Research

I use geometric and topological tools both within pure Mathematics and Computer Science. My main area of research is Machine Learning but I also work on certain problems in Contact/Symplectic Geometry as well as Computational Geometry. In Machine Learning, I'm especially interested in interactions with Neuroscience - understanding theoretical principles governing both artificial and biological learning, and taking inspiration from biological learning to develop machine learning algorithms. I also regularly collaborate with researchers in other disciplines, including neuroscience, to tailor machine learning tools to their applications.

In recent years, I've been increasingly engaged in discussions around the use of AI in research mathematics (including as a guest editor for a double special issue of the Bulletin of the AMS on the theme `Will machines change mathematics?', and other News below). Regarding the impact of AI more generally, in recent work I use key ideas from the mathematics of learning systems, i.e. the mathematical tradition in AI, to offer accessible conceptual tools for non-experts to make sense of the rapid development of AI—and, especially, to help consider questions of AI safety from the point of view of societies and other living systems. I am curently writing a book on this topic.

Links:

• INTER-MATH-AI (IMA), an NSERC-CREATE-funded collaborative training program in Math and AI that I started in 2022. Applications for Fall 2025 admission open late February.
• Major Thematic Program on the Mathematics of Neuroscience, at the Fields Institute, July-Dec 2025. Still accepting applications!

News

• I was interviewed in ‘Brutal’ math test stumps AI but not human experts, by Zack Savitsky, Science, 03/12/2024.

• Co-moderated a Panel Discussion, at the Fields Institute 16/08/2024. Panelists: Mikhail Belkin (UCSD), Heather MacBeth (ICL), Ravi Vakil (Stanford), Moderators: Colin McLarty (Case Western), Maia Fraser. This panel, which mainly discussed the use of formalization and AI for research mathematics, was part of the New Paradigms session I chaired at the event Forward from the Fields Medal (FFFM) 2024, organized by the Fields Institute to celebrate the centenary of the 1924 International Congress of Mathematicians (ICM) held in Toronto. Coverage of FFFM in the Globe and Mail: Mathematicians to mark a century of progress in Toronto by Ivan Semeniuk, 11/08/2024.

• Co-organizer of The changing face of Mathematics, 2022 Fields Medal Symposium in honour of Akshay Venkatesh, at the Fields Institute. Organizing committee: Kevin Buzzard, Maia Fraser, Michael Harris, Alma Steingart.

• Co-applicant on The self as agent-environment nexus-crossing disciplinary boundaries to help human selves and anticipate artificial selves, one of 10 successful Canada-UK AI Initiative grants awarded. Georg Northoff (U. Ottawa, The Royal Hospital) and I assembled a team of researchers in Canada and the UK, to propose and carry out this work that combined development of clinical tools as well as machine learning theory. For more of the story, an announcement at my institution.

Publications and preprints

For a complete and up-to-date list of publications, please see my Google Scholar page.

Selected publications and non-archived papers that give an overview of my work across research areas since starting at U. Ottawa:

1. V Létourneau, M Fraser, RL in context - towards a framing that enables cybernetics-style questions. Finding the Frame workshop, Reinforcement Learning Conference (RLC), 2024. (pdf)

2. T Poggio, M Fraser, Compositional sparsity of learnable functions. Bull. Amer. Math. Soc. 61(3), 438-456, 2024. (pdf)

3. C Beeler, X Li, C Bellinger,M Crowley, M Fraser, I Tamblyn, Dynamic programming with partial information to overcome navigational uncertainty in POMDPs. Proceedings of the Canadian Conference on Artificial Intelligence, 2024.

4. M Fraser, S Sandon, B Zhang, Contact non-squeezing at large scale via generating functions. Under review, 2024. (arxiv version)

5. V Létourneau, C Bellinger, I Tamblyn, M Fraser, Time and temporal abstraction in continual learning: tradeoffs, analogies and regret in an active measuring setting. Conference on Lifelong Learning Agents (CoLLAs), 2023.

6. C Mirmiran, M Fraser, L Maler, Finding food in the dark: how trajectories of a gymnotiform fish change with spatial learning. Journal of Experimental Biology 225 (23), jeb244590, 2022.

7. V Létourneau, M Fraser, Inexperienced RL Agents Can’t Get It Right: Lower Bounds on Regret at Finite Sample Complexity. Conference on Lifelong Learning Agents (CoLLAs), 327-334, 2022.

8. G Northoff*, M Fraser*, J Griffiths, D Pinotsis, P Panangaden, RJ Moran, K Friston, Augmenting human selves through artificial agents–lessons from the brain. Front. Comput. Neurosci. 63, 2022. (*=equal contribution)

9. M. Golesorkhi, J. Gomez-Pilar, S. Tumati, M. Fraser, G. Northoff, Temporal hierarchy converges with spatial hierarchy: Intrinsic neural time scales follow core-periphery organization. Nature Communications in Biology, 4(1), 1-14, 2021.

10. M. Fraser, G. Northoff, Temporospatial hierarchy and reinforcement learning - towards more general AI. NAISys, 2020.

11. M. Fraser, Contact non-squeezing at large scale in ℝ²ⁿ x S¹. International Journal of Mathematics, (27)13, pp. 60-85, 2017. (arxiv version)

12. M. Fraser, Multi-step learning and underlying structure in statistical models. NIPS2016. (proceedings pdf)

13. M. Fraser, Contact non-squeezing via generating functions: A low-tech proof in the language of persistence modules. Poster in Summer School 2016 on Symplectic Topology, Sheaves and Mirror Symmetry, Paris IJM-PRG, 2016. (poster)

During studies at U. Chicago and U. Toronto, prior to faculty position at U. Ottawa:

1. M. Fraser, Contact spectral invariants and persistence, preprint 2014. (2015 arxiv version)

2. M. Fraser, Group Actions in Topological Data Analysis and Hierarchical Learning. PhD Thesis, Dept. of Computer Science, University of Chicago, August 2013.

3. M. Fraser, Tight Linear Lower Memory Bound for Local Routing in Planar Digraphs. In Proceedings of Canadian Conference on Computational Geometry (CCCG12), August 2012. (proceedings pdf)

4. M. Fraser, Persistent Homology of filtered covers. 2012. (arxiv version)

5. M. Fraser, Local Routing in Graphs Embedded on Surfaces of Arbitrary Genus. 2012. (arxiv version)

6. M. Fraser, Structural Observations on Neural Networks for Hierarchically Derived Reproducing Kernels. University of Chicago Master's thesis, November 2011.

7. M. Fraser, Two Extensions to Manifold Learning Algorithms Using α-Complexes. Dept. of Computer Science, University of Chicago, Technical Report TR-2010-07, 2010.

Teaching

• MAT4373/5314: Mathematical Machine Learning
• MAT4155: Elementary Manifold Theory
• MAT3555/MAT3155: Géométrie Différentielle/Differential Geometry
• MAT3775: Analyse de la régression
• MAT3153: Introduction to Topology
• MAT2355: Introduction to Geometry
• MAT2143: Algebraic Structures: Intro. to Group Theory
• MAT2122: Multivariable Calculus
• MAT1741/1341: Algèbre linéaire/Linear Algebra

Previously at University of Toronto:

• CSC263: Data Structures and Analysis
• MAT102: Intro. to Mathematical Proofs

TA'ing at University of Chicago:

• CMSC28100 Introduction to Complexity Theory
• CMSC25010 Introduction to AI
• CMSC15300 Foundations of Software