수님 (인하대)  5시

발표 제목: Decomposition theorems for surface vector fields and an application to spectral theory of the Neumann-Poincare operator

초록:  I will explain two decomposition theorems for vector fields defined on the boundary of the bounded domain. I will also explain an application of the decomposition to spectral theory of the Neumann-Poincare operator, aka the double layer potential, which motivated this research. 

The first decomposition theorem says that the space of vector fields on the boundary is decomposed into three subspaces orthogonal to each other: the first one consists of vector fields that extend to the inside of the domain as divergence-free and rotation-free vector fields, the second one those to the outside as divergence-free and rotation-free vector fields, and the third one is a subspace of the finite codimension of the subspace of vector fields which extend to both the inside and the outside as divergence-free harmonic vector fields. The second one does that the space of vector fields is decomposed into three subspaces orthogonal to each other: the first one consists of vector fields that extend to the inside as gradients of harmonic functions, the second one those to the outside as gradients of harmonic functions, and the third one those to both the inside and the outside as divergence-free harmonic vector fields. The codimensions of various subspaces appearing in the decompositions are determined by the first Betti numbers of corresponding domains and the boundary. The decompositions are derived using layer potential theory.

This talk is based on two recent joint papers with Shota Fukushima (Inha) and Yong-Gwan Ji (KIAS).

 2024년 1월 3일  

연사: 오창근 박사 (MIT)  4시

발표 제목: Discrete restriction estimates for manifolds avoiding a line.

초록: Fourier introduced the idea that every periodic function can be represented as an exponential sum. There have been a lot of efforts to understand exponential sums in algebraic geometry, additive combinatorics, number theory, and Fourier analysis. Although a group of people from different areas tried to have a better understanding of exponential sums, it is surprising to me that basic questions for exponential sums still remain open. I'll introduce some backgrounds on exponential sums and present a recent work with Larry Guth and Dominique Maldague.

 11월 28일 

연사: 박민석 (서울대)  4시

발표 제목: $L^(d+2)/2$ bound for Nikodym maximal function

초록: The Kakeya or the Nikodym maximal conjecture is that the Kakeya or the Nikodym maximal function is bounded from $L^p$ to $L^q$ for $1 \leq p \leq d, q=(d-1)p$. There have been results in improving the range of p. In this talk, We prove the estimate for $ p\leq (d+2)/2$. We focus on the Nikodym maximal function in $R^3$ and shortly remark on the general cases. This talk is based on “An improved bound for Kakeya type maximal functions” by T. Wolff. 

연사: 김유현 (서울대) 5시 

발표 제목: Mixed Norm $l^2$ Decoupling for Paraboloids

초록: We review the paper "Mixed Norm $l^2$ Decoupling for Paraboloids" by Dasu, Jung, Li, and Madrid. In this paper, Bourgain and Demeter's sharp $(l^2, L^p_{x,t})$ decoupling for the paraboloid is generalized to the sharp $(l^2, L^q_tL^r_x)$ decoupling for the paraboloid. A sketch of the proof will be given in this talk.

 11월 21일  

연사: 양창훈 (충북대) 4시

발표 제목: The modified scattering for Dirac equations of scattering-critical nonlinearity 

초록:  In this talk, we consider the Maxwell-Dirac system in 3 dimensions under zero magnetic field. We prove the global well-posedness and modified scattering for small solutions in the weighted Sobolev class. Imposing the Lorenz gauge condition, and taking the Dirac projection operator, it becomes a system of Dirac equations with Hartree-type nonlinearity with a long-range potential as 1/|x|. We perform the weighted energy estimates. In this procedure, we have to deal with various resonance functions that stem from the Dirac projections. We use the spacetime resonance argument of Germain-Masmoudi-Shatah, as well as the spinorial null structure. On the way, we recognize a long-range interaction that is responsible for a logarithmic phase correction in the modified scattering statement. This talk is based on joint work with Yonggeun Cho, Soonsik Kwon, and Kiyeon Lee.

연사: 이정진 (UNIST) 5시 

발표 제목: Local smoothing estimates for the wave equation in higher dimensions.

초록: Guth devised k-broad estimates to improve the restriction conjecture in higher dimensions. Following his arguments Ou and Wang improved the cone restriction estimates as well.  Based on their cone restriction estimates Gao, Liu, Miao, and Xi obtained some improved local smoothing estimates for the wave equation in higher dimensions. In this seminar, we will talk about the differences between the cone restriction estimate and the local smoothing estimate, and the difficulties caused by the differences.

