2024-09-03
(announcements and presentation by NTHU representative)
Course / colloquium archive · Fall 2024
NYCU Math Colloquium
Fall 2024 NYCU Math Colloquium
National Yang Ming Chiao Tung University
Scheduled talks
Unless otherwise announced, all talks take place between 14:00 and 15:00, in room SA213.
2024-09-10
Tien Nguyen (NTU)
Singularity formation in the complex Ginzburg-Landau equation
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.
2024-09-17
(holiday, no talk)
2024-09-24
Jenn-Nan Wang (NTU)
Stability and instability estimates for inverse problems
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 Calderon’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 Calderon’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.
2024-10-01
Sze Yiu Chau (CUHK)
RSA Signature Forgery Attacks Against Weak Implementations
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.
2024-10-08
TBD
2024-10-15
Daniel Spector (NTNU)
The DiPerna-Majda 2D Gap Problem
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 ω := curl u : ℝ² × [0, T] → ℝ the vorticity and ω₀ := curl u₀ : ℝ² → ℝ the initial vorticity. Briefly, they proved that for vorticities bounded in an α log-Morrey space one does not have any concentration for α > 1, while for α ≤ 1/2 one may have concentration-cancellation. The interval α ∈ (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 α = 1.
2024-10-22
TBD
2024-10-29
TBD
2024-11-05
(canceled)
2024-11-12
Yat Hin Marco Suen (NCKU)
An introduction to mirror symmetry
Mirror symmetry is a duality between complex and symplectic geometry. In 1994, Kontsevich proposed a mathematical definition for mirror symmetry which is now known as homological mirror symmetry (HMS). HMS predicts that the Fukaya category of a symplectic manifold is quasi-equivalent to the derived category of its mirror complex manifold. Despite HMS has been proven in many interesting cases, it's usually hard to give an exact geometric correspondence between objects due to its homological nature. Two years after Kontsevich's proposal, Strominger-Yau-Zaslow introduced an entirely geometric approach to mirror symmetry, which is now known as the SYZ proposal. SYZ suggested that mirror pairs can be obtained by taking dual torus fibration and the mirror functor in HMS can be obtained by a Fourier-Mukai-type transform. In this talk, I would like to introduce mirror symmetry from the SYZ perspective. If time permits, I will talk about realization problems in tropical geometry.
2024-11-19
Pak-Yeung Chan (NTHU)
Ricci flow and Ricci soliton
Ricci flow was introduced by Hamilton in 1982 to study the topology of 3-manifolds. It can be viewed as a heat flow of the Riemannian metric which evolves in the direction of its Ricci curvature. Ricci soliton is a self similar solution to the Ricci flow and often models the singularity of the flow. In this talk, we will discuss some basic concepts on the Ricci flow and Ricci soliton. Some of the results are based on joint works with Zilu Ma and Yongjia Zhang, also with Man-Chun Lee and Luke Peachey.
2024-11-26
Yen-Chang Huang (NUT)
From Theory to Practice: Industrial Applications of Integral Geometry
數學在科學、工程、醫學、經濟和金融等領域有著廣泛的應用，這些應用通常被歸類為應用數學。它不僅能幫助其他學科解決實際問題，有時還會促成新的數學分支或成果的誕生。本次演講將分享個人的應用經驗，涵蓋如何將凸幾何的重要定理結合影像視覺辨識技術，用於產業中的物件量測，為業主需求提供理論基礎。同時，也會討論利用變分法（variational method）解決影像邊框勾勒的實際案例，並分享與產業合作中面對的技術挑戰及解決方法的經驗。
Nicolau Aiex (NTNU)
Minimal surfaces and Calculus of Variations
We will go over many different flavours of the basics of minimal surfaces and see how it connects to calculus of variations. In doing so we will touch on Morse theory of infinite dimensional spaces and the relation between topology and critical points of geometric functionals. We will present many examples and eventually reach some of the modern open questions in the theory.
2024-12-03
Chee Han Tan (NSYSU)
An isoperimetric sloshing problem in shallow containers
The classical isoperimetric inequality asserts that the sphere encloses the largest volume with a given surface area, but we can pretty much ask the problem of optimising any quantity with a given geometrical constraint. We will review existing results for the problem of optimising the kth-eigenvalue of the Laplacian with various boundary conditions subject to volume constraint. We will then discuss the problem of optimising the first nonzero eigenvalue of a certain coupled ODE that is motivated by the sloshing problem in fluid dynamics.
