twistor theory and hodge theory
Twistor theory was proposed by Sir. Penrose. Since then，Penrose correspondence becomes an important tool in differential geometry and algebraic geometry。In this talk，we will introduce various applications of twistor theory and recent progresses related to Hodge theory.
Dynamic behavior of stochastic epidemic models with Markov switching
Epidemic diseases, including plague, cholera, AIDS and bird flu, have always been a major problem affecting the life and survival of all human beings. By using deterministic differential equations to establish a mathematical model, epidemiology describes and studies the internal transmission laws of diseases, predicts disease trends and gives the optimal control strategy. However, there always exists noise in the environment. Whether it is the change of light intensity in a day or the change of temperature and humidity caused by the alternation of seasons, it will affect the transmission rate of diseases. Random noise is generally divided into two types: one is the sum of many independent tiny random noises, which are usually described in mathematics by white noise; the other is random noise with a small number but a large intensity, such color noise will cause the system to switch between different states and is usually described by continuous-time Markov chains. Based on deterministic epidemic models, scholars established the stochastic epidemic model driven by a mixture of white noise and color noise. They achieved many important results by using the theories of stochastic differential equation and properties of hybrid diffusion system. By mathematical modeling, stochastic analysis and other methods, the SIRS stochastic epidemic models under different noises with nonlinear transmission rate and vaccination are studied in this paper. We further study the SIS epidemic model in complex network under the influence of color noise, and give an improved numerical method. This dissertation is a preliminary study on the dynamic modeling of epidemic with Markov switching.