 11월 14일  

연사: Robert Schippa (KIAS) 2시 30분

발표 제목: Non-existence of radial eigenfunctions for perturbations of the Heisenberg sublaplacian.

초록: In this talk we show uniform resolvent estimates in weighted $L^2$-spaces for radial solutions of the inhomogeneous Helmholtz equation for the sublaplacian in the Heisenberg group. The approach is based on a multiplier method with appropriate weights (a generalisation of those of Morawetz for the Klein--Gordon equation) and Hardy inequalities in the Heisenberg group. 

연사: Kalachand Shuin (서울대) 3시 30분

발표 제목:Generalized Radon transforms on fractal measures.

초록: In the setting of a general Borel measure $\mu$ on $\mathbb{R}^d$ with the natural ball size condition $\mu(B(x,r))\leq Cr^s$, we establish the $L^{p}(\mu)-L^{q}(\mu)$- estimate for the generalized Radon transform. 

This talk is based on a paper "Generalized Radon transforms on fractal measures" by Shengze Duan.

 11월 7일  

연사: Luz Roncal (Basque Center for Applied Mathematics - BCAM, Spain) 4시 

발표 제목: Non-existence of radial eigenfunctions for perturbations of the Heisenberg sublaplacian.

초록: In this talk we show uniform resolvent estimates in weighted $L^2$-spaces for radial solutions of the inhomogeneous Helmholtz equation for the sublaplacian in the Heisenberg group. The approach is based on a multiplier method with appropriate weights (a generalisation of those of Morawetz for the Klein--Gordon equation) and Hardy inequalities in the Heisenberg group. 

연사: 이성철(서울대)  오후 5시 

발표 제목: L^2 Fourier restriction for α-Salem measure

초록: Stein-Tomas theorem states that Fourier restriction on a sphere defines a 

bounded operator from Lp(Rd) to L2(dσ) for 1 ≤ p ≤ 2d/(d+1). As a generalization of

the theorem, we may replace the sphere surface measure σ with an α-Salem measure µ. 

It is known that the Fourier transform cannot yield a bounded operator from Lp(Rd) 

to L2(dµ) for p > 2d/(2d−α), due to Mitsis. In this talk, we will review the Xianghong

Chen and Andreas Seeger’s construction of an α-Salem measure µ, for which the Fourier 

transform defines a bounded operator from Lp(Rd) to L2(dµ) within the optimal range

1 ≤ p ≤ 2d/(2d − α).

 10월 31일  

연사: 오세욱 박사(KIAS)  오후 4시 

발표 제목:  Restriction estimates for quadratic manifolds of arbitrary codimensions

초록: In this talk, we review the paper with same title written by Gan, Guth, and Oh. They use broad-narrow analysis with new ingredient which they call covering lemma. This talk focuses on introducing briefly their result and methods.

 10월 24일  3rd HARMONIC ANALYSIS WORKSHOP IN SEOUL

 10월 17일 

연사: 이진봉 박사(서울대)  오후 3시 

발표 제목: Improved bounds for Stein's square functions

초록: We review the paper "Improved Improved bounds for Stein's square functions" by Lee, Rogers, and Seeger. By the square functions, we mean the square function for the Bochner-Reisz mean. The authors make use of bilinear adjoint restriction estimates to show $L^2$ weighted norm inequalities, which yields the improved bounds for the square functions. 

연사: 고혜림 박사(서울대)  오후 4시 

발표 제목: A Sharp square function estimate for the moment curves in R^d

초록: We review the recent paper of Guth–Maldague for the sharp square function estimate of moment curve in R^d. We briefly sketch the proof of the main inductive estimates for the moment curves, cones over moment curves, and more general m-th order Taylor cones.

 10월 10일  

연사: 함세헌 박사(서울대)  오후 4시 

발표 제목: A sharp Mizohata-Takeuchi-type estimate for the cone. 

초록: In this talk, we continue to review Mizohata-Takeuchi estimates for the cone. As a motivation, we study weighted inequalities for the Fourier extension operator for the parabola, which is a variant of the Mizohata-Takeuch conjecture for the parabola. Then we discuss the main estimate for the cone and related maximal estimate. 

