**What are the best topics to research in Pure and Applied Mathematics?**

Selecting the best Pure and Applied Mathematics project topic is important to the success of any final year student or scholar. After all, the more engaging and interesting the project topic, the more likely it is that students will be able to stay motivated on the task for the entire project. However, deciding which is right for you can be tough with many options.

We have made a list of simple pure and applied mathematics project topics and ideas for final year students to help you get started. From Differential Forms and Applications to Floquet Theory and Applications, these ideas will surely challenge and interest you. And if it’s hard for you to keep up with real-world project ideas, consider reading *Dealing With Problems in Mathematics: 14 Tips for Solving Maths* while thinking about simple, pure and applied mathematics project topics and ideas. This is because the project is often changed, so there is always a need to learn how to deal with maths problems.

**Simple Pure and Applied Mathematics Project Topics and Ideas**

**1. Sobolev Spaces and Variational Method Applied to Elliptic Partial Differential Equations **

Variational methods have proved important in studying the optimal shape, time, velocity, volume or energy. Laws exist in mechanics, physics, astronomy, economics and other fields of natural sciences.

**2. Characteristic Inequalities in Banach Spaces and Applications**

The contribution of this project falls within the general area of nonlinear functional analysis and applications. We focus on an important topic within this area: Inequalities in Banach spaces and applications.

**3. Differential Forms and Applications**

This project deals mainly with Differential Forms on smooth Riemannian manifolds and their applications through the properties of their classical Differential and Integral Operators. The calculus of Differential Forms.

**4. Evolution Equations and Applications**

This project concerns Evolution Equations in Banach spaces and lies at the interface between Functional Analysis, Dynamical Systems, Modeling Theory and Natural Sciences.

**5. Sobolev Spaces and Linear Elliptic Partial Differential Equations**

The cardinal goal of the study of the theory of Partial Differential Equations (PDEs) is to insure or find out properties of solutions of PDE that are not directly attainable by direct analytical means.

**6. Spectral Theory of Compact Linear Operators and Applications**

This Project primarily falls into Linear Functional Analysis and its Applications to Eigenvalue problems. It concerns the study of Compact Linear Operators (i.e., bounded linear operators)

**7. The Mountain Pass Theorem and Applications**

This project lies at the interface between Nonlinear Functional Analysis, unconstrained Optimization and Critical point theory. It concerns mainly Ambrosetti-Rabinowitz’s Mountain Pass Theorem which is a min-max.

**8. Isoperimetric Variational Techniques and Applications**

This project is at the interface between Nonlinear Functional Analysis, Convex Analysis and Differential Equations. It concerns one of the most powerful methods often used to solve optimization problems with constraints.

**9. Floquet Theory and Applications**

This project is at the interface between Analysis, Natural Sciences and Modeling Theory. It deals with Floquet Theory (also referred to as Floquet-Lyapunov theory) which is the main tool of the theory of periodic ordinary.

**10. Semigroups of Linear Operators and Application to Differential Equations**

This work concerns one of the most important tools to solve well-posed problems in the theory of evolution equations.

**11. Controllability and Stabilizability of Linear Systems in Hilbert Spaces**

Questions about controllability and stability arise in almost every dynamic system problem. As a result, controllability and stability are one of the most extensively studied subjects in system theory.

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