# 학과 세미나 및 콜로퀴엄

구글 Calendar나 iPhone 등에서 구독하면 세미나 시작 전에 알림을 받을 수 있습니다.

We introduce a distribution-theoretic conjecture of Roert Coleman of the 1980's and prove the conjecture in a recent joint work with Burns and Daoud. This accordingly gives an explicit description of the complete set of Euler systems for the multiplicative group over Q together with a connection to other conjectures in number theory.

I will give an introduction to the Monstrous moonshine conjectures of 70's-80's, which are on remarkable relations between Klein's j-invariant in number theory and the Monster sporadic simple group. I will only assume mild basic knowledge of complex analysis and group theory. I will start from a brief introduction to modular forms and Hauptmoduln, then connect it to finite simple groups. If I can manage the time, I will briefly explain a hint to a connection to the 3d gravity theory.

Singular perturbations occur when a small coefficient affects the highest order derivatives in a system of partial differential equations. From the physical point of view, singular perturbations generate thin layers near the boundary of a domain, called boundary layers, where many important physical phenomena occur. In fluid mechanics, the Navier-Stokes equations, which describe the behavior of viscous flows, appear as a singular perturbation of the Euler equations for inviscid flows, where the small perturbation parameter is the viscosity. In general, verifying the convergence of the Navier-Stokes solutions to the Euler solution (known as the vanishing viscosity limit problem) remains an outstanding open question in mathematical physics. Up to now, it is not known if this vanishing viscosity limit holds true or not, even in 2D for which the existence, uniqueness, and regularity of solutions for all time are known for both the Navier-Stokes and Euler. In this talk, we discuss a recent result on the boundary layer analysis for the Navier-Stokes equations under a certain symmetry where the complete structure of boundary layers, vanishing viscosity limit, and vorticity accumulation on the boundary are investigated by using the method of correctors. We also discuss how to implement effective numerical schemes for slightly viscous fluid equations where the boundary layer correctors play essential roles. This is a joint work in part with J. Kelliher, M. Lopes Filho, A. Mazzucato, and H. Nussenzveig Lopes, and with C.-Y. Jung and H. Lee.

The new infectious disease are emerging around the world. Coronavirus disease 2019 (COVID-19) caused by a novel coronavirus has emerged and has been rapidly spreading. The World Health Organization (WHO) declared the COVID-19 outbreak a global pandemic on March 11, 2020. Mathematical modelling plays a key role in interpreting the epidemiological data on the outbreak of infectious disease. Moreover, mathematical modeling can give us an early warning about the size of the outbreak. First, we construct a mathematical model to estimate the effective reproduction numbers, which assess the effect of control interventions. Second, we forecast the COVID-19 cases according to the different effect of control interventions. Finally, the most effective intervention can be suggested in terms of modeling approach. In this talk, I’d like to briefly introduce the main results of recent research on the mathematical modeling for various infectious diseases.

Liouville quantum gravity (LQG) surfaces are random topological surfaces which are important in statistical mechanics and have deep connections to other mathematical objects such as Schramm–Loewner evolution and random planar maps. These random surfaces are too singular and fractal in the sense that the Hausdorff dimension, viewed as a metric space equipped with its intrinsic metric, is strictly bigger than two. I will talk about the interesting geometric structure and recent progress on LQG surfaces.