# Mathematics Colloquia and Seminars

The stable homotopy groups of spheres form one of the most fundamental objects in homotopy theory. I will focus on two large trends in homotopy theory which I have used in computations: chromatic homotopy theory gives a sequence of approximations for the stable homotopy groups, while motivic homotopy theory over $\mathbb{C}$ and $\mathbb{R}$ gives a rich comparison object with a well-understood relationship to the homotopy groups of spheres. First I will discuss a comparison between the sphere and the topological modular forms spectrum which produces new infinite families in the homotopy groups of spheres at the prime 3 in chromatic filtration 2. I will also present some computations in the $\mathbb{R}$-motivic homotopy groups of spheres, which has implications for both classical and $C_2$-equivariant homotopy theory. Finally, I will discuss a new comparison theory which relates to $C_p$-equivariant homotopy theory for odd primes.