Matthew Burke
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Table of Contents


Macquarie University, Sydney

Doctor of Philosophy in Mathematics 2011-2015

Christ's College, University of Cambridge

Part III Mathematics (MMATH) 2010-2011

Bachelor's degree in Mathematics (BA) 2007-2010

Work experience

Lyryx Learning

Jan 2020 - Present

Senior software developer (11 months); Director of technology (Current)

  • Designed, developed and distributed a cross-platform mobile and web application to combine existing textbook content with new interactive questions. Used TypeScript, webpack and Capacitor to create web, iOS and Android applications. Authentication and authorisation using AdonisJs.
  • Created a local development environment for an existing Java web application. Used Docker, NGINX and Firejail to locally develop features that spanned multiple production servers.
  • Constructed a Web API for automatically scheduling examinations. Used Java Servlets and MySQL.
  • Designed a grade book interface which allows instructors to specify a custom weighted marking algorithm.

Cybera Data Science Fellowship

Aug 2019

Data science fellow

  • Collaborated with an industry partner to clean, explore and analyse 7 years of live events data stored in a PostgreSQL database whose largest table had 13 million entries.
  • Used Python scripts and Jupyter notebooks for data cleaning and exploration. Used the R programming language for data analysis and modelling.
  • Used the plotly express visualisation library for histograms, scatter plots, box plots, choropleth maps etc.. Constructed a generalised linear model (GLM) for analysing count data and a log linear model for analysing sales data.

University of Calgary

Sep 2017-Sep 2019

Postdoctoral scholar

  • Designed and completed projects in pure mathematics and computer science leading to publication in a peer-reviewed journal.
  • Used Coq programming language in seminar series on HoTT and proof of result in category theory.
  • Mentored two PhD students. Reviewed two papers for mathematics journals.
  • Organised the University of Calgary Peripatetic Seminar (Dec 2017-May 2019) and chaired a session of the Alberta Mathematics Dialogue 2018.
  • Used Jupyter notebooks to lecture 4 classes of around 230 students each.

MathSpire Ltd.

Jun 2016-Aug 2017

Software engineer (5 months); Chief technology officer (9 months)

  • Developed a cross-platform mobile and desktop application to teach A-level mathematics using interactive graphs, videos and integrated testing. Used F#, .NET and Xamarin.
  • Created a web front-end and API for teachers to track student progress.
  • Showcased application at the BETT education technology conference.

Debate Chamber Ltd.

Jul 2016-Aug 2016

Summer school tutor

  • Guided classes of around 14 A-level students through university level material using a combination of presentations, exercises and group work.

Masaryk University, Brno

Oct 2015-Nov 2015

Visiting postdoctoral researcher

  • Plenary speaker at the multi-disciplinary Eduard Cech Institute Workshop.

Macquarie University, Sydney

Jan 2012-Dec 2013


  • Demonstrated solutions on the whiteboard for three undergraduate mathematics courses.
  • Guided and motivated first year students at the drop-in centre.

Blue Tutors

Jun 2010

GCSE Tutor


Professional development and prizes

2018: Mitacs online workshop: Managing Project Timelines

2016: University of Michigan on Coursera: Using Python to Access Web Data

2016: University of Michigan on Coursera: Using Databases with Python

2008: Christ's College Whelan Prize for First Class Examination Performance

Research papers

  • Tangent infinity-categories and Goodwillie calculus. Joint with Kristine Bauer and Michael Ching. Available at arXiv.
  • Involution algebroids: a generalisation of Lie algebroids for tangent categories. Joint with Benjamin MacAdam. Available at arXiv.
  • A Synthetic Version of Lie’s Second Theorem. Applied Categorical Structures, 2018.02.06. Available at arXiv and publisher.
  • Connected Lie Groupoids are Internally Connected and Integral Complete in Synthetic Differential Geometry. Symmetry, Integrability and Geometry: Methods and Applications, 2016.06.29. Available at arXiv and publisher.

Essay and thesis

Part III essay: Synthetic differential geometry pdf

PhD thesis: Synthetic Lie theory pdf