School Colloquium

Tuesday 8 September 2026
Mike Meylan, University of Newcastle
Computational Functional Analysis: Squeezing the Infinite into the Computer

I will talk about my work on making ideas from functional analysis work on a computer. I will discuss the importance of angles (i.e. inner products) and how Lebesgue and Sobolev spaces can be represented numerically. The emphasis will be on the ideas behind the calculations and the methods I am using.

Tuesday 25 August 2026
Arunima Ray, University of Melbourne
What topology can tell us about the shape of space and time

I will use the provocative title as an excuse to discuss classification results for 3- and 4-manifolds, and which of these might correspond to our physical universe and space-time.

Tuesday 11 August 2026
Xu-Jia Wang, ANU
Optimal transportation and its link with Monge-Ampere equation

Optimal transportation is a versatile tool for measuring differences between data sets, with applications in machine learning, data science, and various other fields.
Over the past three decades, the theory has been extensively studied. The existence of optimal mappings follows from Kantorovich's duality. With regard to regularity, one is naturally led to the associated Monge-Ampère type equation, which serves as a prototypical
example of a fully nonlinear partial differential equation.
In this lecture, we will introduce optimal transport and review the regularity theory for the Monge-Ampère equation

Tuesday 28 July 2026
Peter Taylor, University of Melbourne
What does `doing mathematics' mean in the age of AI?

I’ve seen it put that 2026 is the ‘tipping point’ in the use of AI to do mathematics. AI is now capable of doing serious things, not just helping with exposition and coding. Furthermore, the capability of the proof verification tool Lean is now extensive and trusted. As someone who is very far from an expert in AI, my purpose in giving this talk is to
raise awareness, and generate a discussion about what the job of a mathematician will be in the age of AI. Here are some relevant quotes:
Lean enables large-scale collaboration by allowing mathematicians to break down complex proofs into smaller, verifiable components. This formalization process ensures the correctness of proofs and facilitates contributions from a broader community. With Lean, we are beginning to see how AI can accelerate the formalization of mathematics, opening up new possibilities for research. - Terry Tao (on the Lean website)
We are all having to keep revising upwards our assessments of the mathematical capabilities of large language models. I have just made a fairly large revision as a result of ChatGPT 5.5 Pro, producing a piece of PhD-level research in an hour or so, with no serious mathematical input from me. - Tim Gowers
Maybe mathematicians will need to pay more attention to convincing people that their work is not only difficult, but good. If it’s truly good, it doesn’t get its value mainly from being difficult. - John Baez in the comments on Gower’s blog
Things are moving quickly, so what is true today will be different to what is true in 1 month – Recently ChatGPT did a genuinely brilliant thing, resolving a hard problem of Erdos in discrete geometry using deep results in number theory (a link which no human had ever noticed). - Kevin Buzzard
If anyone wants to do some preparation, I suggest that you watch Kevin Buzzard’s talk at the Newton Institute Workshop on AI and the Mathematical Sciences on March 31 this year (https://www.newton.ac.uk/seminar/50266/).