An Operator Semigroup in Mathematical Genetics 1 Introduction 2 Genetic background 3 Motivating example 4 Mathematical tools 5 Master Equation 6 Epilogue This authored monograph presents a mathematical description of the time evolution of neutral genomic regions in terms of the differential Lyapunov equation. The qualitative behavior of its solutions with respect to different mutation models and demographic patterns can be characterized using operator semi group theory. Mutation and drift are two of the main genetic forces which act on genes of individuals in populations. Their effects are influenced by population dynamics. This book covers the application to two mutation models: single step mutation for microsatellite loci and single base substitutions. The effects of demographic change to the asymptotic of the distribution are also covered. The target audience primarily covers researchers and experts in the field but the book may also be beneficial for graduate students