김준일 (연세대학교)
Title: Discrete Two-parameter Exponential Sums
Abstract: In the first part of the lectures, we will investigate the decay properties of oscillatory integrals and exponential sums associated with frequencies ξ. To build a foundational understanding, we will explore the Dirichlet Approximation Theorem, which allows us to approximate ξ by a rational number a/q. A key insight lies in recognizing that the size of the denominator q in such approximations plays a critical role in determining these decay properties of the exponential sums. This understanding naturally leads us to examine Weyl sums and Gauss sum estimates, essential tools for exponential sums.
In the second part, we turn our attention to Bourgain’s groundbreaking generalization of the Birkhoff pointwise ergodic theorem. We will study his framework emphasizing the main roles of major and minor arcs in the Hardy-Littlewood-Ramanujan circle method. This method provides a powerful approach for handling exponential sums associated with discrete maximal and singular Radon transforms associated with curves.
Finally, we will extend our exploration to the circle method in a two-parameter setting, focusing on exponential sums over lattice boxes of arbitrary size. With this setting, we can prove the boundedness property of two-parameter maximal functions and Hilbert transforms. As a final goal of this lecture series, I will present a sketch of the proof for a recent result (joint work with Hoyoung) on the multi-parameter exponential sums associated with discrete maximal functions and double Hilbert transforms. This result highlights the intricate interplay between oscillatory behavior and the geometry of lattice structures.
오창근 (서울대학교)
Title: Discrete restriction estimate for the parabola
Abstract: In these series of talks, I'll present a discrete restriction estimate for the parabola. For the first part, I'll present Bourgain's lower bound of the discrete restriction estimate for the parabola. As an ingredient of Bourgain's proof, we'll study basic properties of Mobius function and Euler phi function. For the second part of the proof, we'll prove an upper bound of the discrete restriction estimate for the parabola. A decoupling method is one of the method to study the discrete restriction estimate. In 2014, a decoupling inequality for the parabola was proved by Bourgain and Demeter. Since their work, many different proofs of the decoupling inequality for the parabola have been found. Among them, the proof using the high-low method is particularly notable because it provides the current best bound for the decoupling constant for the parabola. I will give a simplified proof of a decoupling inequality for the parabola using a high-low method.
고혜림 (전북대학교)
Title: Maximal estimates for orthonormal systems of wave equations
Abstract: In this talk, we study the maximal estimates for wave operators defined on orthonormal families of initial data. We explore necessary conditions for these estimates and present some partial progress in two and three dimensional settings. This is joint work with Shinya Kinoshita and Shobu Shiraki.
오세욱 (고등과학원)
Title: Maximal estimates for averages over degenerate hypersurfaces
Abstract: In this talk, we study the Lp boundedness of maximal averages over hypersurfaces. I will explain the connection between L^p maximal bounds and the Fourier decay. Additionally, I will present my recent findings, which establish that the maximal average is L^p-bounded when the Fourier decay rate is 1/2.
유재현 (고등과학원)
Title: Oscillatory integral operators of degeneracies
Abstract: We study the boundedness properties of oscillatory integral operators whose associated canonical relations have singularities in their projections onto the right and left cotangent bundles. These singularities include one-sided and two-sided fold singularities and Morin singularities. While L^p estimates for operators with folding canonical relations are now well understood, the behavior of operators with more general types of singularities remains largely unexplored. We also discuss the applications of these results to restricted X-ray transforms along curved lines.
조주희 (서울대학교)
Title: On dimension-free variational inequalities for averaging operators in R^d
Abstract: In this talk, we review the paper "On Dimension-Free Variational Inequalities for Averaging Operators in R^d" by Jean Bourgain, Mariusz Mirek, Elias M. Stein, and Błażej Wróbel. We introduce variation norm inequality and discuss dimension-free L^p inequalities for r-variations of Hardy–Littlewood averaging operators over symmetric convex bodies.