Project description

Barry Mazur posited a mysterious parallel between arithmetic and low-dimensional topology in which primes correspond to knots, and arithmetic duality is interpreted as Poincaré duality. By combining stratified homotopy theory with recent advances in the geometrisation of Galois groups, the SHIFTED project aims to realise Mazur's vision. Armed with this new take on the geometry of number fields, we are launching a study of arithmetic factorisation homology and the arithmetic quantum field theories envisioned by Minhyong Kim.


Below please find our research works associated to the SHIFTED project as well as forthcoming work.

Pyknotic objects, I.
Basic notions

C. Barwick & P. Haine
Last updated April 2019

Exodromy for stacks

C. Barwick & P. Haine
Last updated January 2019

Extended étale homotopy groups from profinite Galois categories

P. Haine
Last updated January 2019

On Galois categories and perfectly reduced schemes

C. Barwick
Last updated November 2018

On the homotopy theory of stratified spaces

P. Haine
Last updated August 2019


C. Barwick, S. Glasman, & P. Haine
Last updated July 2019


  1. Pyknotic objects, II. Exodromy for ℓ-adic sheaves, C. Barwick & P. Haine
    A paper extending the classification of constructible sheaves with finite coefficients from Exodromy to Q- and Q-coefficients.
  2. The branched topology, C. Barwick, P. Haine, & T. Schlank
    A paper defining a new topology on schemes ‘dual’ to the étale topology.
  3. Stratified homotopy theory, C. Barwick & P. Haine
    A book devoted to the systematic study of stratified homotopy theory.


Below please find presentations on stratified homotopy invariants, field theories, exodromy, and duality that we have given, with videos available when possible.

Clark Barwick

Principal Investigator
Reader in Mathematics, University of Edinburgh

Clark received his 2001 BS from the University of North Carolina. His 2005 PhD from the University of Pennsylvania preceded a string of postdoctoral positions – Göttingen, Oslo, the Institute for Advanced Study, and finally Harvard. Clark joined the faculty at the Massachusetts Institute of Technology in 2010. In 2017 Clark moved to the University of Edinburgh. His research interests include homotopy theory, arithmetic geometry, and bureaucracy-avoidance strategies.

Peter Haine

Graduate student, MIT

Peter Haine grew up in Northern New Hampshire; he is an enthusiastic trail runner. He was an undergraduate at MIT and is now a third year PhD candidate also at MIT, under the direction of Clark Barwick. His research interests include homotopy theory, arithmetic geometry, and oriented fibre products of étale topoi.


Emotional support dog/permanent distraction

Available positions

Below please find positions available in our research group.

PhD Student, University of Edinburgh

Clark Barwick is currently accepting PhD students at the University of Edinburgh. See the School of Mathematics' PhD Application page for more information about the applciation process.