Akhil Mathew

I am a PhD student in mathematics at Harvard University. My advisor is Jacob Lurie. My research is mostly focused on higher categorical and homotopy-theoretic methods in algebraic topology, algebraic K-theory, and algebraic geometry, and much has revolved around the ideas of nilpotence and descent and their applications.

From May-August 2015 and October-November 2016, I was a guest at the Hausdorff Institute of Mathematics in Bonn. For the 2014-2015 academic year, I was a graduate student at the University of California at Berkeley. Before that, I was an undergraduate student at Harvard.

**Postal address:** Department of Mathematics, One Oxford Street, Cambridge, MA 02138

**Email address:** amathew (at) math (dot) harvard (dot) edu

**CV:** .pdf

- Examples of descent up to nilpotence (last updated Dec. 2016).

A (mostly expository) survey of some examples of the ideas of descent and nilpotence, highlighting various questions. -
A short proof of telescopic Tate vanishing, with Dustin Clausen (last updated Jan. 2017). To appear in
*Proceedings of the AMS*.

We give a short proof of a theorem of Kuhn, based on the Kahn-Priddy theorem and the Bousfield-Kuhn functor, that Tate constructions for finite groups vanish in telescopically localized homotopy theory. -
Descent in algebraic K-theory and a conjecture of Ausoni-Rognes, with Dustin Clausen,
Niko Naumann and Justin Noel (last updated June 2016).

We prove various descent results in the algebraic K-theory of ring spectra. Our main results state that, after periodic localization, algebraic K-theory (and general additive invariants) satisfy descent for a wide class of morphisms, including many Galois extensions in the sense of Rognes. This confirms several cases of a conjecture of Ausoni-Rognes. -
Torus actions on stable module categories, Picard groups, and localizing subcategories (last updated Dec. 2015).

Studies a natural torus action on the stable module category of an abelian p-group, whose homotopy fixed points can be described using Galois descent. This leads to a new proof of Dade's theorem on the Picard group and gives an approach to the classification of localizing subcategories (which is a slight modification of Benson-Iyengar-Krause's proof). -
Picard groups of higher real K-theory spectra at height p-1, with Drew Heard and Vesna Stojanoska (last updated Nov. 2015). To appear in
*Compositio Mathematica*.

Using a spectral sequence based on Galois descent, we show that the Picard group of the higher real K-theory spectra of Hopkins-Miller at the height p-1 is always cyclic. This confirms a general expectation about such Picard groups in this case. -
Derived induction and restriction theory, with Niko Naumann and Justin Noel
(last updated Jan. 2017).

We show that the F-nilpotence of a G-spectrum implies forms of Artin and Brauer induction in equivariant cohomology theories as well as variants of Quillen's F-isomorphism results. Many more examples of F-nilpotence are given. In particular, the best possible families ("derived defect bases") are determined for numerous classes of equivariant ring spectra. -
Nilpotence and descent in equivariant stable homotopy theory, with Niko Naumann and Justin Noel.
*Adv. Math.***305**(2017), 994--1084.

For G a finite group and F a family of subgroups, introduces a class of G-spectra called "F-nilpotent." These G-spectra can be recovered from their restrictions to subgroups in F in a strong homotopical sense. In the paper, decompositions of categories of equivariant module spectra are developed and used to show (following an argument of Quillen) that complex orientability often leads to nilpotence for the family of abelian subgroups. -
THH and base-change for Galois extensions of ring spectra (last updated June 2016). To appear in
*Algebraic and Geometric Topology*.

Studies when THH satisfies a base-change formula for Galois extensions of ring spectra. We show that many natural Galois extensions (such as KO to KU) do satisfy base-change but give a counterexample over F_p. We also give a categorical interpretation of the results of Weibel-Geller and McCarthy-Minasian for \'etale extensions. -
Residue fields for a class of
rational E_\infty-rings and applications
*J. Pure Appl. Algebra***221**(2017), no. 3., 707--748.

