12 problems

Let $E_n$ be the height-$n$ Morava $E$-theory spectrum, let $G_n$ be its Morava stabilizer group, and let $I$ range over a cofinal collection of ideals defining quotients $E_n/I$.…

Let $E_h$ be Morava $E$-theory at a prime $p$, with coefficient ring … Let $\mathbb{S}_h$ denote the Morava stabilizer group, and let $G\subseteq\mathbb{S}_h$ be a finite subgroup.…

Assume $p\geq 5$. Let $E_{p-1}$ be the height-$(p-1)$ Morava $E$-theory spectrum with its $\mathbb{Z}/p$-action induced by the corresponding subgroup of the Morava stabilizer group…

Non-splitting conjecture. These sequences do not split.

Rognes's conjecture. This map should be a nice' map in sufficiently high dimensions; in particular, after smashing with a finite $p$-local $CW$-spectrum of chromatic type $n+1$, it…

Chromatic Vanishing Conjecture. The following assertions hold for all heights $h$, primes $p$, and (co)-homological degrees $s$:

Reduced Homological Vanishing Conjecture. For the parameters under consideration, this map is an isomorphism

Let $E_*$ be the coefficient ring of Morava $E$-theory at height $n$, let $E_*E$ be its cooperations, and let $v_0=p,v_1,\ldots,v_{n-1}$ denote the usual periodicity elements. For…

Let $E_2$ be the height-$2$ Morava $E$-theory spectrum at the prime $2$, let $\mathbb{G}_2$ be its Morava stabilizer group, and let $G_{24}$, $C_6$, and $C_4$ be the finite subgrou…

Let $K\subset \mathbb{S}_n$ be a finite subgroup. Let $\rho\in E_n$ be placed in degree $-2$, let $S(\rho)$ denote the graded symmetric algebra, let $N$ be a trivial representation…

Ando–Hopkins–Strickland's conjecture. There is a small resolution of length $n$ for modules over $\Gamma$; equivalently, the ring of power operations of a height $n$ Morava $E$-the…

Let $E$ be a Morava $E$-theory associated to a one-dimensional formal group, and let $\mathcal{P}$ be its ring of power operations. A ring is Koszul when its residue field admits a…