陽明交通大學應用數學系

Abstract: In a series of influential papers in the 1980s, DiPerna and Majda introduced a rigorous framework of approximate solutions of the Euler equations and proved several results concerning concentration of solutions related to hypothesis on $\omega:= \operatorname*{curl}u:\mathbb{R}^2 \times [0, T] \to \mathbb{R}$ the \emph{vorticity} and $\omega_0:= \operatorname*{curl}u_0:\mathbb{R}^2 \to \mathbb{R}$ the \emph{initial vorticity}. Briefly, they proved that for vorticities bounded in an $\alpha$ log-Morrey space one does not have any concentration for $\alpha>1$, while for $\alpha\leq 1/2$ one may have \emph{concentration-cancellation}. The interval $\alpha \in (1/2,1]$ remained an open question in their paper and subsequent papers, whether one can rule out concentration or find sequences which admit concentration-cancellation. In this talk I discuss a recent result in collaboration with Oscar Dominguez in which we resolve this question, closing the gap, showing in particular that one may have concentration-cancellation up to $\alpha=1$.

Abstract
RSA signature is a cornerstone of network security, widely used by different systems and applications as the means of achieving cryptographic authentication guarantees. In this talk, we will revisit the problem of verifying RSA signatures. Using different techniques, our recent research revealed many instances of unwarranted leniency in implementations of RSA signature verifiers. Critically, our findings suggest that many systems are susceptible to variants of the Bleichenbacher-style RSA signature forgery attack. We will look at how this attack, enabled by weak implementations, can nullify the security guarantees promised by the underlying cryptography, and discuss how this threat can be mitigated in practice.

Abstract
According to Hadamard’s definition, a well-posed problem satisfies three criteria: existence, uniqueness, and continuous dependence on the data. Most of forward problems (e.g., the boundary value problem or Calder´on’s problem) can be proved to be well-posed. However, many inverse problems are known to be ill-posed, for example, the inverse boundary value problem in which one would like to determine unknown parameters from the boundary measurements. The failure of the continuous dependence on the data in Hadamard’s sense makes the feasible determination of unknown parameters rather difficult in practice. However, if one restricts the unknown parameters in a suitable subspace, one can restore the continuous dependence or stability. Nonetheless, the ill-posedness nature of the inverse problem may give rise a logarithmic type modulus of continuity. For Calder´on’s problem, such logarithmic stability estimate was derived by Alessandrini and Mandache showed that this estimate is optimal by proving an instability estimate of exponential type. When we consider the time-harmonic equation, it was first proved by Isakov that the stability increases as the frequency increases. In this talk, I would like to discuss a refinement of Mandache’s idea aiming to derive explicitly the dependence of the instability estimate on the frequency. If time allows, I also want to discuss the increasing stability phenomenon from the statistical viewpoint based on the Bayes approach. The aim is to show that the posterior distribution contracts around the true parameter at a rate closely related to the decreasing instability estimate derived above.

Abstract
Many central problems in geometry, mathematical physics, and biology are reduced to questions regarding the behavior of solutions of nonlinear evolution equations. The global dynamical behavior of bounded solutions for large times is of significant interest. However, in many real situations, solutions develop singularities in finite time. The singularities must be analyzed in detail before attempting to extend solutions beyond their singularities or to understand their geometry in conjunction with globally bounded solutions. In this context, we have been particularly interested in qualitative descriptions of blowup. This talk considers the complex Ginzburg-Landau equation as a particular model for which I will present constructive methods based on spectral analysis and energy-type estimates for the existence and stability of blowup solutions.

Abstract
Given a set of lines in R^d, a joint is a point contained in d linearly independent lines. Guth and Katz showed that N lines can determine at most O(N^{3/2}) joints in R^3 via the polynomial method.
Yu and I proved a tight bound on this problem, which also solves a conjecture proposed by Bollob\'as and Eccles on the partial shadow problem. It is surprising to us that the only known proof to this purely extremal graph theoretic problem uses incidence geometry and the polynomial method.

Abstract. Nonlinear Schrödinger equations are a widely studied topic due to their various applications in nonlinear optics and Bose–Einstein condensates. The semiclassical analysis serves as a link between quantum mechanics and classical mechanics, providing insight into how classical physics emerges from quantum mechanical principles, with Planck's constant playing a crucial role.

In this talk, I will first apply Madelung's transform to the Schrödinger equation and formally derive the hydrodynamic equations. Then, I will introduce a mathematical method for investigating the semiclassical limits of Schrödinger equations. The focus will be on introducing a modulated energy functional to govern mass density and linear momentum, particularly in relation to the compressible Euler equations. Finally, I will introduce my ongoing work about a nonlocal Schrödinger equation, drawing connections to a nonlocal compressible Euler equation.

Abstract. We consider the hard-capacitated facility location problem with uniform facility cost (CFL-UFC). This problem arises as an indicator variation between the general CFL problem and the uncapacitated facility location (UFL) problem, and is related to the profound capacitated k-median problem (CKM). In this work, we present a rounding-based 4-approximation algorithm for this problem, built on a two-staged rounding scheme that incorporates a set of novel ideas and also techniques developed in the past for both facility location and capacitated covering problems. Our result
improves the decades-old LP-based ratio of 5 for this problem due to Levi et al. since 2004. We believe that the techniques developed in this work are of independent interests and may further lead to insights and implications for related problems.

Abstract
Pressure-induced drag and lift are key to the performance of wings, rotors and propellers; undulating fins and flapping wings generate forces that are key to locomotion in fish, birds and insects; time-varying fluid dynamic forces drive flutter and flow-induced vibrations of flexible structures in engineering and biology, and these same forces enable the extraction of energy from flow via devices such as wind-turbines. Pressure on a body immersed in a flow is however induced simultaneously by vortices, acceleration reaction (a.k.a. added mass) effects associated with body and/or flow acceleration, and viscous diffusion of momentum, and determining the relative contribution of these different mechanisms on surface pressure remains one of the most important and fundamental issues in fluid dynamics. I will describe the force partitioning method (FPM), a new data-enabled method that partitions pressure forces into components due to vorticity, acceleration reaction and viscous diffusion. FPM has been used to gain new insights into a variety of vortex dominated flows including dynamic stall in pitching foils, vortex-induced vibration of bluff-bodies, hydrodynamics of schooling fish and rough-wall boundary layers, and results from these analyses will be presented. Application of FPM to data generated from experiments will also be described. Finally, FPM has been extended to aeroacoustics, and applications of the aeroacoustic partitioning method (APM) to dissect aeroacoustic noise in engineering and biological flows will be presented.