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Paul Pearson

Research

My field is algebraic topology, and I study the relationship between certain moduli spaces arising in algebraic geometry and their applications to the homotopy of spectra. Specifically, I use the stratification of the moduli stack of formal groups by height to study the chromatic stratification of the stable homotopy groups of spectra.

Calculations

• Ext_A(2)^s,t(F₂,F₂) => π_*(tmf) (Click on the chart to make it large enough to view.)
  The Adams spectral sequence for the homotopy of the spectrum of topological modular forms at the prime 2. Based on work by Hopkins and Mahowald. This chart is |v₂³²| = 192 periodic. Differentials are red, hidden extensions are green, and many infinite h₀ towers have been truncated. Updated August 26, 2008


• The mod p cohomology of Brown-Gitler spectra (pdf) (dvi) (ps) (djvu)
  We calculate the mod 2 cohomology and mod 3 cohomology of some Brown-Gitler spectra, and describe the mod p cohomology of the Brown-Gitler spectra. The role these cohomology calculations play in determining the mod p Dieudonné ring of a cohomology theory E with cup products is briefly discussed in the introduction. Some calculations of the anti-automorphism of the mod p Steenrod algebra are given as a reference. Updated September 24, 2005


• The Witt Vector Affine Ring Scheme (pdf) (dvi) (ps) (djvu)
  We describe the ring of Witt vectors as an affine ring scheme, and then describe the objects that corepresent this affine ring scheme together the structure homomorphisms that make it corepresent a ring-valued functor. We call the corepresenting object the co-Witt ring. Further, we describe how to make calculations in the ring of Witt vectors and give some sample calculations. Updated September 25, 2005


• Calculating Formal Group Laws (pdf) (dvi) (ps) (djvu)
  We calculate the formal group laws associated to complex cobordism (the universal formal group law), Brown-Peterson theory (the universal p-typical formal group law), the Johnson-Wilson theory E(n) and its connective version which is the truncated Brown-Peterson theory BP<n>, Morava E_n-theory (also known as Lubin-Tate theory), Morava K(n)-theory, the elliptic curve Hopf algebroid, and some specific supersingular elliptic curves. Formal group laws classified by the right unit map in Hopf algebroids are also constructed, as they represent the most general change of coordinates. Updated September 25, 2005


• Calculating the right unit, coproduct, and conjugation in the Hopf algebroid associated to Brown-Peterson theory (pdf) (dvi) (ps) (djvu)
  We describe how to calculate the right unit, coproduct, and conjugation in the Hopf algebroid associated to Brown-Peterson theory, give examples of calculations in terms of the Araki and Hazewinkel generators, and supply Maple source code for making these calculations. Updated September 25, 2005


• PRTensor (html) (mws)
  Maple procedures for making calculations in the tensor product of polynomial rings. Updated October 4, 2005


• Cocycle representatives in Ext(BP_*) at p=3
  Cocycle representatives in the Adams-Novikov spectral sequence at the prime 3. Updated December 22, 2006


• Calculations in the Steenrod Algebra (pdf) (dvi) (ps) (djvu)
  We calculate the admissible basis for the mod 2 and mod 3 Steenrod algebras, the Adem relations, the anti-automorphism of the Steenrod algebra, and the Milnor Bockstein operations. Updated July 15, 2005


• Adams operations in representation rings of some finite groups using Maple:
  
  • Adams Operations on the Representation Ring of the Dihedral Group of Order 8 (html) (mws). Updated October 4, 2005
  • Adams Operations on the Representation Ring of the Quaternion Group of Order 8 (html) (mws). Updated October 4, 2005
  • Adams Operations on the Representation Ring of the Symmetric Group on 3 Elements (html) (mws). Updated October 4, 2005
  • Adams Operations on the Representation Ring of the Symmetric Group on 4 Elements (html) (mws). Updated October 4, 2005