연사: 유재현 박사(KIAS)  오후 5시 

발표 제목: Sharp local $L^p$ estimates for the Hermite eigenfunctions

초록: In this talk, we review the paper "Sharp local $L^p$ estimates for the Hermite eigenfunctions" by Wang and Zhang. The local $L^p$ bounds of eigenfunctions over geodesic balls on manifolds have been studied by many authors in connection with an investigation of the concentration of the eigenfunctions. We study sharp $L^p$ bounds of the Hermite eigenfunctions over balls. To prove the nontrivial part of the result, we take an approach introduced in the work of Jeong-Lee-Ryu. We study the strategy of the proof by describing the main ideas.

 10월 3일  개천절

 9월 25일 월요일 장소: 27동 220호 

연사: Wenjuan Li (Northwestern Polytechnical University)  오후 4시 

발표 제목: A survey on generalized Schr\"{o}dinger operators along curves

초록:  In this survey, we review the historical development of the Carleson problem about a.e. pointwise convergence in five aspects: a.e. convergence for generalized Schr\"{o}dinger operators along vertical lines; a.e. convergence for Schr\"{o}dinger operators along arbitrary single curves; a.e. convergence for Schr\"{o}dinger operators along a family of restricted curves; the upper bounds of $p$ for the $L^p$-Schr\"{o}dinger maximal estimates; a.e. convergence rate for generalized Schr\"{o}dinger operators along curves. Finally, we list some open problems which need to be addressed.

연사: 함세헌 박사(서울대)  오후 5시 

발표 제목: A sharp Mizohata-Takeuchi-type estimate for the cone

초록: In this talk, we review the paper entitled "A sharp Mizohata-Takeuchi type estimate for the cone in $\mathbb R^3$" by A. Ortiz. We begin with Mizohata-Takeuch conjecture for the parabola. Then we study the main estimate and related estimates such as weighted Fourier extension estimates, or fractal local smoothing estimates for the wave equation. The key ingredients in proof of the main estimate are pointwise decay for the Fourier transform of the surface measure on the cone and sharp $L^3$ estimates for Wolff's maximal function.

  9월 19일 

연사: 이주영 박사(서울대)  오후 4시 

발표 제목: Critical weighted inequalities of the spherical maximal function

초록: Weighted inequality on the Hardy-Littlewood maximal function is completely understood while it is not well understood for the spherical maximal function. For the power weight $|x|^{\alpha}$, it is known that the spherical maximal operator on $\R^d$ is bounded on $L^p(|x|^{\alpha})$ only if $1-d\leq \alpha<(d-1)(p-1)-1$ and under this condition, it is known to be bounded except $\alpha=1-d$. In this paper, we prove the case of the critical order, $\alpha=1-d$.

연사: 정인지 교수(서울대)  오후 5시 

발표 제목: Degenerate dispersive equations and wavepackets

초록: We will first explain how degenerate dispersive PDEs appear in various areas of physics. Unlike their non-degenerate counterparts, wellposedness for those is a very subtle issue. This issue can be understood in terms of wavepackets, which are roughly speaking highly oscillatory approximate solutions. We explain the main ideas that go into the construction of wavepackets, and how to understand the behavior of solutions from them.

 2024년 4월 16일  

연사: 이주영 박사 (서울대) 4시 

발표 제목: A weighted decoupling inequality and its application to the maximal Bochner-Riesz problem

초록: We review the recent paper "A weighted decoupling inequality and its application to the maximal Bochner-Riesz problem" by Shengwen Gan and Shukun Wu. They obtained a weighted $l^p$-decoupling inequality when $p=2n/(n-1)$. Precisely, they restricted the domain in a smaller set and obtained a smaller decoupling constant. The authors obtained a sharp estimate when $n=2$, and some partial results when $n\geq 3$. As an application, we will see how this refined decoupling inequality gives an improvement on the maximal Bochner-Riesz problem.

 2024년 4월 16일  

연사: 이주영 박사 (서울대) 4시 

발표 제목: A weighted decoupling inequality and its application to the maximal Bochner-Riesz problem

초록: We review the recent paper "A weighted decoupling inequality and its application to the maximal Bochner-Riesz problem" by Shengwen Gan and Shukun Wu. They obtained a weighted $l^p$-decoupling inequality when $p=2n/(n-1)$. Precisely, they restricted the domain in a smaller set and obtained a smaller decoupling constant. The authors obtained a sharp estimate when $n=2$, and some partial results when $n\geq 3$. As an application, we will see how this refined decoupling inequality gives an improvement on the maximal Bochner-Riesz problem.