Homepage of Daniel G. Davis

homepage last modified: 4/25/08
(a) [Publications and submissions] (b) [Preprints, etc.] (c) [Exposition related to research]
update on (3) in (a); added (xiii) and (xii) in (c); (0) in (b) added.
Daniel G. Davis
Assistant Professor
Department of Mathematics
University of Louisiana at Lafayette
e-mail: dgdavis AT louisiana DOT edu
(replace AT and DOT with their symbols; it's to reduce spam)
Research Interests: Algebraic Topology: (a) stable homotopy theory, especially from the chromatic perspective; (b) spectra with continuous actions by profinite groups, and their homotopy fixed points; (c) the Multiplicative Ring Spectra project of Paul Goerss and Mike Hopkins; and (d) Morava E-theory.
Curriculum Vitae [dvi] [pdf] - last updated: August 29, 2007.

Research Statement [dvi] [pdf] - posted on November 3, 2006.

ULL Topology Seminar: Spring, 2008.

Teaching: Math 250, Math 536, and teaching that I've done at ULL and at Wesleyan and Purdue Universities.

Publications and submissions

  • (1) "Homotopy fixed points for L_{K(n)}(E_n \wedge X) using the continuous action" [dvi] [pdf] Journal of Pure and Applied Algebra, 206(3): 322-354, 2006.

  • (2) "The E_2-term of the descent spectral sequence for continuous G-spectra" [dvi] [pdf] New York J. of Math. 12 (2006), 183-191. This paper lives at http://nyjm.albany.edu/j/2006/12-11.html. (The version available here is identical to the published version, except for changes required by the journal's style file.)

  • (3) To appear in the Ravenel/Wilson HHA proceedings: "Explicit fibrant replacement for discrete G-spectra" [dvi] [pdf] - (13 pp.) Submitted on 12/31/07, accepted on 4/2/08. Given a profinite group G with finite vcd, a closed subgroup H, and a discrete G-spectrum X, we construct an explicit fibrant replacement for X in the category of discrete H-spectra. Several applications of this fibrant model are given. The only differences between the version here and the published version are: changes required by the journal's latex guidelines; adding a thanks to the referee; and adding an entry to the bibliography. For a little more on the topic of this paper, please see (xii) in (c).

  • (4) Submitted for publication: "The homotopy orbit spectrum for profinite groups" [dvi] [pdf] - (13 pp.) Submitted on 8/8/06. Last updated: 8/8/06.

  • (5) Submitted for publication: "Epimorphic covers make R^+_G, for profinite G, a site" [pdf] - (10 pp.) Submitted on 10/24/06. Resubmitted on 3/14/08. Last updated: 3/14/08. Devinatz and Hopkins present the action of G_n and its quotients by closed normal subgroups K on E_n and E_n^{hK}, respectively, by constructing a functor from the category R^+_{G_n} to commutative S-algebras. In this paper, we show that R^+_G is a site when equipped with the pretopology of epimorphic covers.

  • (6) Submitted for publication: "Iterated homotopy fixed points for the Lubin-Tate spectrum" [pdf] - (18 pp.) Submitted on 10/26/06. Last updated: 3/11/07. Extends a result of Devinatz and Hopkins and relates to the former's paper "A Lyndon-Hochschild-Serre spectral sequence for certain homotopy fixed point spectra."

    Preprints, etc.

  • (5) (joint with Mark Behrens) "The homotopy fixed point spectra of profinite Galois extensions" [pdf] - (33 pp.) Paper in preparation. Current preliminary version last updated on: 9/17/05. (Current length of the actual working draft: 46 pp.) We are working on proving a theorem about certain function spectra - a special case deals with the relationship between (E_n[[G_n/H]])^{hG} and F(E_n^{hH}, E_n^{hG}), where H and G are closed subgroups of the extended Morava stabilizer group.

