Abstract:$\mathbb{A}^{1}$-Euler characteristic is an invariant from the motivic
homotopy theory which associates to a smooth algebraic variety over
a field $k$ a non-degenerate symmetric bilinear form over $k$ (more
precisely, an element of the Grothendieck–Witt ring of symmetric
bilinear forms over $k$). This invariant generalizes the topological
Euler characteristic in the sense that over the field of complex
numbers it recovers the topological Euler characteristic of the
manifold of complex points. In the talk I will recall the
construction of this invariant and give an overview of some
computations. Then I will discuss what is known about the $\mathbb{A}^{1}$-Euler
characteristic of the variety of maximal tori in a reductive group
and present the corresponding generalized splitting principle.