We will cover various topics in geometry. There may be a slight bias towards (complex) algebraic geometry, to be more relevant to number theory, but there will be plenty of differential geometry, symplectic geometry and topology.

This will be * nothing like* an undergraduate course, nor a book. There will be no comprehensive notes. There will be gaps in the theory.
Often there will be no theory at all -- just

Mathematics is built on a few key ideas and examples. Those get turned into 1000 page books of general theory. But having read the book, you often find that
everything is encapsulated in one or two key examples or key concepts. If you understand them, you understand how the theory *should* go.

The best mathematics also usually involves ideas from all over the subject. It is not feasible to read all the 1000 page books across all these subjects. So we cannot learn all of the theory.

So as a graduate student, you will slowly learn how to pick up on key ideas and examples, without learning every last detail of the full theory. (Then when it comes to writing your thesis you will specialise again, learning every last detail of the theory in the particular areas you need them. But let's worry about that in a few years' time!)

So consider the topics we cover as taster, to give you a feeling for various topics, and to give you some of the fundamental ideas that recur throughout the subject.

The plan is to cover most of these topics over 2 terms. Some will appear instead in the Junior Geometry seminar.

You can click on many topics to see lecture notes made by previous years' CDT students.

**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 I.** Polytopes and symplectic toric varieties.

**Toric varieties II.** Givental's quotient construction, line bundles and divisors.

**Toric varieties III.** Fans and algebro-geometric 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.

You might already know you want to do number theory, so why should you learn symplectic geometry ?

Maybe you think you hate number theory, and came here to study analysis on manifolds.

So why should you be interested in many of the topics in this course ?

The best mathematics is usually **broad**, using ideas from all over the subject.

**Number theory:** much of the language, philosophy and inspiration in this subject comes from geometry.
We will learn some geometry for its own sake, not adapted to number theory, but the ideas and way of thinking will help number theorists enormously.
Observe how many of the greatest number theorists began as geometers (Grothendieck, Deligne, ...) or topologists (Serre, Mazur, ...).

**Geometry:** Number theory has inspired a lot of modern geometry (eg geometric Langlands, arithmetic 3-manifolds, etc)
as well as motivating a lot of the abstraction and general theory of schemes in algebraic geometry (in order to make it general
enough to include number theory) -- working relative to a base, rationality questions, singularities, cohomology theories... .

Basic number theoretic objects
such as modular forms crop up in all kinds of geometric contexts (Donaldson invariants, Gromov-Witten invariants, etc) and
should be part of the toolkit of any educated mathematician.

**Geometric analysis:** Usually the applications run from analysis to geometry, but not always. For instance, algebraic subvarieties give minimal submanifolds, so a good knowledge of algebraic geometry is crucial to study many aspects of geometric analysis. And many of Donaldson's analytical breakthroughs are based on applying ideas from symplectic geometry in infinite dimensions to PDE.

So: in an algebraic geometry proof, for example, we often use ideas from topology and differential geometry, passing backwards and forwards between the algebraic and analytic categories (using "GAGA"), using the easiest or most suitable methods, and changing cohomology theory from line to line!

Even if the Deligne paper you are reading has now been expressed entirely algebraically, I can assure you that came later! Initially he would have worked using Kähler geometry and analysis (eg for Hodge theory), symplectic geometry and algebraic topology (eg for vanishing cycles), and so on.

Ideas from one area always find application from others. Eg moduli spaces appear all over differential geometry, symplectic geometry, algebraic geometry and number theory.

So don't think "this isn't my subject" -- anything you learn will be useful. The harder you find it, the better it is for you!

We'll have one lecture a week for 90 minutes or so. **Fridays 3pm, Teaching room 3, Imperial CDT space**. See here for updates.

There will be a wrap-up session the following **Friday 11am** in Huxley 408, Imperial, to go over exercises, examples, things you didn't understand, etc.

We will continue next term, but only every 2 weeks.

One person 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.

(In previous years the owner gave a short lecture or summary at the start of the wrap up, and also produced LaTeXed notes of the lecture. But we won't do this from now on as I want to encourage non-experts to own topics, and more emphasis on examples.)

We will learn some cool maths. And together we will learn a common language.

That way you will have the most valuable resource for the rest of your PhD --
colleagues who you can talk to and learn from, who understand you and who you understand.

Please stop me a lot and ask lots of questions.

**Make a fool of yourself** -- the most important skill you can learn this year is to lose your fear of showing that you
don't know something that you think you "should" know, or that you think the others know.

Despite the internet, almost all
good research and PhD theses come from people at top universities -- because they
have access to expertise and people to *talk* to.

