Math4AI/AI4Math Workshop

Mathematics has long guided the evolution of AI, and in recent years AI has begun to shape mathematical discovery in return. In this one-day workshop, we bring together researchers at the intersection of "math4AI" and "AI4math" to share results, highlight emerging directions, and identify pressing challenges. This exchange aims to inspire fresh ideas and collaborations for advancing both fields.

Speakers:

Baran Hashemi (ORIGINS Excellence Cluster at TUM, Germany)
Moa Johansson (Chalmers Universtiy of Technology, Sweden)
Michał Lipiński (Institute of Science and Technology, Austria)
Steve Oudot École Polytechnique & Inria, France)
Bastian Rieck (University of Fribourg, Switzerland)
Melanie Weber (Harvard University, USA)

Registration ist closed.

Program

08:30 - 09:00	Registration & Welcome Coffee
09:00 - 09:40	Steve Oudot (École Polytechnique & Inria) Learning through module categories. Modern machine and deep learning models spend much of their resources computing adequate representations of the data for the task at hand. These representations are obtained through embeddings that are learned from the training data, either in a supervised or in an unsupervised way. The idea advocated in this talk is that it may be fruitful to use embeddings that factor through some module category. These are very different in nature from the usual classes of embeddings, therefore they may capture information about the data that is different, and, to some extent, complementary. There are several important challenges to using such embeddings though, including: how to compute, differentiate, and optimize them. I will explain how these challenges are overcome in the case of persistence modules coming from topological data analysis, and through this special example I will provide some general guidelines that might apply to other module categories as well. The content of the talk is based on arXiv preprint 2411.00493.
09:40 - 10:20	Bastian Rieck (University of Fribourg) Of Shapes and Spaces: Geometry, Topology, and Machine Learning The machine-learning revolution is still in full swing, but so far, there is no large-scale adoption of machine-learning methods in mathematics. I believe the opacity of the field to be one of the driving forces behind this state of affairs. This talk, geared at an audience of people interested but potentially not yet proficient in machine learning, aims to serve as an invitation for more mathematics and mathematical methods in the field. As part of this invitation, I will demonstrate how concepts from geometry and topology can serve to enrich the machine-learning ecosystem and lead to improved models. Moreover, I will describe one possible road towards collaborative mathematical research based on deep-learning methods. Given my own background and proclivities, the focus will remain on geometry and topology, but the suggestions remain very much transferable to other domains of mathematical inquiry.
10:20 - 10:40	Coffee Break
10:40 - 11:20	Melanie Weber (Harvard University) A Geometric Lens on Challenges in Graph Machine Learning: Insights and Remedies Graph Neural Networks (GNNs) are a popular architecture for learning on graphs. While they achieved notable success in areas such as biochemistry, drug discovery, and material sciences, GNNs are not without challenges: Deeper GNNs exhibit instability due to the convergence of node representations (oversmoothing), which can reduce their effectiveness in learning long-range dependencies that are often crucial in applications. In addition, GNNs have limited expressivity in that there are fundamental function classes that they cannot learn. In this talk we will discuss both challenges from a geometric perspective. We propose and study unitary graph convolutions, which allow for deeper networks that provably avoid oversmoothing during training. Our experimental results confirm that Unitary GNNs achieve competitive performance on benchmark datasets. An effective remedy for limited expressivity are encodings, which augment the input graph with additional structural information. We propose novel encodings based on discrete Ricci curvature, which lead to significant gains in empirical performance and expressivity thanks to capturing higher-order relational information. We then consider the more general question of how higher-order relational information can be leveraged most effectively in graph learning. We propose a set of encodings that are computed on a hypergraph parametrization of the input graph and provide theoretical and empirical evidence for their effectiveness.
11:20 - 12:00	Michał Lipiński (Institute of Science and Technology Austria) Conley-Morse barcodes - an algebraic signature of a combinatorial bifurcation Bifurcation theory is one of the major topics in dynamical systems. Its goal is to understand the nature of qualitative changes in a parametrized dynamical system. In this project, we study (combinatorial) bifurcations within the framework of combinatorial multivector field theory - a young but already well-established theory providing a combinatorial model for continuous-time dynamical systems. In this talk, I will explain the idea behind Conley-Morse barcodes, a compact algebraic descriptor of (combinatorial) bifurcations. The barcodes capture structural changes in the dynamical system at the level of Morse decomposition and provide a characterization of the nature of observed transitions in terms of the Conley index. The construction of Conley-Morse barcodes builds upon ideas from topological data analysis. Specifically, we construct a persistence module obtained from a zigzag filtration of topological pairs (formed by index pairs defining the Conley index) over a poset, which can be decomposed into simple intervals.
12:00 - 13:00	Lunch Break
13:00 - 13:30	Scientific Discussion
13:30 - 14:10	Moa Johansson (Chalmers University of Technology) Symbolic, neural and neuro-symbolic methods for lemma discovery in proof assistants The idea of getting computers to automatically discover mathematical conjectures is about as old as AI itself. With the advent of Large Language Models in combination with Proof Assistants, perhaps the time is now ripe to develop systems that can both capture some mathematical intuitions while retaining formal guarantees of correctness. I will introduce the area of theory exploration: summarizing methods for conjecture discovery in symbolic, neural, and neuro-symbolic AI-systems, contrasting the pros- and cons of each, and discuss some ongoing work in my group.
14:10 - 14:50	Baran Hashemi (ORIGINS Excellence Cluster at TUM) Can Transformers do Enumerative Geometry? How can Transformers model and learn enumerative geometry? What is a systematic procedure for using Transformers in abductive knowledge discovery within a mathematician-machine collaboration? In this work, we introduce an AI approach to computational enumerative geometry, specifically targeting the computation of ψ-class intersection numbers on the moduli space of curves. By reformulating the problem as a continuous optimization task, we compute intersection numbers across a wide value range from 10e-45 to 10e45. To capture the recursive nature inherent in these intersection numbers, we propose the Dynamic Range Activator (DRA), a new activation function that enhances the Transformer's ability to model recursive patterns and handle severe heteroscedasticity. Given precision requirements for computing the intersections, we quantify the uncertainty of the predictions using Conformal Prediction with a dynamic sliding window adaptive to the partitions of equivalent number of marked points. To the best of our knowledge, there has been no prior work on modeling recursive functions with such a high variance and factorial growth. Beyond simply computing intersection numbers, we explore the enumerative "world-model" of Transformers. Our interpretability analysis reveals that the network is implicitly modeling the Virasoro constraints in a purely data-driven manner. Moreover, through abductive hypothesis testing, probing, and causal inference, we uncover evidence of an emergent internal representation of the the large-genus asymptotic of ψ-class intersection numbers.
14:50 - 15:10	Coffee Break

Math4AI/AI4Math Workshop

Program

Organizers

Diaaeldin Taha

Marzieh Eidi

Administrative Contact

Katharina Matschke