In autumn, we will have one lecture a week for approximately 90 minutes from 3 pm which will be delivered live online by a geometry faculty member from King's, UCL or Imperial. In the spring, the course will be divided into two strands. The students should only attend one of these strands and the assessment will reflect that. Attending to both strands is likely to be overwhelming but is allowed. The lectures will be hosted at our server running the open source software Big Blue Button. The link to each lecture will be shared with you on the day of the lecture. In addition, there will be a wrap-up session the following Friday to go over exercises, examples, things you didn't understand, etc. (There is an ongoing effort to arrange a room so that the wrap-up session can be in a physical location at King's.) One student will own each topic and be responsible for finding people to present examples in the wrap-up session. That person will also supervise the wrap-up session for the 1st year students next year. The owner gives a short lecture or summary at the start of the wrap up, and also produces notes of the lecture in TeX. Previously this course was arranged by a 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.) There is also a Junior Geometry Seminar organized by Jaime Roche, John McCarthy and Mohammed Shafi which meets weekly and covers 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 joining the room called Topics in Geometry. 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. **