Related papers: The Multiplication on BP

We equip $\mathrm{BP} \langle n \rangle$ with an $\mathbb{E}_3$-$\mathrm{BP}$-algebra structure, for each prime $p$ and height $n$. The algebraic $K$-theory of this ring is of chromatic height exactly $n+1$, and the map…

We characterize ring spectra morphisms from the algebraic cobordism spectrum $\QTR{Bbb}{MGL}$ (\QCITE{cite}{}{Vo1}) to an oriented spectrum $\QTR{Bbb}{E}$ (in the sense of Morel \QCITE{cite}{}{Mo}) via formal group laws on the…

We prove the multiplicity formula for the automorphic discrete spectrum of the metaplectic group $\mathrm{Mp}_4$ of rank $2$.

We compute the homotopy groups of spectra associated by a theorem of Lurie to the Shimura curves of discriminants 6, 10, and 14, beginning with a computation of integral rings of automorphic forms on these curves. As an application, we find…

We show that the odd-primary Brown-Peterson spectrum $\mathrm{BP}$ does not admit the structure of an $\mathbb{E}_{2(p^2+2)}$ ring spectrum and that there can be no map $\mathrm{MU} \to \mathrm{BP}$ of $\mathbb{E}_{2p+3}$ ring spectra. We…

Inspired by Stewart Priddy's cellular model for the $p$-local Brown-Peterson spectrum $BP$, we give a construction of a $p$-local $E_\infty$ ring spectrum $R$ which is a close approximation to $BP$. Indeed we can show that if $BP$ admits an…

We describe an iterable construction of THH for an E_n ring spectrum. The reduced version is an iterable bar construction and its n-th iterate gives a model for the shifted cotangent complex at the augmentation, representing reduced…

Let R be an E_2 ring spectrum with zero odd dimensional homotopy groups. Every map of ring spectra MU to R is represented by a map of E_2 ring spectra. If 2 is invertible in pi_0(R), then every map of ring spectra MSO to R is represented by…

BPS spectrum with finite number of states are found for higher rank four dimensional N=2 theory engineered from six dimensional A_{N-1} (2,0) theory on a Riemann surface with various kinds of defects. The wall crossing formula is…

We produce refinements of the known multiplicative structures on the Brown--Peterson spectrum $BP$, its truncated variants $BP\langle n \rangle$, Ravenel's spectra $X(n)$, and evenly graded polynomial rings over the sphere spectrum.…

We uncover a somewhat surprising connection between spaces of multiplicative maps between $A_\infty$-ring spectra and topological Hochschild cohomology. As a consequence we show that such spaces become infinite loop spaces after looping…

We determine the BP-module structure, mod higher filtration, of the main part of the BP-homology of elementary abelian 2-groups. The action is related to symmetric polynomials and to Dickson invariants.

This paper begins with an exposition of the author's research on the category of BP_*BP-comodules, much of which is joint with Neil Strickland. The main result of that work is that the category of E(n)_*E(n)-comodules is equivalent to a…

Every homology or cohomology theory on a category of E-infinity ring spectra is Topological Andre-Quillen homology or cohomology with appropriate coefficients. Analogous results hold for the category of A-infinity ring spectra and for…

In this article we show that $\mathbb{S}/8$ is an $\mathbb{E}_1$-algebra, $\mathbb{S}/32$ is an $\mathbb{E}_2$-algebra, $\mathbb{S}/p^{n+1}$ is an $\mathbb{E}_n$-algebra at odd primes and, more generally, for every $h$ and $n$ there exist…

When $p$ is an odd prime, we prove that the $\mathbb F_p$-cohomology of $\mathrm{BP}\langle n\rangle$ as a module over the Steenrod algebra determines the $p$-local spectrum $\mathrm{BP}\langle n\rangle$. In particular, we prove that the…

Making use of a droplet-epitaxial technique, we realize nanometer-sized quantum ring complexes, consisting of a well-defined inner ring and an outer ring. Electronic structure inherent in the unique quantum system is analyzed using a…

Let E=BP<n> denote the Johnson-Wilson spectrum, localized at p. It is proved that if E_*(X) is locally finite, then there is an isomorphism of right E_*-modules E^*(X) = (E_*(Sigma^{D+n+1}X))^V, where D=Sum |v_i| and M^V=Hom(M,Q/Z) is the…

We calculate the mod (p, v_1, v_2) homotopy V(2)_* TC(BP<2>) of the topological cyclic homology of the truncated Brown--Peterson spectrum BP<2>, at all primes p\ge7, and show that it is a finitely generated and free F_p[v_3]-module on 12p+4…

The group ring of the automorphism group of a p-group is studied using the automorphism groups of subgroups and quotient groups of P.