In autumn, we will have one lecture a week for approximately 90 minutes from 3 pm which will be delivered by a geometry faculty member from King's, UCL or Imperial. The first lecture is on Friday, September 29th at Imperial in Huxley 140. Afterwards, this course will run in Huxley 658. In the second term, the lectures are back at Huxley 140. In addition, there will be a wrap-up session the following Friday between 11am - 1pm to go over exercises, examples, things you didn't understand, etc. These will be in Huxley 341. One student will "own" each topic and be responsible for finding people to present examples in the wrap-up session. The owner gives a short lecture or summary at the start of the wrap up, and leads the discussion. Every week you may receive a set of exercises based on the course material. Solving exercises yourself is important. Previously this course was arranged by Richard Thomas. There is an impressive amount of useful information and good advice in the previous website which you should check out by clicking this. (I have freely moved some of the content from there to this website but some good advice by Richard remains on his website.) Last year, this course was arranged by Ed Segal, see here for the course website. There are also Junior Geometry Seminar at Imperial and Junior Geometry Seminar at UCL/KCL which meet weekly and cover supplementary topics.

There is a Matrix chat room associated with this course. You can access it by creating an account at the Freemath chat server (make sure to choose Other so that your account is created in freemath.xyz, and don't worry about entering any phone or email information. If you forget your password, just create a new account and join again.) and joining the room called Topics in Geometry (LSGNT). This is a platform for informal discussions related to the course among the participants. I will be available and happy to chat about the topics of this course in an asynchronous manner.

Here are some of the topics that may be covered in this course. Some of these were taught in the past years and you can get the previously produced notes associated with these lectures by clicking the links.

**Spec and Proj.** Affine and projective varieties and schemes.

**Complex manifolds and the Kähler condition.** Levi-Civita and Chern connections. (GAGA?)

**Poincaré duality**. Cohomology, differential forms, currents, de Rham theory, Thom isomorphism.

**Morse theory and the Witten complex.**

**Chern Classes.**

**Classifying spaces, equivariant cohomology, localisation.**

**Blowing up.** Blow ups and blow downs. Symplectic blow ups. Topology.

**Line bundles and the minimal model programme.** Kodaira embedding, bend and break.

**Toric varieties** Polytopes and symplectic toric varieties.

**Koszul resolutions and Koszul duality.**

**Hodge theory.**

**The Weil conjectures.** Cohomology, motives, point counting.

**Tropical curves.**

**The ordinary double point.** Vanishing cycles, symplectic geometry. The simple flop.

**Geometric Invariant theory.**

**Symplectic reduction.** Moment maps and the Kempf-Ness theorem.

**Stable bundles and simple bundles.** Moduli spaces.

**Lefschetz pencils in algebraic and symplectic geometry.**

**Mixed Hodge structures.**

**Spectral sequences.**

**Atiyah - Singer index theorem.**

**Feynman path integrals.**

**Higgs bundles.**

**Deformation theory.** Perhaps also virtual cycles.

**Mirror symmetry. **

**Floer homology and Fukaya category. **

**Seiberg-Witten Invariants. **

**Loop groups. **

**K3 surfaces.**

**Stable homotopy theory. **

**Springer resolution. **

**McKay Correspondence. **

**Weinstein manifolds. **

**Constructible sheaves. **

**General Relativity. **

**Teichmüller Theory. **

**Special Holonomy. **