Matthew Burke
My home page is here.


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

Jan 2020 - Present: Lyryx Learning

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, SQLite and Capacitor.
  • 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 and implemented a grade book which allows instructors to specify a custom weighted marking algorithm.

Aug 2019: Cybera Data Science Fellowship

Data science fellow

  • Collaborated with an industry partner to clean, explore and analyse 7 years of live events data stored in a PostgreSQL database with over 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.

Sep 2017-Sep 2019: University of Calgary

Postdoctoral scholar

  • Designed and completed projects in pure mathematics and computer science leading to a publication in a peer-reviewed journal and another paper under review.
  • Used the Coq programming language in seminar series on HoTT.
  • Created a formal proof of a well-known result in category theory using Coq.
  • Provided mentoring support for 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.

Jun 2016-Aug 2017: MathSpire Ltd.

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.

Jul 2016-Aug 2016: Debate Chamber Ltd.

Summer school tutor

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

Jul 2016-Aug 2016: Masaryk University, Brno

Visiting postdoctoral researcher

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

Jan 2012-Dec 2013: Macquarie University, Sydney


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

Jun 2010: Blue Tutors

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