PSD - Prof Omer Bobrowski

1. Mathematical foundations for AI

Supervisors: Prof Omer Bobrowski and Prof Primoz Skraba

Project description:

This PhD position is part of the “Erlangen Programme for AI,” a prestigious multi-university initiative focused on developing a rigorous mathematical foundation for Artificial Intelligence. The project emphasizes the integration of concepts from topology, geometry, and probability, with the overarching goal of enhancing the interpretability, robustness, and generalization of AI models.

Potential Research Directions:

Understanding Deep Neural Networks

DNNs represent a cutting-edge approach in machine learning and AI, but there remains a significant gap in understanding the intrinsic mechanisms behind their powerful performance. This research aims to combine topological and geometric tools with probabilistic analysis to unveil hidden structures in neural networks. By investigating how these structures arise during training, how information flows through layers, and what vulnerabilities exist, we expect to gain insights that will drive future advancements in model design, optimization, and resilience.

Understanding Large Language Models

LLMs have shown to capture (encode) both the semantics and structure (grammar) of language within their learned parameters. However, the methods used to access this knowledge (decoding) remain basic, typically involving the representation of textual objects (e.g., words, sentences) as continuous vectors in Euclidean space. This project aims to leverage geometry and topology to explore the internal representations and latent spaces within the LLMs parameters that go beyond simple vectors analysis. We will develop advanced methods for decoding meaning and structure from LLMs, enabling richer and more diverse access to the linguistic knowledge they encode, and test it in a range of linguistic tasks (polysemy, cross-lingual transfer, among others). This approach holds the potential for breakthroughs in both AI theory and practical applications.

Further information:

For this specific project students of all nationalities are welcome to apply; however please note that successful students with Overseas fees status are expected to cover the difference between Oversees and Home fees (approx. £20K per year).

How to apply

Entry requirements

Fees and funding

Applications of Universality in Topological Data Analysis

Supervisor: Dr Omer Bobrowski

Project description:
Recently discovered, the phenomenon of universality in Topological Data Analysis (TDA) offers a new direction to data analysis. This project will explore its applications and develop new methodologies on top of this phenomenon. There are many possible directions but the initial focus will be on:

1. Dimensionality estimation: One of the key challenges in machine learning and data analysis is figuring out how many important features are in a dataset (i.e., its intrinsic dimension). However, traditional methods, such as principal component analysis (PCA), may not be suitable for high-dimensional or noisy data. The project will develop statistical tools for applying universality for estimating dimension from topological features (including mixed dimensional spaces, relaxing the manifold hypothesis).
2. Topological clustering: The connection between clustering and topology is well established with a substantial amount of previous work. This will build on this work to provide a complete framework for proving consistency in different clustering schemes as well as providing a provable approach to estimating the number of clusters from data. This will be driven by applications in a wide range of areas.
3. Quantifying disorder: While TDA is often concerned with global structure, there are many cases where the distributions of smaller features in cases such as quasicrystals or other types of materials plays an important role. The goal of this application is to leverage universality to quantify the amount of order (as a form of regularity) in a point set. This will connect to existing work in sampling and discrepancy theory.
   
   

Further information: 
How to apply 
Entry requirements
Fees and funding