I’ll be giving a talk at Tohoku University (Algebra Seminar) next week.

**Title: **Towards a motivic homotopy theory without A¹-invariance.

**Abstract: **Motivic homotopy theory as conceived by Morel and Voevodsky is based on the crucial observation that the affine line A¹ plays in algebraic geometry the role of the unit interval in algebraic topology. Inspired by the work of Kahn-Saito-Yamazaki, we constructed an unstable motivic homotopy category “with modulus”, where the affine line is no longer contractible. In the talk, we will sketch this construction and we will explain why this category can be seen as a candidate environment for studying representability problems for non A¹-invariant generalised cohomology theories.