The topic for the week-long event was 4-manifolds. A 4-manifold is a space that locally looks like R^4, in the same way that a sphere locally looks like R^2. One surprising phenomenon is that there is a 4-manifold M that is homeomorphic to R^4 but not diffeomorphic to R^4 (meaning that we can identify M and R^4 through continuous functions but not differentiable functions). In fact, there are uncountably many different such M. Even more surprisingly, this only works for R^4 and no other R^n.

Major open questions in the field are: Which 4-manifolds admit multiple smooth structures (like R^4)? If the algebraic topology of a 4-manifold is simple, is the 4-manifold also geometrically simple?

The summer school featured mini-courses given by leaders in the field from MIT, Max-Planck-Institut für Mathematik in Bonn, and the Alfréd Rényi Institute of Mathematics in Budapest. The mini-courses featured a blend of pre-recorded material and synchronous lectures, as well as Zoom office hours led by TAs. The videos of the mini-courses are available online, and to date have hundreds (or in some cases, thousands!) of views.

The mini-courses were complemented by research talks given by researchers from across the country. In addition, junior participants were invited to submit lightning talks, 5 minute pre-recorded talks giving an introduction or overview of their research. During the summer school, there were several different breakout room sessions, allowing participants to discuss material ranging from the mini-courses to the research and lightning talks, as well as whatever else was on people’s minds.

“Math is so collaborative, and one of the things that got somewhat lost during the pandemic was the opportunity for spontaneous conversation and collaboration,” said co-organizer Jen Hom. “One of our main goals when pivoting to the online format was to capture a small part of that magic that happens at conferences, when people from different institutions and with different perspectives on a topic come together and combine their knowledge to lead to new directions of research.”

The next iteration of the Tech Topology Summer School will be held in-person in Summer 2023.