Author: erlangenaihub

In this Erlangen Hub Q&A we spoke to Marika Taylor, Co-Investigator and Theme C Deputy Lead. Marika is a Professor of Mathematics, Physics and AI at the University of Southampton. She trained in theoretical physics under Stephen Hawking, and is currently interested in geometric ML for fundamental physics applications and physics-inspired methods for ML. Marika was a Turing Institute Fellow and recipient of the “Dutch ERC” Vidi. She has a long track record with start-ups, including in encryption and fintech.  

Can you share a bit about your background and your current research focus?

My main research background is in mathematical and theoretical physics – particularly string theory, quantum theory and gravity. In parallel with this work in fundamental science, I have always been involved in mathematical modeling for real world problems, particularly finance, and have used neural nets for many years in that context. In recent years I’ve seen a convergence between my fundamental science research and my applied work: concepts from my areas of physics are being used within AI, while AI is increasingly being applied within fundamental physics too. A nice example would be graph neural networks in non-Euclidean geometry. Non-Euclidean geometry underpins our understanding of Einstein’s theory of general relativity (gravity). Many of the physical insights obtained from studying particularly geometries for gravity lead to insights into GNNs embedded into such geometries. Another example would be around symmetries. Physicists always build in their understanding of the underlying symmetries (exact or approximate) into their modelling of a system. One can similarly build symmetry equivariance into neural networks – for example, if you are classifying images of 3d objects and are agnostic about the orientation of the objects, then a network with rotational equivariance built into it will be more efficient in classification. My group is currently exploring more general symmetry equivariant networks, drawing from physics insights; this enables us to reduce substantially the number of parameters that need to be learned, and also to understand conceptually patterns found in previous algorithms. We are also interested in using physics understanding of time dependent systems to develop spiking neural networks; the latter are nature inspired, in that neurons only fire when a threshold is met, making them much more energy efficient.

What inspired you to pursue this area?

Throughout my career I have always worked on the frontiers of fundamental science. String theory is a “theory of everything”. It uses concepts from right across mathematics and also leads to new insights and ideas in mathematics – topological quantum field theory, for which Ed Witten won the Fields Medal, is a notable example. In parallel I’ve always enjoyed using the breadth of my knowledge in mathematical sciences for real world applications and I’ve often found that I get new insights into fundamental science from the applied work I’ve done. Over the last few years I’ve gradually moved more and more into AI, for both fundamental science and applied work, because there are so many exciting developments.

Which themes are you connected to within the Erlangen AI Hub and how does your work within the hub intersect with your research background?

The main theme that I am connected with is “Understanding Learning”, but I link with all the themes. Much of what I do relates to understanding conceptually hidden structures in data, and how geometry and topology can be used to characterize these. My physics insights into geometry also facilitate relating geometric and topological insights to real world phenomena.

What attracted you to the Erlangen AI Hub and what do you hope to see it achieve?

The hub is exploring the mathematical foundations of intelligence – this is essential to develop better, more efficient and safer models, which in turn will allow us to use AI in more contexts.

What’s been the most surprising or exciting finding in your work so far?

I don’t think that I would have predicted ten years ago that my two parallel streams of research would become so closely connected, with physics giving insights into developing AI and AI started to be used more in physics. (For AI to really be adopted more widely in physics, we will need robustness and accuracy.)

What challenges have you faced in your research, and how did you overcome them?

I like to take on research problems that are quite open ended and conceptually challenging. Inevitably that means that at times I get stuck or can’t quite see where to go next! Then I take a break, think about something else, and also talk to others, to get new ideas on where to go next.

What advice would you give to someone just starting out in your field?

I would advise somebody to follow their interests, and see where these take them!

Continuing our Q&As, where hub members kindly answer a set of questions to share more about them and their work, we introduce Hub Co-Director and Imperial Lead Anthea Monod.

Can you share a bit about your background and your current research focus?
My undergraduate degree is in pure mathematics, while my PhD is in statistics. I now work at the intersection of pure mathematics and statistics, data analysis, and machine learning. This uses aspects of my formal training and also provides plenty of opportunities to learn more and explore more ways to use pure mathematics in computation and data science, as well as use computational and data-centric approaches to questions in pure mathematics.

