Fokker-Planck and Kolmogorov (backward) equations can be interpreted as linearisations of the underlying stochastic differential equations (SDE). It turns out that, in particular, on infinite dimensional spaces (i.e. for example if the SDE i...

Counting circles in Apollonian circle packings and beyond

We consider different growth rates associated with the geometry (distance, volume, heat kernel) on a cover of a compact Riemannian manifold. We present general inequalities. We discuss the rigidity results and questions in the case of negati...

The talk will review a new approach to the limits of the quantum N-body dynamics leading to the Hartree equation (in the large N limit) and to the Liouville equation (in the small Planck constant limit). This new strategy for studying both l...

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...

Modern mathematics with axiomatic systems has been developed to create a complete reasoning system. This was one of the most exciting mathematical experiments. However, even after the failure of the experiment, mathematical research is still...

In this talk, we discuss some results of convex and non-convex optimization methods in image processing. Examples including image colorization, blind decovolution and impulse noise removal are presented to demonstrate these methods. Their a...

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...

Gromov introduced the analytic method of pseudoholomorphic curves into the study of symplectic topology in the mid 80's and then Floer broke the conformal symmetry of the equation by twisting the equation by Hamiltonian vector fields. We sur...

We introduce a conformally invariant nonlinear sigma model on the bulk of contact manifolds with boundary condition on the Legendrian links in any odd dimension. We call any finite energy solution a contact instanton. We also explain its Ha...

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...

Introduction to Non-Positively Curved Groups

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

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 consider the problem of identifying the material properties from boundary measurements. For the conductivity case, this is known as Calderon problem: “Is it possible to determine the electrical conductivity inside a domain from the bounda...

Magnetic Resonance Electrical Impedance Tomography (MREIT) is a late medical imaging modality visualizing static conductivity images of electrically conducting subjects. When we inject current into the object, it produces internal distributi...

Diffusion of information, rumors or epidemics via various social networks has been extensively studied for decades. In particular, Kempe, Kleinberg, and Tardos (KDD '03) proposed the general threshold model, a generalization of many mathemat...

Studies on PDEs are mostly focused on ?nding properties of PDEs within a speci?c discipline and on developing a technique specialized to them. However, ?nding a common structure over di?erent disciplines and unifying theories from di?erent s...

Normal form method is a classical ODE technique begun by H. Poincare. Via a suitable transformation one reduce a differential equation to a simpler form, where most of nonresonant terms are cancelled. In this talk, I begin to explain the not...

The notion of essential dimension was introduced by Buhler and Reichstein in the late 90s. Roughly speaking, the essential dimension of an algebraic object is the minimal number of algebraically independent parameters one needs to define the...