Dynamics of many particle system can be described by PDE of probability density function. The Boltzmann equation in kinetic theory is one of the most famous equation which describes rarefied gas dynamics. One of main property of the Boltzman...

Machine learning (ML) has achieved unprecedented empirical success in diverse applications. It now has been applied to solve scientific problems, which has become an emerging field, Scientific Machine Learning (SciML). Many ML techniques, h...

This lecture explores the topics and areas that have guided my research in computational mathematics and deep learning in recent years. Numerical methods in computational science are essential for comprehending real-world phenomena, and dee...

학부에서 왜 abstract algebra I, II 를 온전히 배워야 하는지를 BC 5세기경 Pythagoras로 부터 시작된 수론 문제가 현재까지 어떻게 발전되어 왔는지를 예를 들어 설명합니다.

We will review a number of topological themes in number theory, starting with homology and ending with a discussion arithmetic homotopy.

The mean curvature flow is an evolution of hypersurfaces satisfying a geometric heat equation. The flow naturally develops singularities and changes the topology of the hypersurfaces at singularities, Therefore, one can study topological pr...

One of the important problems in understanding large and complex data sets is how to provide useful representations of a data set. We will discuss some existing methods, as well as topological mapping methods which use simplicial complexes ...

After a review of various types of tilings and aperiodic materials, the notion of tiling space (or Hull) will be defined. The action of the translation group makes it a dynamical system. Various local properties, such as the notion of "Finit...

I will talk in general about theory and applications of partial differential equations. A recent progress in the regularity theory for nonlinear problems will be also discussed, including uniform estimates of solutions in various function sp...

The significance of dimensions in mathematics

Creating information and knowledge from large and complex data sets is one the fundamental intellectual challenges currently being faced by the mathematical sciences. One approach to this problem comes from the mathematical subdiscipline cal...

The revolution of molecular biology in the early 1980s has revealed complex network of non-linear and stochastic biochemical interactions underlying biological systems. To understand this complex system, mathematical models have been widely ...

In many applications such as X-ray Crystallography, imaging, communication and others, one must construct a function/signal from only the magnitude of the measurements. These measurements can be, for example, the Fourier transform of the den...

Since Bose and Einstein discovered the condensation of Bose gas, which we now call Bose-Einstein condensation, its mathematical properties have been of great importance for mathematical physics. Recently, many rigorous results have been obta...

The Lagrange spectrum is the set of approximation constants in the Diophantine approximation for badly approximated numbers. It is closely related with the Markov spectrum which corresponds the minimum values of indefinite quadratic forms ov...

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Soon after Gromov’s applications of pseudo-holomorphic curves to symplectic topology, Floer invented an infinite-dimensional Morse theory by analyzing moduli spaces of pseudo-holomorphic curves to make substantial progress on Arnold&r...

Symplectic geometry arose from the study of classical mechanics, and later many interesting symplectic invariants has been found since Gromov introduced techniques of J-holomorphic curves. Miraculously, such invariants are closely related wi...

Symmetry Breaking in Quasi-1D Coulomb Systems

It is usually a difficult problem to characterize precisely which elements of a given integral domain can be written as a sum of squares of elements from the integral domain. Let R denote the ring of integers in a quadratic number field. Thi...