What inspired you to pursue this area?
During my PhD, I missed the pure mathematics that I studied in undergrad but, in undergrad, I yearned for my work to have real-world impact and significance. When I was searching for a postdoc after my PhD, I stumbled on this area by accident. I wanted to work more on the geometry of random fields, and contacted Robert Adler (Technion, Israel) for a postdoc in this area, as he is the world expert in it and literally wrote the book on it. But he informed me that his latest interests are in topological data analysis (TDA), which adapts algebraic topology to the computational setting for data analysis and statistics. Although I had hoped to spend some of our discussions on the geometry of random fields during my postdoc, it turns out that I ended up finding TDA more inspiring and motivating and have been working in that area ever since, also expanding to other intersections as well, including algebraic statistics. Algebraic statistics uses techniques from algebraic geometry to study statistical models, which I am exploring in the extension to machine learning methods and neural networks, mostly using tropical geometry, which is a piecewise linear, combinatorial, and polyhedral variant of algebraic geometry.

Which themes are you connected to within the Erlangen AI Hub?
Initially, while constructing the proposal, I felt most connected to Themes A and B (Understanding Data, Understanding Machine Learning Models). However, recent work that was born out of the hub framework has also produced some work in collaboration with other hub members as well as new connections to industry. So, I have also dabbled in Themes C and D (Understanding Learning, Understanding Decision-Making), especially where we have used algebraic topology to try and understand latent spaces of large language models under adversarial influence.

What attracted you to the Erlangen AI Hub and what do you hope to see it achieve?
I feel very fortunate to be one of the team who founded the idea from the moment that the EPSRC call for proposals went live, together with Jeff, Heather, Jacek, Omer, Primoz, and Michael. Jeff, Heather, Omer, Primoz, and myself have primarily been working on applications of topology to data science, mostly persistent homology, and were interested in developing other areas of algebraic topology to data science, such as K-theory. Heather and I also work both in algebraic statistics and wanted to push further to include algebraic geometry as well, which then grew when Omer proposed a probabilistic grounding to the research with his background, Jeff who suggested category theory, and Michael who has worked a lot in adaptations of smooth and continuous geometry to machine learning as well. Over the lifetime of the hub, I would really like to see more ways that more “exotic” mathematics can be adapted to understand deep learning and AI better and, in that sense, establish and ground the position of mathematics in modern data processing techniques and learning theory. I would also like to see the reach of our work extend beyond academia, and I would like for concepts in pure mathematics that might have been inaccessible or intimidating so far to become more widely-used, or even household concepts!

What’s been the most surprising or exciting finding in your work so far?
I think currently what I’ve been most excited about is the reach of tropical geometry to data settings that I’ve been interested in. I started off with an interest in understanding, analysing, and comparing evolutionary relationships in biology captured by phylogenetic trees. It turns out that tropical geometry is a very powerful framework for doing this, and over the past 7 years that I’ve gone into this direction, together with some wonderful collaborators that I have had the great fortune to meet, I’ve done quite a lot of work in providing guarantees for data analytic questions in tropical geometry. The more I learn about tropical geometry, the more I am realising that it also has connections to other discrete mathematical objects that I am interested in, such as metric graphs, which are also very important mathematical structures to model urban road networks, for example. Perhaps most relevant to the hub, it turns out that tropical geometry also has connections to neural networks, which can be thought of as the engine to modern AI systems. I’ve had the great fortune of supervising wonderful PhD students in this area, one of whom is now a hub PDRA in Durham! And another who is also carrying forward the mission of the hub as a Digital Futures Fellow at KTH Sweden, collaborating with some other well-known researchers in both algebraic geometry and algebraic topology in machine learning, such as Kathlén Kohn (a plenary speaker at the hub’s first public conference) and Martina Scolamiero.

What challenges have you faced in your research, and how did you overcome them?
In such an interdisciplinary field, it’s always a challenge to have the expertise in all of the required areas to make a real contribution. Fortunately, I’ve built a large network of fantastic collaborators from all backgrounds with different expertise. This collaborative spirit of exchange and openness is something that we’ve built into the hub by construction. Nobody knows everything, nobody is an expert in everything, so it’s important to talk to each other and work together, which I’m happy to report has worked well for me so far in the hub, and I hope for this to grow and expand over the lifetime of the Hub.

What advice would you give to someone just starting out in your field?
The area is quite vast and fast-moving, so it can be difficult to know where to start. However, fortunately, there is also a lot of content out there: blog posts, tutorials, vlogs, as well as opportunities for in-person interactions, such as talks, workshops, and conferences. Don’t be afraid to sit in and listen to these talks, to talk to the speakers, or reach out to researchers. Don’t be afraid of not understanding everything, hardly anyone does! Interest is what motivates you to learn, and we need to learn continuously to do the work that we do, so keep up the interest and jump on opportunities to learn more, but also make the time to get serious and do some work as well.

