Mathematical Data Science and Foundations of AI

Mathematical data science is an interdisciplinary field that harnesses advanced mathematical tools to extract meaningful information from data, including but not limited to artificial intelligence (AI) and machine learning (ML) models. It employs a wide range of tools from statistical theory and other areas of mathematics to ascertain relationships and patterns within datasets, ensuring that inferences and predictions are both robust and interpretable.

1. Inverse problems are fundamental mathematical challenges where the effects are observed and the causes or parameters need to be inferred. They appear in numerous and widespread applications, such as medical imaging and in geophysics, deducing earth properties from seismic data. These problems often require sophisticated regularization techniques to overcome ill-posedness, ensuring stable and meaningful solutions.
2. There are too many applications to list, but interpreting neural networks is a critical area of research. Neural networks, particularly deep learning models, have been notoriously labelled as "black boxes" due to their intricate structures and the opaque nature of their internal workings.

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