Motywiczna teoria homotopii

I will introduce motivic spaces, using infinity-categorical techniques. I will illustrate the power of the theory with a couple of results, then I will very quickly make the link to motivic spectra and Morel--Voevodsky's big category of motives. Some smooth varieties with torus actions admit nice Białynicki-Birula-type decompositions there; I'll then say two words about the non-smooth case, which is the starting point of joint work with J. Jelisiejew.