Fall 2026

Oriented cohomology theories provide a general framework to perform intersection-theory style calculations. Prototypical examples include Chow ring, $K$-theory, and algebraic cobordism, while Annala--Hoyois--Iwasa recently constructed the category $\mathsf{MS}_S$ for studying other non-$\mathbb A^1$-invariant theories. In this talk, we prove a blowup formula for oriented cohomology theories in $\mathsf{MS}_S$. We then give a ring presentation version of this formula and use it to compute oriented cohomology rings of blowups such as the moduli space of complete conics and $\overline M_{0,n}$. In particular, we demonstrate that one can recover the number $3264$ of conics simultaneously tangent to five given ones in $\mathbb P^2$ in algebraic cobordism. This is based on joint work with Arkamouli Debnath.