What’s something people might be surprised to learn about you outside of research?
I almost didn’t go to university at all. I almost went to study at a conservatoire instead, but quickly got intimidated by how difficult a career in music is, so settled for the “easier” option of mathematics!

In the latest of the hub’s seminar series, Raphaël Tinarrage of the Institute of Science and Technology Austria visited Imperial College London on 11 November to give a talk on Linear orbits of compact Lie groups and machine learning.

When a problem involves continuous symmetries, such as rotations, one naturally expects a Lie group action. In some cases, this action is linear, that is, made of rigid Euclidean motions. As a matter of fact, linear actions arise in several corners of data analysis: in image processing, where standard embeddings commute with Euclidean isometries; in equivariant neural networks, where one structurally forces linear actions or favors them via optimization; or in physical systems, where representations are found sometimes through Noether’s theorem, and sometimes more unexpectedly.

However, most of the time, the representation is not observed directly, but only through its orbits. Recovering the underlying representation from a single orbit would not only allow one to verify the Lie linear orbit hypothesis, but also to improve existing data analysis techniques.

In his talk, Raphaël presented such an orbit-regression algorithm, developed with Henrique Ennes, PhD student at the Inria Centre at the Université Côte d’Azur. Building on previous work by Cahill, Mixon and Parshall, they tackle the problem at the level of Lie algebras, where it can be reformulated as a discrete-continuous optimization over the orthogonal group. In addition to presenting the algorithm and its theoretical guarantees, Raphaël’s talk also delved into the applications mentioned above.

City St George’s, University of London, played host to a highly successful Topological Data Analysis seminar supported by the Erlangen AI Hub, in association with the London Mathematical Society, on 6-7 November.

The London – Oxford – Paris TDA Seminar, whose organising team included Hub Co-Director Anthea Monod and Hub Co-I Omer Bobrowski of Imperial College London (pictured left and centre above), brought together researchers from across the UK and France working in and around the field of algebraic topology, geometry and topological data analysis.

This special edition of the seminar included a number of high profile speakers, including academics from École Polytechnique, University of Oxford, Imperial College London, King’s College London, University of Southampton, Jussieu Institute of Mathematics, and Northeastern University London.

View more information about the event and speakers.

Xinyu Li, a new Postdoctoral Research Associate based at Oxford’s Mathematical Institute, delivered the latest in the hub’s seminar series on 23 October. The seminars provide a great opportunity for the newest members of the hub’s teams to present their research to the community.

Xinyu’s talk, entitled Markov α-Potential Games: A Framework to study Multi-Agent Reinforcement Learning, proposed a new framework of Markov α-potential games to study Markov games. It showed that any Markov game with finite-state and finite-action is a Markov α-potential game, and established the existence of an associated α-potential function. Any optimizer of an α-potential function is shown to be an α-stationary Nash equilibrium.

Xinyu studied two important classes of practically significant Markov games, Markov congestion games and the perturbed Markov team games, via the framework of Markov α-potential games, with explicit characterisation of an upper bound for α and its relation to game parameters. She also provided a semi-infinite linear programming-based formulation to obtain an upper bound for α for any Markov game. Furthermore, Xinyu studied two equilibrium approximation algorithms, namely the projected gradient- ascent algorithm and the sequential maximum improvement algorithm, along with their Nash regret analysis.

Our hub members have been kindly answering a set of questions so that we can share more about them and their work. We start with Hub Co-Director Jeffrey Giansiracusa of Durham University.

Can you share a bit about your background and your current research focus?
I started off as a very pure mathematician, working in topology and homotopy theory. From there I drifted towards algebraic aspects of tropical geometry, but over the past 5 years I’ve become increasingly interested in applications of topological data analysis to quantum field theory data, as well as machine learning in non-archimedean and tropical geometry.

What inspired you to pursue this area?
By now I’ve worked in several very different areas of mathematics. In each case it was the incluence of mentors and a supportive community that brought me into learning and doing new things.

Which themes are you connected to within the Erlangen AI Hub?
Theme A: Understanding Data
Theme B: Understanding Machine Learning Models

What attracted you to the Erlangen AI Hub and what do you hope to see it achieve?
As one of the architects of the hub, I was very excited about the opportunity to help develop the already impressive community of people in the UK doing topological data analysis, encouraging them to connect to ML and AI and some of the really big questions around right now.