Describes certain arithmetic invariants (thick subcategories, Galois groups, and partially Picard groups) of a class of rational E_\infty-rings, by constructing residue fields and proving an analog of the nilpotence theorem. Latest version includes counterexamples in the non-noetherian case. -
The Picard group of topological modular forms via descent theory, with Vesna Stojanoska.
*Geom. Topol.***20**(2016), no. 6, 3133--3217.

Describes descent-theoretic methods in the study of Picard groups of structured ring spectra. The primary application is the calculation of the Picard groups of TMF and Tmf. The former Picard group is cyclic, but we find exotic elements in the latter. -
Torsion exponents in stable homotopy and the Hurewicz homomorphism.
*Algebr. Geom. Topol.***16**(2016), no. 2, 1025--1041. arXiv version.

Determines bounds on the torsion exponents for the kernel and cokernel of the integral Hurewicz homomorphism of a connective spectrum. The bounds are provably the best possible up to small constants at each prime. Applications include exponent theorems for the equivariant stable stems. -
The homology of tmf.
*Homology Homotopy Appl.***18**(2016), no. 2., 1--29. arXiv version.

An approach, based on the identification of a certain vector bundle on the moduli stack of elliptic curves, to the calculation of the mod 2 homology of (connective) tmf. -
The Galois group of a stable homotopy
theory.
*Adv. Math.***291**(2016), 403-541. arXiv version.

Formulates a version of the Galois correspondence for E_\infty-ring spectra, following ideas originally introduced by Rognes, and describes several homotopy-theoretic Galois groups. In addition, discusses various ideas in descent theory for ring spectra. My undergraduate thesis. -
Fibers of partial totalizations of a
pointed cosimplicial space, with Vesna Stojanoska.
*Proc. Amer. Math. Soc.***144**(2016), no. 1, 445--458. arXiv version.

Shows that fibers between totalizations can be delooped in a certain range. -
On a nilpotence conjecture of J.P. May, with Niko Naumann and Justin Noel.
*J. Topol.***8**(2015), no. 4, 917-932. arXiv version.

Proves that integral homology is sufficient to detect nilpotence of elements in H_\infty-ring spectra. -
A thick subcategory theorem for modules over certain ring spectra.
*Geom. Topol.***19**(2015), no. 4., 2359-2392. arXiv version.

Stratifies thick subcategories of perfect modules over TMF and related ring spectra in terms of the relevant moduli stack. -
Affineness and chromatic homotopy theory, with Lennart Meier.
*J. Topol.***8**(2015), no. 2, 476-528. arXiv version.

Shows that certain derived stacks in chromatic homotopy theory (such as the derived moduli space of elliptic curves that leads to TMF) have an affineness property at the level of quasi-coherent sheaves.

*Note*: There is a small mistake in the published version in the results on Tmf with level structures. This is corrected in the arXiv version (last updated June 2016). -
Categories parametrized by schemes and representation theory in complex
rank.
*J. Algebra***381**(2013), 140-163. arXiv version.

A study of certain continuous families of categories in representation theory at "generic" points.

- Notes for my mini-course "Nilpotence, descent, and algebraic K-theory" at the 2016 Young Topologists' Meeting in Copenhagen.
- Video for my talk "Galois extensions and Picard groups of ring spectra" at the Fields Institute Workshop on Group Actions - Classical and Derived.
- Video for my talk "On a nilpotence conjecture of J.P. May" at the BIRS workshop on Operations in Highly Structured Homology Theories.
- Video for my talk "Descent in algebraic K-theory" at the BIRS workshop on Equivariant Derived Algebraic Geometry.
- Notes for my talk "Descent and nilpotence in stable homotopy theory" at the 2015 Oberwolfach "Homotopy Theory" workshop.
- Slides for my talk "Descent for ring spectra and applications" at the 2015 Joint Mathematics Meetings.

- Expository writings
- Notes from selected courses
- The CRing project: an open-source, collaborative commutative algebra textbook
- Past seminars organized: Classical topics in topology seminar, 2012. Derived algebraic geometry seminar, 2011
- Blog