# Homepage of Daniel G. Davis

homepage last modified (though not up-to-date): 9/15/19 ... ..

(a) [**Publications and submissions**]
(b) [**Preprints, etc.**]
(c) [**Documents related to research**]

- **In preparation**: (the following title refers to (not all of) what's currently in the document, which is a shadow of what's in prep.) "\hat{L}^{h\Gamma_m} is dualizable," joint with Thomas Credeur; as of 9/15/19: 2 pages; here, \hat{L} is the K(n)-local sphere; uses the Jardine/Thomason framework for homotopy fixed points.

- **Accepted** for publication (joint with Hans-Werner Henn, Rick Jardine, Mark W. Johnson, Charles Rezk): Volume 729 in the AMS CONM series; a conference proceedings (from the 2017 conference "Homotopy theory: tools and applications," at UIUC). Estimated publication date: May, 2019.

- (joint with Gereon Quick and Takeshi Torii) the manuscript "A theory of continuous G-spectra, where G is profinite: ..." is currently 82 pages long.

- [Here] are the slides for my 6/2/15 Univ. of Chicago Topology Seminar (Note about p. 32: "Draft theorem" means that I want to go over the texed-up proof in the manuscript before removing the word "Draft.")

- [Here] are the slides for my 5/24/15 talk at the 2015 Lehigh University Geometry and Topology Conference. These slides are a shorter version of the above "Chicago slides."

- Ph.D. students:

Vojislav Petrovic: currently (Fall, 2017) an instructor at UL (no longer a student); Dec. '17 is expected date of degree.

Thomas Credeur

- in (a): (15) was published on Sept. 12, 2016 in Algebraic & Geometric Topology;

- in (b): updated entry (3) (the pdf is currently 30 pages);

- in (a): (14) was published in Geometry & Topology;

- in (a): (12) was published online in the Glasgow Mathematical Journal;

- in (a): (11) was published;

- very much a work in progress: [Homotopy Theory at ULL];

- in (a): (8), (9), and (10) are now published; (13) is still under revision.

**Daniel G. Davis**

**Associate Professor** (Aug. 2013 - present)

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** [pdf] - last updated:
January 17, 2013. A document not listed on my CV: my notes from the Fall '11 Math 635 (Algebraic Varieties) and Spring '12 Math 636 (Sheaves and Cohomology) courses are partly contained in a 56-page pdf file (a decent rough draft; last updated: 1/27/13). I would like to improve and expand these notes as the years go by, although this expository project is in the background.

On December 14th, 2018, Jonathan Steven Prieto Cubides (University of Bergen) gave a talk in the Topology Seminar titled "Dependent paths in homotopy type theory." The main slides for his talk are here: for optimal viewing, you need to rotate them or perhaps it is best to just download them.

Here is the webpage for the conference "Homotopy theory: tools and applications," at the University of Illinois at Urbana-Champaign, July 17-21, 2017. This conference is sponsored by National Science Foundation grant DMS-1719242, as well as the UIUC Department of Mathematics.

**Some information** about the Nov. 8-10, 2013 **Lloyd Roeling ULL Topology Conference** that I co-organized follows. The slides for the talk of Andrew Salch are here, and the slides for the talk of Takeshi Torii can be viewed here. Here is the conference's restaurant webpage.

**Research Statement** [pdf] - This has not been updated in a long time: posted on November 3, 2006.

**Seminar organization** that I've done: the UL Topology Seminar, in **Spring**, 2008; **Spring**, 2009; **Spring**, 2010; a brief "Stacks and homotopy theory seminar"; a "learning seminar" in algebraic geometry in Summer, 2010.

**The ULL Topology Seminar** from Jan. 2008 to May 2015: Here is a slightly tweaked version of the schedule for the seminar. This schedule was not produced by me; its copyright is at the bottom; it's here for archival purposes. Please excuse those links (just a few!) that are broken.

**Teaching:** Spring '13: Math 250, 301, and 536. Fall '12: Math 301, 440, and 535. Fall '11: Math 250, Math 635. Old: Spring '09: Math 270, 538, and 591. Here's the **teaching** that I've done at ULL and at Wesleyan and Purdue Universities.

## Publications and submissions

**(15)** (joint with Gereon Quick): **"Profinite and discrete G-spectra and iterated homotopy fixed points"** [pdf]. Algebraic & Geometric Topology, 16 (4), 2016, 2257-2303. The tex file for (this webpage's) pdf used the AGT style file. A quick introduction to this paper is given in the long abstract in the original version.

**(14) To appear** in Geometry & Topology (joint with Tyler Lawson): **"Commutative ring objects in pro-categories and generalized Moore spectra"** [pdf], 39 pp. Submitted on 8/22/12. Last updated on 7/5/13. (The tex file for this pdf file used a style file that is a modification (done by me) of the style file for Geometry & Topology.)

