Abstract: 

In filtration 1 of the Adams spectral sequence, using secondary cohomology operations, Adams computed the differentials on the classes h_j , resolving the Hopf invariant one problem.  In Adams filtration 2, using equivariant and chromatic homotopy theory, Hill–Hopkins–Ravenel proved that the classes h_j² support non-trivial differentials for j ≥ 7, resolving the celebrated Kervaire invariant one problem.

I will talk about joint work with Robert Burklund: In Adams filtration 3, we prove an infinite family of non-trivial d₄-differentials on the classes  h_j³ for j ≥ 6, confirming a conjecture of Mahowald.  Our proof uses two different deformations of stable homotopy theory – C-motivic stable homotopy theory and F₂-synthetic homotopy theory – both in an essential way.

About the Speaker:
Zhouli Xu is a Chinese mathematician specializing in topology.  He is currently an Associate Professor of Mathematics at the University of California, San Diego.  Xu earned both his B.S. and M.S. in Mathematics from Peking University and his Ph.D. from the University of Chicago in 2017.  Before joining UCSD, Xu was a C.L.E. Moore Instructor at MIT. 
His research accomplishments include collaborations that established the unique smooth structure of the 61-dimensional sphere, a “10/8 + 4”-theorem addressing the geography problem in 4-dimensional topology, the development of the motivic deformation method and the Chow t-structure, and the computation of classical and motivic stable homotopy groups of spheres in previously uncharted dimensions.
Xu was a recipient of the Harper Dissertation Fellowship by the University of Chicago in 2016, a recipient of the K-Theory prize in 2022, an invited speaker at the International Congress of Mathematicians 2022, and a Fellow of the American Mathematical Society of class 2023.