Evolutionary games of cyclic competitions have been extensively studied
to gain insights into one of the most fundamental phenomena in nature:
biodiversity that seems to be excluded by the principle of natural selection.
The Rock-Paper-Scissors (RPS) game of three species and
its extensions [e.g., the Rock-Paper-Scissors-Lizard-Spock (RPSLS) game] are
paradigmatic models in this field.
In all previous studies, the intrinsic symmetry associated with the cyclic competitions imposes
a limitation on the resulting coexistence states, leading to only selective types of such states.
We investigate the effect of nonuniform intraspecific competitions on coexistence and
find that a wider spectrum of coexistence states can emerge and persist.
This surprising finding is substantiated using three classes of cyclic game models through stability analysis,
Monte Carlo simulations and patterns of continuous spatiotemporal dynamical evolution.
Our finding indicates that intraspecific competitions or alternative symmetry-breaking mechanisms
can promote biodiversity to a broader extent than previously thought.

재미로 풀어보는 퀴즈에나 등장할 법한 추상적인 수학적 개념이 기계공학(예, 응용역학) 연구에 도움을 줄 수 있을까? 수학과 역학 사이의 간극이 가장 좁았던 때는 언제였고, 수학과 역학이 만나는 지점에서 두 학문을 두루 섭렵했던 수리과학자는 누구였을까? 이와 같은 질문에 대한 답변의 일환으로, 본 발표의 전반부에서는 수학과 역학(유체역학, 고체역학, 열역학, 파동학)의 역사가 공존했던 시절을 인물 중심으로 살펴보고자 한다. 본 발표의 후반부에서는, 역학적 파동과 메타물질에 관한 발표자의 연구주제(음향 투명망토, 음향 블랙홀, 생물음향학 등)를 간략하게 소개한다.

Many problems in control and optimization require the treatment

of systems in which continuous dynamics and discrete events

coexist. This talk presents a survey on some of our recent work on such

systems. In the setup, the discrete event is given by a random

process with a finite state space, and the continuous component is the

solution of a stochastic differential equation. Seemingly similar

to diffusions, the processes have a number of salient features

distinctly different from diffusion processes. After providing

motivational examples arising from wireless communications,

identification, finance, singular perturbed Markovian systems,

manufacturing, and consensus controls, we present necessary and

sufficient conditions for the existence of unique invariant

measure, stability, stabilization, and numerical solutions of

control and game problems.

Liquid crystal is a state of matter between isotropic fluid and crystalline solid, which has properties of both liquid and solid. In a liquid crystal phase, molecules tend to align a preferred direction and molecules are described by a symmetric traceless 3x3 matrix which is often called a second order tensor. Equilibrium states are corresponding to minimizers of the governing Landau-de Gennes energy which plays an important role in mathematical theory of liquid crystals. In this talk, I will present a brief introduction to Landau-de Gennes theory and recent development of mathematical theory together with interesting mathematical questions.