My research is grounded in the interplay between Geometry, Integrable Systems, and Mathematical Physics, with recent expansions into Quantum Computing and Machine Learning. Building upon the mathematical foundations developed during my PhD, I wish to apply this knowledge to explore new applications and interdisciplinary connections that integrate both theoretical and practical advancements.
During my PhD, I focused on the geometric aspects of Topological Field Theory, particularly on how geometric structures give rise to integrable systems and their associated solutions. Central to this research is the role of Dubrovin Frobenius manifolds, which not only provide a rich geometric framework but also connect to various integrable systems, such as isomonodromic deformations and integrable hierarchies of topological type.
Dubrovin Frobenius Manifolds: These manifolds play a key role in the structure of integrable systems, particularly through their relationship to isomonodromic systems, which arise from the deformation theory of differential equations.
Integrable Hierarchies of Topological Type: These are hierarchies of integrable partial differential equations (PDEs) that naturally emerge from topological field theories, forming a deep connection between physics and geometry.
Hamiltonian Perturbations of Hyperbolic Systems: These perturbations are related to the integrable structure of the underlying topological field theory and provide further insight into the physical correlators and mathematical structures.
Theta-Functions and Nonlinear Waves: The solutions to integrable hierarchies are closely related to theta-functions on Riemann surfaces, which play a crucial role in understanding the behavior of nonlinear waves and their underlying mathematical structures.
Recently, I have expanded my research into Quantum Computing and Machine Learning. This transition is inspired by the natural connection between these fields and Information Geometry, which provides a common ground for exploring these disciplines.
2.1 Quantum Computing
Quantum computing presents an exciting domain where geometric and integrable systems perspectives can offer deep insights. My specific interests include:
Quantum Machine Learning: An emerging field that merges quantum computing with machine learning algorithms, offering the potential for quantum speed-ups in data analysis and optimization tasks.
Quantum Information Theory and Geometry: This focuses on understanding the geometric structure of quantum information spaces, drawing parallels to the information geometry explored in classical contexts.
Quantum Error Correction: Studying how geometric insights can contribute to more robust error correction schemes, which are critical for reliable quantum computation. I am particularly interested in Toric/ Surface codes.
Quantum Chemistry: Applying quantum computing to solve complex problems in chemistry, particularly those related to the simulation of molecular structures and dynamics.
2.2 Machine Learning
In machine learning, the geometric frameworks I studied during my PhD continue to play a role, especially through Information Geometry, which shares many structural similarities with Dubrovin Frobenius manifolds. My focus areas include:
Information Geometry: This field studies the geometric properties of statistical models and provides tools for understanding learning algorithms, optimization, and model generalization. The connection between information geometry and the Dubrovin Frobenius manifold structure has yet to be fully explored but holds significant potential for uncovering new synergies.
Deep Learning and Neural Networks: Information geometry can offer new perspectives on the curvature of the parameter space in deep learning models, potentially leading to more efficient training algorithms and better understanding of neural network dynamics.
The extension of my research into Quantum Computing and Machine Learning is a natural progression of my previous work, where Information Geometry acts as a unifying theme. Both classical and quantum information geometry provide a geometric structure to information theory, and by extension, to quantum computing and machine learning.
Information Geometry and Dubrovin Frobenius Manifolds: Both classical and quantum information geometry share a structure akin to that of Dubrovin Frobenius manifolds. The geometric methods developed in the context of topological field theories could therefore provide new approaches to problems in quantum information theory and machine learning, particularly at their intersection, as in Quantum Machine Learning.