1. Martin Hairer, EPFL, Switzerland and Imperial College London, UK
TBA
2. Phan Dương Hiệu, Institut Polytechnique de Paris, France
Title: Cryptography under the Two-View Principle
Abstract: In this talk, I would like to share a personal perspective on how cryptography has developed through what I call the two-view principle: different observations on the same object. In particular: The design of cryptographic schemes often arises from the distinction between the mathematical view and the computational view of algebraic structures. Security is often achieved when an insider’s view can be simulated to appear indistinguishable from an outsider’s view, ensuring that no secret information is leaked. Breaking a scheme can be understood as a process of learning by an adversary, revealing a kind of duality between machine learnability and cryptographic hardness. Finally, I will discuss how the two-view principle has shaped my own recent research, particularly in the context of protecting user privacy against powerful adversaries - what we refer to as dictators - with a focus on emerging cryptographic primitives such as Anamorphic Cryptography.
3. Minhyong Kim, University of Edinburgh, UK
Title: Diophantine Equations in Two-Variables
Abstract: The equation y³ = x⁶ + 23x⁵ + 37x⁴ + 691x³ − 19450902x² + 5188193639 obviously has the solution (1, 1729). Are there any other solutions in rational numbers? The study of integral or rational solutions to polynomial equations, sometimes known as the theory of Diophantine equations, is among the oldest pursuits in mathematics. This lecture will give an idiosyncratic survey of the remarkable advances made in the 20th and 21st century for the special case of equations of two variables. The emphasis will be on the techniques of arithmetic geometry, the study of spaces built up of finitely-generated systems of numbers.
4. Phạm Tiến Sơn, Dalat University, Vietnam
Title: Generalized derivatives at infinity with applications in optimization
Abstract: In this talk, the notion of generalized derivatives at infinity is introduced. Various calculus rules for this notion are established. The main well-posedness properties (the linear openness, the metric regularity, and the Lipschitz-like continuity) at infinity as well as Mordukhovich's criterion at infinity are discussed. The obtained results are aimed ultimately at applications to diverse problems of optimization, such as optimality conditions, weak sharp minima, coercive properties and stability results. The talk is based on recent works in collaboration with Do Sang Kim, Nguyen Minh Tung and Nguyen Van Tuyen.
5. Lê Quý Thường, VNU University of Science, Hanoi, Vietnam
Title: Non-Archimedean geometry and applications to singularity theory
Abstract: It is known that common foundations for motivic integration and Berkovich spaces are provided by several theories on geometry over non-Archimedean valued fields. Main insights in this perspective come from modern model theory, along with methods from algebraic geometry. In particular, geometry over valued fields is closely combined with that over the residue field and the value group, which correspond to stable and o-minimal theories, respectively. This approach gives rise to surprising arguments to solve some fundamental open problems in algebraic geometry and singularity theory. In this talk, motivic integration, Berkovich geometry and model theory of algebraically closed valued fields are mentioned. Applications to singularity theory are various, from Hrushovski-Kazhdan’s to Cluckers-Loeser’s motivic integrals, from our equivariant motivic integration for varieties to that for formal schemes, which inherits Denef- Loeser’s idea on the classical motivic integration, and Berkovich’s nearby cycles functor together with comparison theorems in different contexts. The combination of these foundations carries out proofs of the integral identity conjecture, the motivic Thom- Sebastiani theorem, and many other interesting results.