What’s been the most surprising or exciting finding in your work so far?
Gradient descent optimisation shouldn’t work in a non-archimedean setting, where small steps can’t add up to a big step. But we found a non-archimedean optimisation procedure that looks a lot like gradient descent which does work!

What challenges have you faced in your research, and how did you overcome them?
My biggest challenge is always balancing my various projects and responsibilities, and balancing work with family commitments. I often have to leave meetings early to collect my kids from school and take them to their various activities.

What advice would you give to someone just starting out in your field?
Find the people you enjoy working with, and then work with them! Don’t waste your time working with people that you don’t like.

What’s something people might be surprised to learn about you outside of research?
My favourite person to do mathematics with is my brother.

Two Imperial PhD graduates under the tutelage of Hub Co-Director Anthea Monod have secured key postdoctoral positions at leading research institutions in Europe.

Inés Garcia-Redondo (pictured left) successfully completed her PhD and begins postdoctoral life as Senior Researcher at the AIDOS (AI for Data-Oriented Science) Lab at the University of Fribourg, led by Professor Bastian Rieck, whose work is closely aligned with the mission of the hub. Inés’ research focuses on topological data analysis, particularly in its use within machine learning systems, to investigate the mathematical foundations of AI. She said:

“I intend to continue my research at the interface of topology and geometry, and deep learning systems, which I initiated with Anthea as a student aligned to the hub. I’m very grateful and excited for the new opportunities to come, and to stay connected to the hub as well!”

Meanwhile, Yueqi Cao (pictured right) has been awarded a Digital Futures Postdoc Fellowship at the Department of Mathematics at KTH Royal Institute of Technology in Stockholm, supervised by Profs Johan Karlsson and Sandra Di Rocco. Yueqi’s research sits at the crossroads of mathematics, statistics, and machine learning. During his PhD, he developed new tools and methods to analyse metric graphs using ideas from tropical geometry and topological data analysis. Yueqi’s postdoctoral research will now see him extend his research in metric graphs, exploring new geometric and topological methods and applications in machine learning and data analysis. He said:

“I am excited to embark a new postdoctoral position at KTH, where I look forward to further developing my research and building new collaborations, and making new connections in Europe to advance the research areas of the hub, strengthening connections between pure mathematics, computation, and machine learning.”

Congratulations to Ines and Yueqi. We wish them the best of luck in their new roles!

It’s conference season and hub members have been busy presenting work across the world. Take a look at a snapshot of activity below:

UK AI Research Symposium – September 2025, Northumbria University

Hub members took part in a fantastic two days at the UK AI Research Symposium (UKAIRS) at Northumbria University.

Congratulations to Oliver Clarke, Edward Pearce-Crump and Qiquan Wang, who presented their research during the poster sessions, and Edward who also gave a lightning talk on his research.

UKAIRS was a hugely inspiring event bringing together and consolidating the UK’s AI research community, with highly engaging talks, demos, panels, posters and keynotes across diverse disciplines, with reflections on the future of AI and emerging challenges. It was also a brilliant platform for our postdocs to showcase their research and meet peers from across the UK, facilitating connections and ideas-sharing with the wider AI research community, including the other EPSRC AI hubs. Many thanks to organisers Responsible Ai UK and the steering committee for their hard work putting the event together.

Applied Algebraic Topology Research Network (AATRN) conference – August 2025, Chicago

Several hub members attended a week-long conference in celebration of the 10-year anniversary of AATRN, the Applied Algebraic Topology Research Network, at the Institute for Mathematical and Statistical Innovation in Chicago.

Speakers included hub members Anthea Monod, Omer Bobrowski and Heather Harrington. They were accompanied by hub PhD students Arne Wolf, Inés Garcia-Redondo and David Lanners.

The event was AATRN’s first in-person meeting, bringing together researchers from mathematics, statistics, computer science, physics, biology, and beyond.

Graph Learning Meets Theoretical Computer Science workshop – August 2025, Berkeley

Anthea Monod was a speaker at the Graph Learning Meets Theoretical Computer Science workshop (co-chaired by Michael Bronstein) at the Simons Institute for the Theory of Computing at the University of California, Berkeley. She offered a Bootcamp on geometry and graph learning.

The workshop brought together researchers to provide a more unified perspective on graph learning within theoretical computer science.

