Michael Ching

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This page contains some references to background on my research topics. See also Publications for a list of my publications and descriptions of some of my current research projects.

Goodwillie Calculus

The main references for the calculus of homotopy functors are Goodwillie's original sequence of papers:

• Calculus I, The first derivative of pseudoisotopy theory, Tom Goodwillie, K-Theory 4 (1990), no. 1, 1--27.
• Calculus II, Analytic functors, Tom Goodwillie, K-Theory 5 (1991/92), no. 4, 295--332.
• Calculus III, Taylor series, Tom Goodwillie, Geometry and Topology 7 (2003) 645-711.

A key role in my work is played by the derivatives of the identity functor on based spaces:

• The derivatives of homotopy theory, Brenda Johnson, Trans. Amer. Math. Soc. 347 (1995), no. 4, 1295--1321.
• The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, Greg Arone and Mark Mahowald, Invent. Math. 135 (1999), no. 3, 743--788.
• Partition complexes, Tits buildings and symmetric products, Greg Arone and Bill Dwyer, Proc. London Math. Soc. (3) 82 (2001), no. 1, 229--256.
• Bar constructions for topological operads and the Goodwillie derivatives of the identity, Michael Ching, Geom. Topol. 9 (2005), 833--933.

Much of my initial motivation for studying calculus of functors was to understand `chain rules' in this context, in particular for higher derivatives.

• A chain rule in the calculus of homotopy functors, John Klein and John Rognes, Geom. Topol. 6 (2002), 853--887.
• The curious history of Faà di Bruno's formula, Warren Johnson, Amer. Math. Monthly 109 (2002), no. 3, 217--234.
• A chain rule for Goodwillie derivatives of functors from spectra to spectra, Michael Ching, Trans. Amer. Math. Soc. 362 (2010), no. 1, 399--426.
• Operads and chain rules for the calculus of functors, Greg Arone and Michael Ching, to appear in Astérisque

The Taylor tower has been approached from a different direction by Randy McCarthy and Brenda Johnson:

• Taylor towers for functors of additive categories, Brenda Johnson and Randy McCarthy, J. Pure Appl. Algebra 137 (1999), no. 3, 253--284.
• Dual calculus for functors to spectra, Randy McCarthy, Homotopy methods in algebraic topology (Boulder, CO, 1999), 183--215, Contemp. Math., 271, Amer. Math. Soc., Providence, RI, 2001.
• Deriving calculus with cotriples, Brenda Johnson and Randy McCarthy, Trans. Amer. Math. Soc. 356 (2004), no. 2, 757--803.

Operads

The theory of Koszul duality for operad has been particularly significant in my work:

• Koszul resolutions, Stewart Priddy, Trans. Amer. Math. Soc. 152 1970 39--60.
• Koszul duality for operads, Victor Ginzburg and Mikhail Kapranov, Duke Math. J. 76 (1994), no. 1, 203--272.
• Operads, homotopy algebra, and iterated integrals for double loop spaces, Ezra Getzler and John Jones, preprint.
• Koszul duality of operads and homology of partition posets, Benoit Fresse, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic $K$-theory, 115--215, Contemp. Math., 346, Amer. Math. Soc., Providence, RI, 2004.

I am interested in the particular case of Koszul duality for operads arising from moduli spaces of Riemann surfaces:

• Operads and moduli spaces of genus $0$ Riemann surfaces, Ezra Getzler, The moduli space of curves (Texel Island, 1994), 199--230, Progr. Math., 129, Birkhäuser Boston, Boston, MA, 1995.
• Modular operads, Ezra Getzler and Mikhail Kapranov, Compositio Math. 110 (1998), no. 1, 65--126.

Algebraic K-theory of ring spectra

The Taylor tower of the algebraic K-theory functor has been studied in various ways by Randy McCarthy and coauthors:

• Stable K-theory and topological Hochschild homology, Bjørn Ian Dundas and Randy McCarthy, Ann. of Math. (2) 140 (1994), no. 3, 685--701.
• The Taylor towers for rational algebraic K-theory and Hochschild homology, Ruth Kantorovitz and Randy McCarthy, Homology Homotopy Appl. 4 (2002), no. 1, 191--212.
• The algebraic K-theory of extensions of a ring by direct sums of itself, Ayelet Lindenstrauss and Randy McCarthy, Indiana Univ. Math. J. 57 (2008), no. 2, 577--625.