Algebraic topology studies spaces by attaching algebraic invariants to geometric shapes. These invariants can take values in classical algebraic objects (e.g. commutative rings), but also in more refined algebraic objects “with topology” (e.g. topological rings or so-called E∞-rings).
In my current research, I study different topological generalisations of classical Lie algebras and use them to examine the chromatic homotopy theory of configuration spaces and the formal geometry of moduli stacks in finite and mixed characteristic.