Mathematics

The tumor invasion paradox relates to the artifact that a cancer that is exposed to increased cell death (for example through radiation), might spread and grow faster than before. The presence of cancer stem cells can convincingly explain this effect. In my talk I will use non-local and local reaction-diffusion type models to look at tumor growth and invasion speeds. We can show that in certain situations the invasion speed increases with increasing death rate - an invasion paradox (joint work with A. Shyntar and M. Rhodes).

In this work, improved swarm intelligent algorithms, namely, Salp Swarm Optimization algorithm, whale optimization, and Grasshopper Optimization Algorithm are proposed for data clustering. Our proposed algorithms utilize the crossover operator to obtain an improvised version of the existing algorithms. The performance of our suggested algorithms is tested by comparing the proposed algorithms with standard swarm intelligent algorithms and other existing algorithms in the literature. Non-parametric statistical test, the Friedman test, is applied to show the superiority of our proposed algorithms over other existing algorithms in the literature. The performance of our algorithms outperforms the performance of other algorithms for the data clustering problem in terms of computational time and accuracy.

This is a guest lecture in the PIMS Network Wide Graduate Course in Differential Equations in Algebraic Geometry.

This is a guest lecture in the PIMS Network Wide Graduate Course in Differential Equations in Algebraic Geometry.

The existence of Nash equilibria in the Mean Field Game (MFG) theory of large non-cooperative populations of stochastic dynamical agents is established by passing to the infinite population limit. Individual agent feedback strategies are obtained via the MFG equations consisting of (i) a McKean-Vlasov-Hamilton-Jacobi-Bellman equation generating the Nash values and the best response control actions, and (ii) a McKean-Vlasov-Fokker-Planck-Kolmogorov equation for the probability distribution of the state of a generic agent in the population, otherwise known as the mean field. The applications of MFG theory now extend from economics and finance to epidemiology and physics.

In current work, MFG and MF Control theory is extended to Graphon Mean Field Game (GMFG) and Graphon Mean Field Control (GMFC) theory. Very large scale networks linking dynamical agents are now ubiquitous, with examples being given by electrical power grids, the internet, financial networks and epidemiological and social networks. In this setting, the emergence of the graphon theory of infinite networks has enabled the formulation of the GMFG equations for which we have established the existence and uniqueness of solutions. Applications of GMFG and GMFC theory to systems on particular networks of interest are being investigated and computational methods developed. As in the case of MFG theory, it is the simplicity of the infinite population GMFG and GMFC strategies which, in principle, permits their application to otherwise intractable problems involving large populations on complex networks. Work with Minyi Huang