International Joint Conference on Artificial Intelligence (IJCAI) – August 2025, Montreal

Guiseppe De Giacomo presented three papers at the International Joint Conference on Artificial Intelligence (IJCAI) 2025 in Montreal.

Read: LTLf+ and PPLTL+: Extending LTLf and PPLTL to Infinite Traces

Read: Solving MDPs with LTLf+ and PPLTL+ Temporal Objectives

Read: Computational Grounding of Responsibility Attribution and Anticipation in LTLf

SIAM (Society for Industrial and Applied Mathematics) conference – July 2025, Wisconsin

Oliver presented his work-in-progress alongside Yue Ren, Jeffrey Giansiracusa, and Julio Quijas-Acaves, with a talk entitled Towards non-Archimedean Machine Learning. The project is concerned with developing machine learning tools, for instance gradient descent, over non-Archimedean fields such as the p-adics. Oliver said:

“I was delighted with the attendance for my talk, presenting to a packed seminar room, which lead to fruitful conversations with experts in p-adics analysis and tropical geometry.”

The conference lasted 5 days, during which time Oliver attended around 50 talks, learning about many of the problems and techniques in applying algebraic geometry to machine learning. He added:

“It was an excellent opportunity and I’m looking forward to presenting some concrete results in the future.”

International Conference on Machine Learning (ICML) – July 2025, Vancouver

A team of researchers including Michael Bronstein won the best paper award at the ICML Generative AI and Biology (GenBio) workshop for FORT: Forward-Only Regression Training of Normalizing Flows.

Uzu Lim presented Cover Learning for Large-Scale Topology Representation at ICML. Authors of the joint paper also included Luis Scoccola and Heather Harrington, the hub’s Oxford Maths lead.

“I’m delighted to have had the opportunity to present my work at ICML 2025 in Vancouver! The feedback I received was incredibly valuable and will guide me in my future research. It was also a pleasure to see old colleagues again and engage in thoughtful discussions about the latest advances in AI.”

Doctoral student Thiziri Nait Saada presented work supported by the hub at ICML. Mind the Gap: a Spectral Analysis of Rank Collapse and Signal Propagation in Attention Layers was authored by Thiziri alongside Alireza Naderi and Jared Tanner.

International Conference on Computer-Aided Verification (CAV) – July 2025, Zagreb

Thom Badings (pictured above) presented his work at CAV in July.

In the joint paper Policy Verification in Stochastic Dynamical Systems Using Logarithmic Neural Certificates his team developed novel techniques for the verification of neural network policies in stochastic dynamical systems. 

The Erlangen AI Hub was a ‘diamond’ sponsor of the 2025 London Geometry and Machine Learning (LOGML) Summer School at Imperial College London this year. 

Every July the summer school brings together mathematicians and computer scientists to collaborate on a range of problems at the intersection of geometry and machine learning. The week-long event features a number of group projects, each overseen by an experienced mentor, talks by leading figures in the field, a poster session, networking with industry, and social events.

As a primary sponsor, the Erlangen AI Hub enjoyed a key presence at this year’s school, with many members, hub-aligned postdocs and PhD students involved as organisers, advisors, project leaders, and participants. The organising team included incoming hub-aligned Postdoctoral Research Associate Daniel Platt and hub-aligned PhD student Arne Wolf. The scientific advisory board included Dr Anthea Monod, Prof Heather Harrington, and Prof Michael Bronstein, one of the original founders of the school during his time at Imperial.  

This year’s vibrant and fruitful event welcomed more than 100 participants from across the world, who collaborated in teams on 19 mentored projects and enjoyed a range of talks and tutorials from high profile speakers including the hub’s Prof Coralia Cartis. It wasn’t all work though as attendees enjoyed a range of social activities including a welcome breakfast at the V&A Museum, a company night, bouldering, live music, and lunch at Chiswick House and Gardens. 

Co-Director of the Erlangen AI Hub, Dr Anthea Monod, was a key advisor to the summer school and co-led a project with fellow hub board member Prof Omer Bobrowski. Anthea said:

“It was a fantastic, first-class summer school, and I am proud of the hard work of the organisers. I had the pleasure of leading a project using topology to study the evolution of high dimensional neural activation patterns. It was so much fun and great to catch up with people on the circuit. Huge thanks to the Erlangen AI Hub for being a diamond sponsor.”

Find out more about the LOGML Summer School at https://www.logml.ai/