Akhil Mathew

Associate Professor
Department of Mathematics, University of Chicago
5734 S. University Ave., Chicago, IL 60637-1514

Email: amathew (at) math (dot) uchicago (dot) edu
Office: 326 Eckhart Hall
Research interests: homotopy theory and arithmetic geometry

Midwest Topology Seminar, Nov. 2-3, 2024.

There was an informal workshop on p-adic geometry at UChicago, September 16--20, 2024. For more information see the conference website. Recordings of (most of) the lectures are available here.

Editorial work: Compositio Mathematica, International Mathematics Research Notices, Journal of Topology, Nagoya Mathematical Journal.

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Teaching

• Fall 2024: Prismatization.


• Spring 2024: Honors Basic Algebra III (tentative syllabus).


• Winter 2024: Honors Basic Algebra II (syllabus) and Algebra II (syllabus) .


• Spring 2023: Introduction to Algebraic Topology (syllabus; more information on on Canvas) and Topics in Commutative Algebra.


• Winter 2023: Algebra II (first-year graduate algebra).


• Fall 2022: Condensed Mathematics, with Matthew Emerton.

Papers and preprints

• The mod p Riemann-Hilbert correspondence and the perfect site. Tunis. J. Math. 5 (2023), no. 2, 369--403.


• The K-theory of perfectoid rings, with Benjamin Antieau and Matthew Morrow. Doc. Math. 27 (2022), 1923--1952.


• Syntomic complexes and p-adic étale Tate twists, with Bhargav Bhatt. Forum Math. Pi 11 (2023), E1.


• K-theory and polynomial functors, with Clark Barwick, Saul Glasman, and Thomas Nikolaus.


• Some recent advances in topological Hochschild homology. Bull. Lond. Math. Soc. 54 (2022), no. 1, 1-44.


• Descent and vanishing in chromatic algebraic K-theory via group actions, with Dustin Clausen, Justin Noel, and Niko Naumann. To appear in Annales Sciéntifiques de l'École Normale Supérieure.


• Purity in chromatically localized K-theory, with Markus Land, Lennart Meier, and Georg Tamme. To appear in JAMS.


• On K(1)-local TR. Compos. Math. 157 (2021), no. 5, 1079--1119.


• On the Beilinson fiber square, with Benjamin Antieau, Matthew Morrow, and Thomas Nikolaus. Duke Math J. 171 (2022), no. 18, 3707--3806.


• Remarks on K(1)-local K-theory, with Bhargav Bhatt and Dustin Clausen. Selecta Mathematica. (N.S.) 26 (2020), no. 3, Paper No. 39, 16pp.


• Faithfully flat descent of almost perfect complexes in rigid geometry. J. Pure Appl. Algebra 226 (2022), no. 5, Paper No. 106938 .


• Counterexamples to Hochschild-Kostant-Rosenberg in characteristic p, with Benjamin Antieau and Bhargav Bhatt. Forum Math. Sigma. 9 (2021), Paper No. e49, 26pp.


• Hyperdescent and étale K-theory, with Dustin Clausen. Invent. Math. 225 (2021), no. 3, 981--1076.


• Deformation theory and partition Lie algebras, with Lukas Brantner.


• The arc-topology, with Bhargav Bhatt. Duke Math. J. 170 (2021), no. 9, 1899--1988.


• Revisiting the de Rham-Witt complex, with Bhargav Bhatt and Jacob Lurie. Astérisque, No. 424, (2021).


• K-theory and topological cyclic homology of henselian pairs, with Dustin Clausen and Matthew Morrow. J. Amer. Math. Soc. 34 (2021), 411--473.


• Kaledin's degeneration theorem and topological Hochschild homology. Geom. Topol. 24 (2020), no. 6, 2675--2708.


• On the Blumberg-Mandell Künneth theorem for TP, with Benjamin Antieau and Thomas Nikolaus. Selecta Mathematica N.S. 24 (2018), no. 5, 4555--4576.


• Monadicity of the Bousfield-Kuhn functor, with Rosona Eldred, Gijs Heuts, and Lennart Meier. Proc. Amer. Math. Soc. 147 (2019), no. 4, 1789--1796.


• Examples of descent up to nilpotence. Geometric and topological aspects of the representation theory of finite groups, 269--311, Springer Proc. Math. Stat., 242, Springer, Cham, 2018.


• A short proof of telescopic Tate vanishing, with Dustin Clausen. Proc. Amer. Math. Soc. 145 (2017), no. 12, 5413--5417.


• Descent in algebraic K-theory and a conjecture of Ausoni-Rognes, with Dustin Clausen, Niko Naumann and Justin Noel. J. Eur. Math. Soc. (JEMS) 22 (2020), no.4, 1149--1200.


• Torus actions on stable module categories, Picard groups, and localizing subcategories (last updated Dec. 2015).


• Picard groups of higher real K-theory spectra at height p-1, with Drew Heard and Vesna Stojanoska. Compos. Math. 153 (2017), no. 9, 1820--1854.


• Derived induction and restriction theory, with Niko Naumann and Justin Noel. Geom. Topol. 23 (2019), no. 2, 541--636.
  


• Nilpotence and descent in equivariant stable homotopy theory, with Niko Naumann and Justin Noel. Adv. Math. 305 (2017), 994--1084.
  


• THH and base-change for Galois extensions of ring spectra. Algebr. Geom. Topol. 17 (2017), no. 2, 693--704.
  


• Residue fields for a class of rational E_\infty-rings and applications. J. Pure Appl. Algebra 221 (2017), no. 3., 707--748.
  


• The Picard group of topological modular forms via descent theory, with Vesna Stojanoska. Geom. Topol. 20 (2016), no. 6, 3133--3217.
  


• Torsion exponents in stable homotopy and the Hurewicz homomorphism. Algebr. Geom. Topol. 16 (2016), no. 2, 1025--1041.
  


• The homology of tmf. Homology Homotopy Appl. 18 (2016), no. 2., 1--29.
  


• The Galois group of a stable homotopy theory. Adv. Math. 291(2016), 403-541.
  


• Fibers of partial totalizations of a pointed cosimplicial space, with Vesna Stojanoska. Proc. Amer. Math. Soc. 144 (2016), no. 1, 445--458.
  


• On a nilpotence conjecture of J.P. May, with Niko Naumann and Justin Noel. J. Topol. 8 (2015), no. 4, 917-932.
  


• A thick subcategory theorem for modules over certain ring spectra. Geom. Topol. 19 (2015), no. 4., 2359-2392.
  


• Affineness and chromatic homotopy theory, with Lennart Meier. J. Topol. 8 (2015), no. 2, 476-528.
  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.
  

Talks

• Notes (part 1 and part 2) for my talk at the conference Derived geometry.
• Video for my talk "p-adic K-theory and topological cyclic homology" at MSRI.
• 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.

Other materials

• The Geometric Langlands seminar at UChicago, fall 2022.
• The Geometric Langlands seminar at UChicago, fall and winter 2021--2022. The spring quarter is here.
• The Geometric Langlands seminar at UChicago, November 2020 (lecture series by Bhargav Bhatt).
• 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

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