  • (4) Paper in preparation: "The homotopy fixed points of E_n and the Devinatz-Hopkins construction" - (4 pp.) Current version: April 28, 2006. Comparison of some of the homotopy fixed points described in paper (1) above with those given by Devinatz and Hopkins. Based on the second part of my thesis. For some details, see (viii) below.

  • (3) (joint with Takeshi Torii) Paper in preparation: "Realizing L_{K(n)}L_{K(n+1)}(X) for finite complexes by using sequentially continuous (G,H)-spectra" - (8 pp.) This paper shows that the stated spectrum is the iterated homotopy fixed point spectrum of a G-spectrum where both G and the spectrum combine chromatic heights n and n+1. Title, number of pages, and this "homepage abstract" were last updated on 10/20/07.

  • (2) In progress: "The site R^+_G, for G profinite, is subcanonical" - (8 pp.), last updated on 3/14/08.

  • (1) Work in progress: "Constructing ( K(KU_p) \wedge V(1) )^{h\mathbb{Z}_p^\times}." This homotopy fixed point spectrum is an ingredient in the generalized Lichtenbaum-Quillen conjecture of John Rognes. This work applies recent unpublished work of Ausoni and Rognes. Note: this work is preliminary - I think the idea for the construction is correct, but certain details have to be nailed down.

  • (0) Just an idea: Use the Joachim-Johnson model category structure to define homotopy fixed points, with respect to a finite group, for certain types of C*-algebras. The associated document is just a 19-line pdf file playing the "standard (-)^{hG} game" with the separable C*-algebras in the aforementioned model category. The hope is that, as in chromatic theory, there might be an important, but mysterious, C*-algebra A that turns out to be weakly-equivalent to B^{hG}, where B and G are better understood, thereby shedding useful light on A. In case there turns out to be anything to the idea, I should mention that my thinking it's worthwhile is aided by helpful dialogue with M.W. Johnson and P.W. Ng.

  • (-1) I am interested in using recent work in Lubin-Tate theory to increase the diameter of the pipe between rigid analytic geometry and chromatic theory, with the specific aim of extending Gross-Hopkins duality.

    Exposition related to research

  • (xiii) "Galois theory, commutative rings, and chromatic homotopy theory" [pdf] - (4 pp.) These are the notes that I texed up for myself, for a colloquium that I gave here at ULL on 4/24/08. These notes are written for a general audience of professors (working in various branches of math) and graduate students (so there are various simplifications, such as writing lim instead of holim).

  • (xii) A supplement to paper (3) in (a) [pdf] - (2 pp.) Posted on 4/5/08. An expert on the content of (3) asked me why the fibrant model constructed in this paper is not given immediately by [11, Proposition 3.3] (see (3)'s bibliography). As far as I know, the model in (3) is not given immediately by Jardine's paper; my reasons for thinking this are explained in this note. The key word here is "immediately" - see (3)'s Remark 2.7.

  • (xi) A letter regarding a research project in chromatic stable homotopy theory [dvi] [pdf] - (7 pp.) Dated: July 17, 2005. This is the text of a letter that I sent to Mike Hopkins about my research on the problem of realizing \pi_*(E_n)/I by a discrete G_n-symmetric ring spectrum. The letter summarizes my approaches to this problem and includes observations that might be relevant to solving the problem. Erratum: (a) Section 1 of the letter assumes that H_\infty is a point-set level notion, when it is actually a notion on the level of the homotopy category; (b) the last sentence of paragraph one in (2.9) claims to show that F_n and E_n/I_n^\infty cannot be identical, but Mark Hovey has shown me that my justification is erroneous.

  • (x) "Rognes's theory of Galois extensions and the continuous action of G_n on E_n" [dvi] [pdf] - (14 pp.) Inactive manuscript. Final update: 5/14/04. Also available at the Hopf archive. We study Galois extensions in the context of continuous G-spectra, where G is a profinite group. We explore the Lubin-Tate scenario from this perspective, assuming the conjecture that \pi_*(E_n)/I can be realized as a discrete G_n-symmetric ring spectrum E_n/I. The definitive reference for Galois extensions of spectra is the beautiful paper by Rognes titled "Galois extensions of structured ring spectra" (available at Rognes's website). Apart from the portions dependent on E_n/I, the content of this manuscript is subsumed by Rognes's paper. The content that deals with homotopy fixed points for towers of spectra is subsumed by paper (3) (with Behrens) described above.

