A PhD in mathematics with a focus on the defense sector, combined with computer science, can open up a wide range of opportunities for interdisciplinary research and innovation. Here are some potential research areas within the defense sector where a combined PhD in mathematics and computer science could be applicable:

Cybersecurity and network defense: Developing advanced mathematical algorithms and computational techniques for securing computer networks, systems, and data in defense and military contexts, including topics such as intrusion detection, threat analysis, and network resiliency.

Artificial intelligence and machine learning for defense applications: Applying mathematical and computational methods, including machine learning and data analytics, to support decision-making, autonomous systems, and situational awareness in defense operations, such as image and signal processing, anomaly detection, and predictive modeling.

Modeling and simulation of defense systems: Developing mathematical models and computational simulations for studying complex defense systems, such as modeling the behavior of military vehicles, weapons systems, or communication networks, to support system design, analysis, and optimization.

High-performance computing and numerical simulations: Applying advanced computational techniques, including numerical simulations, parallel computing, and optimization algorithms, to solve complex problems in defense applications, such as simulating physical processes, optimizing resource allocation, or analyzing large-scale data sets.

Cryptography and secure communications: Conducting research on advanced mathematical algorithms for encryption, authentication, and secure communications in defense and military systems, including quantum cryptography, post-quantum cryptography, and secure key management.

Human-computer interaction in defense: Investigating the design and evaluation of user interfaces, decision-support systems, and human-computer interaction technologies for defense applications, such as command and control systems, training simulators, and situational awareness tools.

Cyber-physical systems and autonomous systems: Researching the design, analysis, and optimization of cyber-physical systems and autonomous systems for defense and military applications, such as unmanned vehicles, robotics, and drones, including topics such as control algorithms, sensor fusion, and mission planning.

These are just some examples of the interdisciplinary research areas where a combined PhD in mathematics and computer science can be applicable in the defense sector. The field is rapidly evolving, and there may be other emerging areas where mathematical and computational expertise can be applied to address defense-related challenges. It's important to consult with potential advisors and explore available resources to identify research areas that align with your interests and career goals in the defense sector with a focus on mathematics and computer science.