"Advances in Mathematics Research" presents original studies on the leading edge of mathematics. Each article has been carefully selected in an attempt to present substantial research results across a broad spectrum. Chapter One summarizes the phase/current generalized measures of the entropy/information content in complex quantum states of molecular systems. Chapter Two reviews the current knowledge regarding Mavridis’ area (MA), with emphasis on the role of applied mathematics in its discovery, as well as to explore its mathematical expression. In Chapter Three a model of fractional difference has been defined by the author as a fractional Newton binomial with respect to the finite difference operator as parameter, therefore they obtained an alternative to fractional derivative, and further, as a by-product, they came across the so-called modified Riemann-Liouville derivative which ascribes a special role to the initial value of the considered function. Chapter Four presents some popular uses of exponential distribution in the context of ordered random variables. Chapter Five gives a comprehensive introduction to the Ricci flow on manifolds of dimension two which can be done in a reasonable fashion when the Euler characteristic is negative or zero. Chapter Six investigates some geometric properties by using the concepts of the geometric function theory and studies the convexity and star-like for the new operator.