  • (ix) "A proof of [1, Theorem 8.5]" [pdf] - (2 pp.) In this note, we provide the details for a proof of Theorem 8.5 in publication (1) listed above. Posted on 8/29/07.

  • (viii) "The Lubin-Tate spectrum and its homotopy fixed point spectra" [dvi] [pdf] - (106 pp.) Ph.D. Thesis. Completed May 9, 2003 at Northwestern University. Thesis Advisor: Paul G. Goerss. Here [dvi] [pdf] is a five-page note that summarizes the results of my thesis (plus slight modifications). I apply work by Devinatz and Hopkins, using machinery developed by Jardine and Thomason, to show that (a) the Lubin-Tate spectrum E_n has a continuous G_n-action by the extended Morava stabilizer group G_n; and (b) for any closed subgroup G of G_n, using the continuous action, I construct homotopy fixed point spectra (E_n)^{hG} with descent spectral sequences that are isomorphic to the K(n)-local E_n-Adams spectral sequences constructed by Devinatz and Hopkins for their homotopy fixed point spectra (defined without a continuous action). Also, we show that the K(n)-localization of a finite spectrum X is the G_n-homotopy fixed point spectrum of (E_n \wedge X).

  • (vii) "E_n as a continuous G_n-spectrum and its homotopy fixed point spectra" [pdf] - (13 slides) These are the slides from an invited talk I gave in the Homotopy Theory session at the National AMS Conference, January 18, 2003, Baltimore. We show that the Lubin-Tate spectrum has a continuous action by the Morava stabilizer group and using this continuous action, we construct homotopy fixed point spectra with descent spectral sequences.

  • (vi) "The Adams-Novikov spectral sequence for L_{K(n)}(X), when X is finite" [dvi] [pdf] - (5 pp.) Let X be a finite spectrum. We give a proof of the following fact: the Gal-fixed points of the continuous cohomology of S_n, with coefficients in \pi_t(E_n \wedge X), is the continuous cohomology of G_n, with the same coefficients. Posted on 9/16/05.

  • (v) "The G_n-action on E_n in the stable category" [dvi] [pdf] - (3 pp.) We provide the details for the well-known fact that G_n acts on E_n, in the homotopy category, by maps of ring spectra. The proof is elementary, and as far as the author knows, the details do not appear in the literature. Posted on 11/17/04.

  • (iv) "A potential definition for the classifying space BG_n, from the chromatic perspective" [dvi] [pdf] - (2 pp.) Since the extended Morava stabilizer group G_n is profinite, we suggest that the appropriate definition for BG_n is a tower of classifying spaces of finite groups, whose inverse limit is G_n. We recall a connection between these finite groups and the K(n)-local sphere.

  • (iii) "The Lubin-Tate moduli space in homotopy theory" [dvi] - (4 pp.) This is a brief introduction to how the Lubin-Tate moduli space, from algebraic geometry, appears in chromatic stable homotopy theory, via the Goerss-Hopkins-Miller theorem. All unnumbered references are to the note below.

  • (ii) "Finite CW-complexes and the chromatic tower" [dvi] [pdf] - (6 pp.) This is an introduction to the chromatic approach to stable homotopy theory, written for a general audience of mathematicians. None of the results discussed are due to the author.

  • (i) "Strongly complete profinite groups and compact p-adic analytic groups" [pdf] - (2 pp.) This short note discusses strongly complete profinite groups, compact p-adic analytic groups, and the Morava stabilizer group. Nothing here is really original, but I've found this collection of facts to be a useful reference for myself in my own research, when I'm working with the Morava stabilizer group.