Opportunities

Postdoc Opportunities

We are looking for postdocs! Click here for more information.

PhD Opportunities

Please contact Natalia Berloff directly at n.g.berloff[at]damtp.cam.ac.uk

The list of potential PhD projects (relevant for 2025 student intake):

1. Characterization of Energy Landscapes in Complex Spin Systems:

   • Problem: While traditional spin glass models with real-valued spins have been extensively studied, the energy landscapes of systems with complex-valued spins (hyperspins) are poorly understood. There's a need to characterize the topology and geometry of these landscapes, including the distribution and nature of local and global minima.

   • Importance: Understanding these landscapes can lead to developing more efficient algorithms for finding global minima in hard optimization problems. It can also provide insights into the fundamental physics of disordered systems and contribute to advancements in material science and information processing.

2. Development of Novel Annealing Schedules for Complex Systems:

   • Problem: Traditional annealing schedules may not be optimal for complex-valued systems. Designing new annealing strategies that are specifically tailored to guide the dynamics of hyperspin systems toward global minima is an open challenge.

   • Importance: Improved annealing schedules can enhance optimization techniques in both classical and quantum contexts. This has direct applications in solving combinatorial optimization problems and improving machine learning algorithms.

3. Extension of Spin Glass Theory to Non-Hermitian Systems:

   • Problem: Explore the theoretical framework of spin glasses when extended to non-Hermitian Hamiltonians, which naturally arise in systems with gain and loss or in open quantum systems. For instance, a network of Stuart-Landau oscillators or a complex Ginzburg-Landau equation with complex couplings captures the dynamics of coupled oscillatory units with amplitude and phase, making it a rich system for exploring non-Hermitian effects.

   • Importance: This can uncover new physical phenomena such as non-Hermitian phase transitions and exceptional points, enriching our understanding of complex systems and potentially leading to novel computational methods.

4. Optimization Algorithms in High-Dimensional Complex Spaces:

   • Problem: Develop optimization algorithms that are effective in high-dimensional spaces with complex-valued variables. Traditional gradient-based methods may fail or become inefficient due to the intricate nature of these spaces.

   • Importance: Efficient optimization in complex spaces is crucial for training complex-valued neural networks and could lead to breakthroughs in fields like signal processing, quantum computing, and cryptography.

5. Analysis of the Learning Dynamics in Complex-Valued Neural Networks:

   • Problem: Investigate how the inclusion of complex numbers in neural network architectures affects learning dynamics, convergence properties, and generalization capabilities.

   • Importance: This could lead to the development of neural networks that are better suited for processing complex-valued data, such as in quantum information, wave physics, and other applications where phase information is critical.

6. Understanding Phase Transitions in Hyperspin Systems:

   • Problem: Study the existence and nature of phase transitions in systems with hyperspins, and how these transitions affect the computational complexity of finding ground states.

   • Importance: Insights from this research could inform the design of algorithms that avoid critical slowing down near phase transitions, thus improving optimization performance.

7. Quantum Annealing with Complex Spins:

   • Problem: Explore the theoretical and practical implications of implementing quantum annealing processes using complex spins, possibly leveraging existing or emerging quantum hardware.

   • Importance: This could enhance the capabilities of quantum annealers to solve a broader class of problems and provide a pathway to realize quantum advantages in optimization tasks.

8. Topological Analysis of Energy Landscapes:

   • Problem: Apply tools from algebraic topology or topological data analysis to study the features of energy landscapes in complex spin systems, such as holes, connected components, and other invariants.

   • Importance: Understanding the topological features can reveal deep insights into the landscape's navigability and the likelihood of optimization algorithms escaping local minima.

9. Statistical Mechanics Approaches to Machine Learning Models:

   • Problem: Use concepts from statistical mechanics to model and analyze the behavior of complex-valued neural networks, including the study of replica symmetry breaking and thermodynamic limits.

   • Importance: This could bridge the gap between physics and machine learning, providing a theoretical foundation for why certain architectures or training methods succeed.

10. Design of Robust Complex-Valued Neural Network Architectures:

    • Problem: Propose new architectures for complex-valued neural networks that are robust to noise and exhibit desirable properties like stability and expressiveness.

    • Importance: Such architectures could be particularly valuable in applications involving wave phenomena, quantum systems, or any domain where data is naturally represented in the complex plane.

11. Investigating the Role of Nonlinearities in Complex Systems:

    • Problem: Analyze how different types of nonlinearities in activation functions affect the dynamics and optimization in complex-valued neural networks.

    • Importance: Nonlinearities are crucial for the expressive power of neural networks. Understanding their role can lead to better-performing models and new activation functions suited for complex data.

12. Convergence Properties of Optimization Algorithms in Non-Convex Complex Spaces:

    • Problem: Study the convergence behaviour of existing optimization algorithms when applied to non-convex, complex-valued energy landscapes, and develop new algorithms with proven convergence guarantees.

    • Importance: This research can improve the reliability and efficiency of training complex-valued neural networks and solving complex optimization problems.

13. Machine Learning Techniques for Predicting Ground States of Spin Glasses:

    • Problem: Use advanced machine learning models to predict the ground states or low-energy configurations of spin glasses, potentially incorporating hyperspins.

    • Importance: Accurate prediction of ground states can have implications in material science, optimization, and understanding the fundamental properties of disordered systems.

14. Cross-disciplinary Applications of Complex Spin Models:

    • Problem: Apply models of complex spin systems to other fields such as biology (e.g., neural activity modeling), economics (e.g., market dynamics), or social sciences (e.g., opinion formation).

    • Importance: This could provide novel insights and tools across various disciplines, demonstrating the broad applicability of complex spin models.