10 December 2025

Finding Their Way to the Mathematical Frontier
Melissa’s path into mathematics began with a surprise. Initially studying chemistry, she soon realised it wasn’t the right fit. Her first pure mathematics class changed everything: “I knew straight away this was what I wanted to do.” With the support of mentors, that spontaneous decision grew into a career she remains grateful for. Today, she works in areas where algebra, geometry, and computation intersect.

Tim grew up surrounded by engineers and mathematicians, where complex ideas never felt far away. He was drawn early to differential geometry, a field exploring how shapes curve and identifying those with “nice curvature properties,” linking deep mathematical questions to phenomena such as the shape of the universe. For Tim, mathematics is “a sophisticated language” for expressing ideas far too intricate for ordinary words.

Eye-Opening Moments
Both researchers highlighted the unique value of working in a truly binational setting made possible by the MATRIX–MFO Tandem Workshop model — a format launched in 2021 that connects participants through coordinated talks and shared discussion sessions. This program brought together participants who normally live in separate research worlds: Machine learning experts interacted with mathematicians; people who write Lean code for formal verification sat next to those who spend time sketching algebraic structures on paper or whiteboards.

For Melissa and Tim, this contrast generated some of the week’s most eye-opening moments. Melissa was particularly struck by a talk on CayleyPy, presented by Dr Alexander Chervov from the MFO side — a new algorithm and Python library that uses artificial intelligence to analyse enormous symmetry graphs. The talk resonated strongly with her own work in group theory. “Group theory is fundamentally about symmetry, and a major computational challenge is to understand how the elements of a large symmetry group can be built from the small set of generators stored on your computer,” she explains. “CayleyPy enables us to explore these large symmetry structures at a scale and speed we’ve never had before on groups that far surpass anything we can currently handle with computer algebra systems.” Seeing a method perform at that level was “extremely exciting” for her, opening possibilities for tackling problems that had long been out of reach.

For Tim, the week offered a vivid glimpse into “what the state of the art is,” even if some of it felt “quite scary” in its ambition and pace, “You could really sense that big changes are coming quickly,” he says. The talk that struck him most came from the MFO side, where Dr Floris Van Doorn and Dr María Inés de Frutos-Fernández presented “The Carleson project: a collaborative formalisation” explaining how they had used Lean to verify “a very complicated harmonic analysis proof.” It challenged assumptions he had never thought to question. “I thought Lean was more a tool for algebraic theorems,” he admits. “I hadn’t fully realised that it can work for harmonic analysis and PDE proofs as well!” Seeing the range and depth of work on display left a strong impression: “It was fantastic to see what a lot of clever people are working on,” he says, an experience that “forced me to rethink where maths is going.”

While nothing replaces being in the same room, Tim valued being able to see what the MFO community is doing, even partially. Melissa adds, “Bringing the perspectives of the MFO group into the room changed the way we discussed things.”

AI, Machine Learning and Mathematics - A Powerful Partnership
AI is rapidly reshaping how mathematicians think about problem solving, and both Melissa and Tim see its power as well as its pitfalls.

For Melissa, AI represents “very fertile ground for new discoveries,” with the potential to spark innovation in areas that have rarely been interdisciplinary. “If we can bring pure mathematicians together with AI or machine-learning engineers, that can be a fantastic combination that really revolutionises mathematics,” she says. Even with impressive breakthroughs already emerging, she feels “we’re right at the beginning of what can be done.” The workshop gave her “the tools to start—playing around and tinkering with machine-learning and AI algorithms to help accelerate my research,” helping her to upskill for a future where AI becomes a natural companion to mathematical insight. In her view, AI isn’t replacing traditional methods but extending a long lineage of computational tools that help mathematicians “test out theories and find examples,” now with far greater reach.

Tim highlighted a remarkable example of how rapidly the field is evolving. The Navier–Stokes smoothness problem, he notes, is “a very famous problem,” unsolved for nearly two centuries, yet “people might be close to solving this.” A team of five U.S.-based mathematicians and seventeen Google DeepMind engineers is using AI-driven methods to identify “an initial fluid configuration that could develop a singularity in finite time”—a key question linked to one of the Millennium Prize Problems. Tim first learned about the project during the MATRIX workshop and found it “cool and impressive” that such a small group is inching toward a potential breakthrough.

But even with progress like this, neither Melissa nor Tim believes AI will replace mathematical reasoning. “AI often gets things wrong—spectacularly wrong,” Tim says with a smile. “We’ll still need mathematicians to distinguish what’s plausible from what’s nonsense.” To him, AI is ultimately “a problem solver” built on layers of mathematics: “machine learning is applied statistics, which is applied optimisation theory, which is applied functional analysis”—a reminder that today’s tools rest on deep mathematical foundations. AI systems will continue to become more complicated but will also continue to make big mistakes; Tim believes that a key job of future mathematicians will be to act as " doctors" for AI systems, who will be charge with “diagnosing how and why these mistakes have occurred”. Meanwhile, he notes, “we should also encourage machine learning researchers to collaborate with mathematicians, as mathematics is a good source of ideas and new knowledge, giving AI interesting problems to solve.”

Melissa agrees, particularly when it comes to discernment. “AI can produce results, but it can’t tell you what’s interesting. That’s where mathematical skills will always be needed.” She expects the training of young mathematicians to evolve quickly as AI becomes “an integral part of how we train the next generation, just like computer algebra systems.” Yet alongside this new tool, she stresses the need for deeper skills: the ability to judge the value of AI outputs, understand their limitations, and recognise when an algorithm has produced something genuinely meaningful versus something “you could have sat down and proved in two minutes.”

Charting the Future
Looking ahead, Melissa is excited about future workshops that bring machine learning experts and mathematicians together, convinced that their combined expertise “is very powerful.” Yet she remains clear-eyed about the challenges: the current skill gap, the risks of over-reliance on AI tools and the responsibility mathematicians carry in guiding AI toward greater interpretability and safety.

Tim is excited about new directions where machine learning and mathematics reinforce each other. His current collaboration with Dr Liam Hodgkinson (University of Melbourne) illustrates this momentum: together, they are exploring how computer-generated approximate solutions to complicated equations can be paired with rigorous mathematical analysis to verify the existence of true solutions. Working side by side during the workshop helped them push the project forward, bringing them closer to what Tim hopes will become their second paper.

For Melissa and Tim, the MATRIX–MFO partnership created a genuinely shared intellectual space — one that broadened perspectives and opened new collaborative pathways. As AI accelerates, they believe research hubs like MATRIX, and the global networks they build, will be essential for helping researchers navigate change, work collaboratively, and drive meaningful scientific breakthroughs.

Their Messages to MATRIX on Its 10^th Anniversary
“It was a week of huge growth for me at MATRIX — expanding my knowledge and meeting new people, which was incredibly valuable. It’s fantastic to have such a strong research institute in the Southern Hemisphere, where so much excellent mathematics is happening. It’s an essential part of mathematics in Australia. I really appreciate that it exists.”
                                                                                                                                                                                                                       – Dr Melissa Lee

“Having a dedicated space to do mathematics is very important for researchers in Australia and New Zealand. MATRIX plays a crucial role in the Australian research environment, and I hope it continues to thrive for many decades to come.”                                                                                                                                                                                                                – Dr Tim Buttsworth