In joint work with Gabriel Angelini-Knoll, Christian Ausoni, Dominic Leon Culver and Eva Höning, we calculate the $\bmod \left(p, v_1, v_2\right)$ homotopy $V(2)_* T C(B P\langle 2\rangle)$ of the topological cyclic homology of the truncated Brown-Peterson spectrum $B P\langle 2\rangle$, at all primes $p \geq 7$, and show that it is a finitely generated and free $\mathbb{F}_p\left[v_3\right]$-module on $12 p+4$ generators in explicit degrees within the range $-1 \leq * \leq 2 p^3+2 p^2+2 p-3$. Our computation is the first that exhibits chromatic redshift from pure $v_2$-periodicity to pure $v_3$-periodicity in a precise quantitative manner.[-]

Hilbert's Nullstellensatz is a fundamental result in commutative algebra which is the starting point for classical algebraic geometry. In this talk, I will discuss a version of Hilbert's Nullstellensatz in chromatic homotopy theory, where Lubin-Tate theories play the role of algebraically closed fields. Time permitting, I will then indicate some of the applications of the chromatic nullstellensatz including to redshift for the algebraic K-theory of commutative algebras. This is joint work with Tomer Schlank and Allen Yuan.[-]

I will explain a construction of the motivic filtration on the topological cyclic homology of ring spectra, generalizing work of Bhatt–Morrow–Scholze and Bhatt–Lurie on the topological cyclic homology of discrete rings. This is joint with Arpon Raksit and Dylan Wilson. As time permits, I will discuss works in progress using the motivic spectral sequence to obtain new calculations in algebraic $K$-theory and prove higher chromatic variants of local Tate duality.[-]