Novel Techniques for Classifying Exotic Spheres in High Dimensions

Discrete calculus deals with developing the concepts and techniques of differential and integral calculus in a discrete setting, often using difference equations and discrete function spaces. This paper explores how differential-difference algebra can provide an algebraic framework for advancing discrete calculus. Differential-difference algebra studies algebraic structures equipped with both differential and difference operators. These hybrid algebraic systems unify continuous and discrete analogues of derivatives and shifts. This allows the development of general theorems and properties that cover both settings. In particular, we construct differential-difference polynomial rings and fields over discrete function spaces. We define discrete derivatives and shifts algebraically using these operators. We then study integration, summation formulas, fundamental theorems, and discrete analogues of multivariate calculus concepts from an algebraic perspective. A key benefit is being able to state unified theorems in differential-difference algebra that simultaneously yield results for both the continuous and discrete cases. This provides new tools and insights for discrete calculus using modern algebraic techniques. We also discuss applications of representing discrete calculus problems in differential-difference algebras. This allows bringing to bear algebraic methods and software tools for their solution. Specific examples are provided in areas such as numerical analysis of discrete dynamical systems defined through difference equations. The paper aims to demonstrate the capabilities of differential-difference algebra as a unifying framework for further developing the foundations and applications of discrete calculus. Broader connections to algebraic modeling of discrete physical systems are also discussed.