Harry Richman (NCTS)
Perspectives on tropical geometry, old and new
Tropical geometry is a way to approach geometry at "extreme magnitudes", and to work with "combinatorial shadows" of algebraic varieties. It is a recent branch of mathematics, but its central ideas can be traced back much farther. I will present some ideas underlying tropical mathematics, starting with two tools for solving for roots of a single-variable polynomial equation: Descartes's "rule of signs" and Newton polygons. In more modern mathematics, these tools are extended to "higher dimension" versions and used for understanding moduli spaces of varieties and their degenerations. I will describe these ideas through examples and cover some current developments.
2024-12-10
Colin McSwiggen (Academia Sinica)
Calibration and continuity
A statistical model is said to be calibrated if it has the appropriate level of confidence in its own predictions: that is, the confidence that it assigns to a predicted outcome should accurately reflect that outcome's likelihood. For example, if a weather model is calibrated, then out of all of the days when the model predicts a 30% chance of rain, we should expect that it actually will rain on 30% of them. Calibration is crucial for managing the risks associated with incorrect predictions, but modern deep learning models are systematically miscalibrated: they are overconfident when they are incorrect. To make matters worse, theorists can't agree about how miscalibration should be quantified! The prevailing miscalibration metric in engineering applications is the expected calibration error (ECE), which has been widely criticized because it is discontinuous: a tiny change in the model can lead to a large change in the error. In this talk, I'll try to convince you that this problem isn't really a problem, that ECE was fine all along, and that engineers should feel free to keep using it the way they always have (at least for binary classification tasks). The argument will require us to answer a strange but fundamental question about the topological properties of the conditional expectation operator.
Charlotte Pollet (NYCU)
Genealogy of Diagrams and Equations in pre-modern China
The study of mathematics in China has often focused on the ‘procedure of the Celestial Source’ (天元術tian yuan shu), which is used to set up polynomial equations. The geometrical ancestors of this procedure are less known. The 益古演段Yigu yanduan, authored by the 13 th century mathematician, 李冶Li Ye (1192–1279), however, presents the procedure alongside its two geometrical counterparts, the ‘Section of Pieces [of Areas]’ (條段tiao duan) and the ‘Old Procedure’ (舊術jiu shu). The three procedures are known to represent three generations of algorithms to set up quadratic equations. This presentation aims to make the geometrical procedure ‘speak’ about its genealogy. That is to say, to attempt the reconstruction of the evolution of the geometrical roots of the famous procedure of the Celestial Source. The construction of negative coefficients plays a pivotal role in this evolution. It also is possible to distinguish several layers of composition that reflect several episodes in the development of the quadratic equation with negative coefficients. In other words, this analysis raises a philological problem pertaining to the question of textual transmission and the nature of authorship. This is the opportunity to show a landscape of history of mathematics in China, from its sources to its connection to Taoism.
2024-12-17
Ying-Jen Yang (Laufer Center, Stony Brook University)
Temporal Symmetry and Convex Conjugacy in Stochastic Processes: Insights from Thermodynamics and Statistical Physics
Theoretical physics has long inspired advancements in both mathematical modeling and the development of mathematics itself. For example, dynamical systems theory emerged from classical mechanics, and information theory was inspired by thermodynamics. In this talk, I will give an overview on the advancements in stochastic processes mathematics from physicists’ recent attempts to generalize Thermodynamics and Statistical Physics to processes that break time-reversal symmetry (nonequilibrium)1, such as those in biology, neuroscience, and ecology. I will then introduce my own work to this endeavor2,3: we formulate Two Laws in analogous to the First and Second laws of thermodynamics, with the first prescribing the fundamental components of stochastic dynamics base on temporal symmetry and conservation law and the second as a variation principle that identifies variables (convex) conjugated to the fundamental dynamical components. I will conclude with examples and outlook.
Student requirements
In order to pass the class, students must attend most lectures (should not miss more than 2 lectures). There will be a sign-in and a sign-out sheet every time in order to record attendance. At the end of the semester students must submit a report of their experience, at least one page long. The report must be written in English.
Original page
Archived course information. Dates and announcements refer to the semester shown above.