Topics in Geometry

Welcome to the LSGNT Topics in Geometry course for 2021/22.

Course structure

This is not a normal lecture course. Every week you'll have a different lecturer, covering some essential aspect of geometry. Lectures don't build on each other directly but later ones might end up using stuff from earlier ones.

The lecturers will set you problems and exercises, of wildly varying difficulty. These are for you to do because doing exercises is how you learn maths. Do as many of the exercises as you can. At least one of them needs to be written up and submitted to Kay within 10 days of the lecture (even if you couldn't solve it completely).

A week after each lecture you'll have a `wrap-up session' where you discuss the exercises and the lecture as a group. There won't be a lecturer at the wrap-up session, instead one of you will act as the `owner' for that week's topic and lead the session. I'll also send along the 2nd year who owned the topic last year for extra guidance.

The website for last year's course has lots of useful things including lecture notes, videos of talks, and further links and advice. I also recommend looking at the 2019 site because it has lots of words of wisdom from Richard Thomas.


Lectures are in Room 139, Huxley building, Imperial College. Lectures will also be streamed live online on BigBlueButton (check your email for the link and access code). Some lectures will just be online.

Lectures will be recorded and the links to the videos will appear here.

Wrap-up sessions are in the Imperial CDT Space, Teaching Room 2 (402B).

Term 2

The lectures for term 2 are divided into two streams, differential/analytic geometry and algebraic/symplectic geometry, which alternate weeks. Since you now have projects to work on you are not expected to attend all lectures and wrap-ups, instead you should pick one of the two streams and attend all of those. Of course you are welcome to attend extra lectures if you wish (especially if a particular topic appeals to you).


Lectures are on Fridays 3pm-5pm. Wrap-up sessions are on the following Friday, 11am-1pm.

Date Topic Stream Lecturer Owner 2nd year
14/1/22Minimal surfaces and the Allen-Cahn equation
Exercises | Video
Diff/AnMarco Guaraco Nick Manriquewin Kobe Marshall-Stevens
21/1/22Hodge Theory
Video (requires Imperial login)
Alg/SympNick Shepherd-Barron Sebastian Monnet Dingli Liang
28/1/22Gauge theory
Notes with exercises
Diff/AnLorenzo Foscolo Enric Solé-Farré
4/2/22Moduli of curves Alg/SympAlessio Corti Mads Christensen Pascale Voegtli
11/2/22The Atiyah-Singer index theorem
Diff/AnMichael Singer Nicolo Cavalleri Jamie Bell
18/2/22The Weil conjectures
Alg/SympDario Beraldo Diego Chicharro Lilybelle Cowland Kellock
25/2/22Geometric measure theory
Diff/AnCostante Bellettini Mel Levin Myles Workman
4/3/22Floer cohomology and Fukaya categories Alg/SympYankı Lekili Yuan Yang Robert Crumplin
11/3/22Riemannian convergence theory I Diff/AnSimon Donaldson Ananya Satoskar
18/3/22Geometric representation theory
Video (requires Imperial login)
Alg/SympTravis Schedler Martin Ortiz
25/3/22Riemannian convergence theory II Diff/AnSimon Donaldson



Date Topic Lecturer Owner 2nd year
01/10/21Spec and Proj
Exercises | Video | Slides | Other notes
Ed Segal David Angdinata Edwin Hollands
08/10/21Complex and Kähler manifolds
Notes and Exercises | Video
Lorenzo FoscoloEnric Solé-Farré Wojtek Wawrów
15/10/21Poincaré duality (online only)
Exercises | Video
Michael SingerMichela BarbieriSara Veneziale
22/10/21Morse theory and Instantons
Notes with exercises | Video
Steven SivekAnanya Satoskar Ilaria Di Dedda
29/10/21Chern classes (online only)
Notes with exercises | Video
Dario BeraldoMartin Ortiz Bence Hevesi
05/11/21Blowing up
Notes with exercises | Video
Richard ThomasAmelia Livingston Pascale Voegtli
12/11/21Positivity in Algebraic Geometry
Notes | Exercises | Video
Paolo CasciniSiqi Yang Dmitry Ulyumdzhiev
19/11/21Toric varieties
Notes with exercises | Video
Johannes Nicaise Thamarai Valli VenkatachalamRobert Crumplin
26/11/21Moment maps and Hamiltonian reduction
Notes with exercises | Video
Yankı LekiliDaniil Mamaev Wei Zhou
03/12/21Special Holonomy (online only)
Slides, exercises and other links | Video
Simon SalamonInés García Francesco Vigano
10/12/21Plateau's problem (online only)
at 1.30pm
Slides with exercises | Video
Giuseppe TinagliaNick Manrique Lucy Phillips