Geometric quantization is an attempt to use the differential-geometric properties of a classical phase pace assumed to be a symplectic manifold M in order to define a corresponding quantum theory. In this talk, I will give an introduction to geometric quantization on symplectic manifold. In particular, I will focus on Kähler manifold endowed with T-symmetry. This is a joint work with Conan Leung.
In applied mathematics, effective problem solving begins with precise problem formulation, highlighting the importance of the initial problem-setting phase. Without a clearly defined problem, identifying suitable tools and techniques for resolution becomes arduous and often futile. This transition from problem setting to problem solving is pivotal within the broader framework of knowledge advancement. Despite the remarkable progress of AI tools, they remain reliant on the groundwork laid by human intelligence. Mathematicians, leveraging their adeptness in discerning patterns and relationships within data and variables, play a crucial role during this phase. This lecture will introduce fundamental mathematical concepts encompassing both traditional machine learning and scientific machine learning. The latter offers an optimal platform for the harmonious fusion of problem setting and problem solving, bolstered by profound domain expertise.
We begin with a simply stated problem in discrete geometry: at least how many distinct dot products must be determined by a large finite set of points in the plane? This is related to some well-studied problems of Erdos about distances. The distance problems have celebrated variants in the fractal setting, such as the Falconer distance problem, which have seen significant progress in recent years. However, the analogous problems for dot products in the fractal setting have not moved past the most fundamental results. We discuss the barriers to current methods, in hopes of motivating new approaches to overcome these barriers.
We derive the general structure for returning to the steady macroscopic nonequilibrium condition, assuming a first-order relaxation equation obtained as zero-cost flow for the Lagrangian governing the dynamical fluctuations. The main ingredient is local detailed balance from which a canonical form of the time-symmetric fluctuation contribution (aka frenesy) can be obtained. That determines the macroscopic evolution. As a consequence, the linear response around stationary nonequilibrium gets connected with the small dynamical fluctuations, leading to fluctuation-response relations. Those results may be viewed as nonequilibrium extension of the well-known structure where the relaxation to equilibrium is characterized by a (dissipative) gradient flow on top of a Hamiltonian motion.
