2-torials

Our “2-torials” feature one-on-one conversations where one participant (the “tutor”) tries to explain a topic they’re passionate about to a volunteer “tutee”.

These are first-and-foremost intended as a tool for Topos staff to explore new ideas, develop creative approaches to science communication, and learn about their coworkers. Secondarily, we record and publish them hoping that they might have pedagogical value as expositions on research-level topics which leave intact some of the confusion, discussion and struggle inherent to learning complicated things.

Quantum information theory
Quantum physics
Computer science

The elements of quantum information theory are not so complicated: some linear algebra over the complex numbers and a bit of classical probability theory. But it doesn’t take long before you run into some intriguing problems that showcase how established experimental results contradict our classical intuition. In this 2-torial we will learn the very basics of quantum theory, see how a Mach–Zehnder interferometer gives a real-world implementation of a classically impossible construction and suggests that we should go all in on complex numbers, and finally how this all applies to the world of theoretical bird watching via the Quantum Zeno effect.

Tim Hosgood
May 2026
Tutee Jason Brown
Prerequisites Complex numbers, (finite) vector spaces and matrices, elementary probability theory.
Concepts Bra-ket notation, unitary matrices, probability amplitudes, beam-splitters, Mach–Zehnder interferometers.

Deformation theory
Algebra
Algebraic geometry

What happens if we take an algebraic structure — say, multiplication in an algebra — and wiggle it a little bit? Formalising what we could actually mean by this and then understanding the consequences will lead us to start exploring the theory of first-order deformations. Following this path will eventually lead us to naturally stumble upon certain chain complexes whose cohomology describe solutions to our stated problems. Eventually we will have all we need to understand why people are so interested in so-called Maurer–Cartan elements and DGLAs, and what any of this has to do with algebra or geometry.

Tim Hosgood
October 2025
Tutee Jason Brown
Prerequisites Associative algebras over a field, tensor product of algebras.
Concepts First-order deformations, Hochschild complex, Hochschild cohomology, Maurer–Cartan elements, DGLAs.
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