**(13) Submitted** for publication: **"The homotopy orbit spectrum for profinite groups"** [dvi] [pdf], 13 pp. Submitted on 8/8/06. Last updated: Spring, 2007.

**(12) Accepted** for publication (joint with Tyler Lawson): **"A descent spectral sequence for arbitrary K(n)-local spectra with explicit E_2-term"**
[pdf], 11 pp. Last updated on 2/16/13, Glasgow Mathematical Journal.

**(11) "Homotopy fixed points for profinite groups emulate homotopy fixed points for discrete groups"** (published here) [pdf]. New York Journal of Mathematics, 19 (2013), 909-924.

**(10) Submitted** for publication: **"Function spectra and continuous G-spectra"** [pdf], 10 pp. Submitted on 1/6/11.

**(9) Submitted** for publication (joint with Takeshi Torii): **"Every K(n)-local spectrum is the homotopy fixed points of its Morava module"** [pdf], 6 pp. Submitted on 12/17/10. Theorem 1.3 in this paper (it's "Theorem 1.1" in the published version) might be my favorite (joint) theorem, out of all of my (sometimes joint) results.

**(8) Accepted** for publication: **"Delta-discrete G-spectra and iterated homotopy fixed points"** [pdf], 40 pp. Last updated on 7/22/11, Algebraic & Geometric Topology.

**(7) "Obtaining intermediate rings of a local profinite Galois extension without localization"** [pdf]. Journal of Homotopy and Related Structures, 5 (1), 2010, 253-268.

**(6)** (joint with Mark Behrens) **"The homotopy fixed point spectra of profinite Galois extensions"** [pdf]. Transactions of the American Mathematical Society, 362 (9), 2010, 4983–5042. A special case of our main result shows that (E_n[[G_n/H]])^hG and F(E_n^hH, E_n^hG) are equivalent, where H and G are closed subgroups of G_n. Also, we show that (a) the Devinatz-Hopkins construction of E_n^hG is equivalent to the construction of the homotopy fixed point spectrum E_n^hG defined with respect to the continuous action of G using the right derived functor of fixed points, and (b) the Adams spectral sequence for the former is isomorphic to the descent spectral sequence for the latter.

**(5) "Iterated homotopy fixed points for the Lubin-Tate spectrum"** with appendix (joint with Ben Wieland) **"An example of a discrete G-spectrum that is not hyperfibrant"** [pdf]. Topology and its Applications, 156 (2009), 2881-2898. 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."

**(4) "Epimorphic covers make R^+_G a site, for profinite G"** (available here). Theory and Applications of Categories, Vol. 22, 2009, No. 16, 388-400. Devinatz and Hopkins define the category R^+_{G_n} to organize the action of G_n and its quotients by closed normal subgroups K on E_n and E_n^{hK}, respectively. In this paper, we show that R^+_G is a site when equipped with the pretopology of epimorphic covers.

**(3) "Explicit fibrant replacement for discrete G-spectra"** (available here). Homology, Homotopy and Applications, 10(3): 137-150, 2008. For a little more on the topic of this paper, please see (xii) in (c).

**(2) "The E_2-term of the descent spectral sequence for continuous G-spectra" ** (available here). New York J. of Math., 12 (2006), 183-191.

**(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.

## Preprints, etc.

**(3)** **Paper in preparation** on constructing ( K(KU_p) \wedge v_2^{-1}V(1) )^{h\mathbb{Z}_p^\times}, 30 pp., last updated on 7/29/15. When p > 3, we construct the aforementioned homotopy fixed point spectrum, an ingredient in the generalized Lichtenbaum-Quillen conjecture of Christian Ausoni and John Rognes, and show that the associated descent spectral sequence is strongly convergent.

**(2)** (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.

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

**(.2)** **Work in progress:** an analogue of the etale fundamental group in the context of Lurie's derived schemes, and also on a point-set level in the context of commutative S-algebras (using the framework of Toen & Vezzosi). (Posted on 3/30/09.)

## Documents related to research

**(xv) "A Lyndon-Hochschild-Serre-type spectral sequence for discrete G-spectra"** [pdf] - (3 pp.) These are the notes that I texed up (and upon which my slides were based) for a 10-minute bell show talk on August 4th, 2011, at the conference "Structured Ring Spectra - TNG," in Hamburg, Germany.

**(xiv) "Arbitrary Morava modules, their Adams spectral sequence, and continuous group cohomology"** [pdf] - (4 pp.) These are the notes that I texed up for myself, for a colloquium that I gave here at UL on 2/10/11. After giving the colloquium, the theorem stated at the end morphed into being part of a project that is joint with Ping Wong Ng.

**(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"** [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 (7) (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 [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"** [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"** [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"** [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"** [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.