16 problems

Let $K(n)$ be Morava $K$-theory, with periodicity generators $p,v_1,\ldots,v_{n-1}$. Its topological Hochschild cohomology is the spectrum $THH^{\widehat{E(n)}}(K(n))$. Morava K-th…

Let $K(n)$ denote the $2(p^n-1)$-periodic Morava $K$-theory. Its topological Hochschild homology, denoted $THH(K(n))$, may depend a priori on the chosen $A_\infty$ structure. Morav…

Let $K(n)^*(-;\mathbb{Z}_p)$ be the specialized height-$n$ Morava K-theory functor, and let $A$ range over flat $\mathbb{Z}_p$-algebras. Consider the functor … from flat $A$-module…

Let $k$ be a field, let $p$ be a prime, and let $\operatorname{K}(n)$ be algebraic Morava K-theory at $p$. Assume that the invertible motives $L_{\alpha}$ exist for the indicated c…

Let $Q$ be an anisotropic quadric over $k$, with defining quadratic form $q$, and let $\widetilde{M}_{\operatorname{K}(n)}(Q)$ be its $\operatorname{K}(n)$-kernel motive. Invertibl…

Let $k$ be a field, let $Q$ be a quadric over $k$, and let $L$ be an invertible direct summand of its $\operatorname{K}(n)$-motive. Descent conjecture for invertible Morava summand…

Let $k$ be a field containing $p^r$-th roots of unity for all $r$. Let $n\ge 1$, and for $m\ge 0$ let $\operatorname{K}(m)$ be the $m$-th algebraic Morava K-theory at a prime $p$.…

Let $K$ be a compact connected Lie group, $T$ its maximal torus, and $\pi\colon K\rightarrow K/T$ the natural projection. Denote by $G=K_{\mathbb C}$ the corresponding split reduct…

Let $C_d({\mathbb R}^m)$ be the cofiber of the inclusion $\operatorname{Gr}_d({\mathbb R}^{m-1})\hookrightarrow\operatorname{Gr}_d({\mathbb R}^m)$, and let…

Let $n\geq 1$, let $\epsilon$ be $0$ or $1$, and let $l\geq 0$. Set $m=2^{n+1}-\epsilon+2l$, and let $k_n(\operatorname{Gr}_d({\mathbb R}^m))$ denote the dimension of…

Let $n\geq 1$, and let $\operatorname{Gr}_d({\mathbb R}^m)$ be the real Grassmannian of $d$-planes in $\mathbb R^m$. Consider the Atiyah–Hirzebruch spectral sequence computing its…

Let $y(n)$ be the connective Adams summand at height $n$, let $TP(y(n))[k]$ denote the $k$th stage of the Greenlees filtration of $TP(y(n))$, and let $K(m)$ denote Morava K-theory…

Sati's conjecture. A twisted form of Morava K-theory and E-theory should describe an extension of the untwisted setting in the study of anomalies at chromatic level two.

Let $\widetilde{Q}_2$ denote the integral lift of the second Milnor primitive, and let $H_7$ denote the twist contribution in degree $7$. Consider the Atiyah–Hirzebruch spectral se…

Let $BP\langle n\rangle$ denote the Johnson–Wilson spectrum at the prime $2$, with coefficient ring $\mathbb{Z}_{(2)}[v_1,\dots,v_n]$. Assume that $BP\langle n\rangle$ admits the s…

Let $K_n$ be the $2$-periodic Morava $K$-theory spectrum associated to a formal group of height $n$ over a perfect field $k$ of characteristic $p$, with homotopy groups … The modul…