John Nicholson
I am currently a Heilbronn Research Fellow at Imperial College London. Until October 2021, I was a PhD student at University College London working under the supervision of Francis E. A. Johnson.
Email:
Research Interests
My research is on classification problems in lowdimensional topology and their interactions with areas of algebra such as homological group theory and integral representation theory.
My current focus is on the development of new algebraic techniques to classify finite 2dimensional cell complexes, i.e. group presentations, up to homotopy equivalence. I also work on applying these techniques to the classification of manifolds and Poincaré complexes in dimensions 3 and 4.
Publications and Preprints

The homotopy type of a finite 2complex with nonminimal Euler characteristic (August 2021)
Submitted, 23 pages (arXiv version)

Homotopy classification of 4manifolds whose fundamental group is dihedral (November 2020)
with Daniel Kasprowski and Benjamin Ruppik
Algebraic & Geometric Topology, to appear, 22 pages (arXiv version)

Cancellation for (G,n)complexes and the Swan finiteness obstruction (May 2020)
Submitted, 29 pages (arXiv version)

Projective modules and the homotopy classification of (G,n)complexes (April 2020)
Submitted, 28 pages (arXiv version)

Connected sum decompositions of highdimensional manifolds (September 2019)
with Imre Bokor, Diarmuid Crowley, Stefan Friedl, Fabian Hebestreit, Daniel Kasprowski, Markus Land
201920 MATRIX Annals (2021), 530 (journal version, arXiv version)

On CWcomplexes over groups with periodic cohomology (May 2019)
Transactions of the American Mathematical Society, 374 (2021), 65316557
(journal version, arXiv version)

A cancellation theorem for modules over integral group rings (July 2018)
Mathematical Proceedings of the Cambridge Philosophical Society, 171 (2021), 317327 (journal version, arXiv version)
Talks
Invited talks

The London Geometry and Topology Seminar (10th December 2021)
Title: The homotopy type of a finite 2complex with nonminimal Euler characteristic (link)

The University of Alabama, Algebra/Topology Seminar (17th November 2021)
Title: The homotopy type of a finite 2complex with nonminimal Euler characteristic

Universidad de Buenos Aires, Topology Seminar (20th October 2021)
Title: The homotopy type of a finite 2complex with nonminimal Euler characteristic (link)

Ludwig Maximilian University of Munich, Arithmetic and Algebraic Geometry Seminar (1st June 2021)
Title: Cancellation for projective modules over integral group rings (link)

Université du Québec à Montréal (CIRGET), Geometry and Topology Seminar (23rd April 2021)
Title: Projective modules and exotic group presentations (link)

Max Planck Institute for Mathematics, Topology Seminar (1st March 2021)
Title: 4manifolds and exotic group presentations (link)

5th Meeting of "New Directions in Group Theory and Triangulated Categories" (8th December 2020)
Title: Projective modules and exotic group presentations

University of Southampton, Pure Mathematics Colloquium (13th November 2020)
Title: Projective modules and exotic group presentations (link)

University of Regensburg, Low Dimensional Topology Seminar (8th October 2020)
Title: Projective modules and the homotopy classification of CWcomplexes

Ohio State University, Topology and Geometric Group Theory Seminar (14th July 2020)
Title: Projective modules and the homotopy classification of CWcomplexes (link)

Durham University, Geometry and Topology Seminar (31st October 2019)
Title: Cancellation theorems in algebra with applications to topology (link)

University of Melbourne, Topology Seminar (27th August 2018)
Title: Wall's D2 problem and stablyfree modules over integral group rings (link)

University of Sydney, Geometry and Topology Seminar (22nd August 2018)
Title: Poincaré 3complexes and stablyfree modules over integral group rings (link)
Selected Junior/Contributed Talks

Imperial College London, Junior London Algebra Colloquium (7th July 2020)
Title: Cancellation for projective modules over integral group rings (link)

At Conference: Four Manifolds: Confluence of High and Low Dimensions (9th July 2019)
Title: On CWcomplexes over groups with periodic cohomology (link)

University of Cambridge, Junior Geometry Seminar (1st March 2019)
Title: FourManifolds and twocomplexes with finite fundamental group (link)

King's College London, Junior Geometry Seminar (14th February 2019)
Title: The topology of fourmanifolds with finite fundamental group (link)

University College London, Junior Geometry Seminar (30th November 2017)
Title: Free actions of finite groups on spheres
Conferences and Workshops

Manifolds and Ktheory: the legacy of Andrew Ranicki
Edinburgh, UK ‐ virtual (June 2021)

British Mathematical Colloquium (BMC)
Glasgow, UK ‐ virtual (April 2021)

2nd International Meeting on Geometric Group Theory and Low Dimensional Topology
Karlsruhe, Germany ‐ virtual (December 2020)

London Mathematical Society Graduate Student Meeting 2020
London, UK ‐ virtual (November 2020)

Four Manifolds: Confluence of High and Low Dimensions
Fields Institute, Toronto (July 2019)

Topology of Manifolds, Interactions Between High and Low Dimensions
MATRIX Institute, Melbourne (January 2019)

Harnessing Higher Structures in Homotopy: Workshop on Manifolds
Isaac Newton Institute, Cambridge (December 2018)

Manifolds, Groups and Homotopy
Isle of Skye (June 2018)
Teaching
202021
201819
201718
Miscellaneous
My Part III essay on Exotic Spheres which was supervised by Oscar RandalWilliams.
From April 2019 to 2020, I was a part of the editorial team of Chalkdust.
I am a coauthor of the textbook Edexcel Alevel Further Mathematics Core Pure Mathematics Book 2 which was published by Pearson in March 2018.
I am a contributor to the material on the website Underground Mathematics, which I worked on during the summer of 2015.
Here are some instructions on How to Set Up a UCL webpage.