You cannot do a PhD alone -- you will get stuck on something that is trivial for
your colleague, or well known to be impossible. You need to master the difficult skill of talking to your colleagues now, so that you can learn from
them in later years.

It is much quicker to learn a subject at a blackboard from a colleague than it is to try to wade through a 1000 page book. They can tell you the key example, or draw the key picture. Or tell you which page to go to in the book, saving you 999 pages.

We will learn some beautiful topics. I hope this will inspire you to learn more -- to fill the gaps in the lectures, to learn a general theory that an example points to, or to fill the gaps in your own knowledge that the lectures show up.

We can't cover everything, and you all have very different backgrounds.

So inevitably there will be plenty that you don't know that is used in lectures.

This is not a problem and is perfectly normal. Use the lectures as inspiration for your own reading -- to suggest which things you could usefully learn next.

But mainly, use this as an opportunity to make a fool of yourself in front of your colleagues and practice talking to them. Some of them may know what you want to know; get them to explain it to you, or direct you to a good reference. Others may also not know it; then you can learn it together.

Maybe you already know some or all of the topics.

Please still come along. Hearing something you already know is always pleasurable and makes you feel good.

Usually it will be presented differently -- so then you can enjoy translating to and from what you've already seen,
and learn more about it that way.

Most importantly, you probably don't know it ** all** anyway.

Students who claim to know the Adams spectral sequence often know fearsome general theory, but can't compute the homotopy groups of a torus.

Students who know all about klt singularities often don't know the basic symplectic geometry or topology of an ordinary double point.

We will focus more on

And you never really know it all until you can *explain it to others*. Some come along and then help the other students afterwards.
Practicing this will improve your understanding, and will mean you get better at talking to older students and postdocs, from whom you can learn a lot.

Pick one topic from "schedule" below and * own* it!

You will be responsible for finding volunteers to make presentations at the board in the wrap-up -- of either an exercise, or some confusion/difficulty for others to discuss and resolve, or (more occasionally) a short piece of theory.

So you do

You will also run the wrap-up session, and keep the level appropriate (i.e. at the level of whoever feels least confident in that week's material). There will also be a 2nd year there to help or intervene if necessary.

Email me with your choice of one of the early lectures...first come first served.

Maths is not a spectator sport. Doing exercises yourself is the most important part of this course. Do as many which challenge you as you can. Please latex up at least one each week and send a pdf to Nicky within 10 days of the lecture.

Don't choose the easiest one (unless it challenges you). Don't only write up exercises that you can do. It is fine to write up something you got stuck on, saying what you tried or why you're stuck. It's also fine to write up something completely different -- your own exercise or an extension of something in the lecture.

The idea is not to get "marks", undergraduate style. It's also not a box ticking exercise; solving problems is the most important thing you'll do all year.

The wrapper-upperer is a 2nd year CDT student -- often the one who owned the same topic last year -- who will help with the wrap-up session that follows a week after each lecture.

Week | Date | Lecture | Lecturer | Owner | Wrap up |
---|---|---|---|---|---|

1 | 6 Oct | Spec and Proj | Thomas | Hanneke Wiersema | Andrea Sartori |

2 | 13 Oct | Differential forms and currents | Bellettini | David Sheard | Galen Voysey |

3 | 20 Oct | Connections and curvature | Lotay | Andrew Graham | Jenny Swinson |

4 | 27 Oct | Chern classes I | Panov | Omri Faraggi | Raffael Singer |

5 | 3 Nov | Chern classes II: axiomatics | Panov | Brad Doyle | Michele Giacomini |

6 | 10 Nov | Complex manifolds and the Kähler condition | Salamon | Daniel Platt | Udhav Fowdar |

7 | 17 Nov | Hodge theory | Singer | Lorenzo La Porta | Giada Grossi |

8 | 24 Nov | Toric varieties I. Polytopes and symplectic toric varieties | Evans | Johannes Girsch | Albert Wood |

9 | 1 Dec | Morse theory and the Thom-Smale-Witten complex | Lekili | Angela Wu | Joe MacColl |

10 | 8 Dec | Geometric Invariant Theory | Thomas | Matthew Habermann | Luigi Lunardon |

11 | 15 Dec | Symplectic reduction | Singer | Pol van Hoften | Tim King |

Junior Geometry seminar | |||||

6 Oct | Riemann surfaces | ||||

13 Oct | Line bundles and divisors | ||||

20 Oct | Betti, de Rham and Dolbeault cohomologies | ||||

27 Oct | Thom isomorphism and duality in cohomology | ||||

3 Nov | Blow-ups and algebraic surfaces | ||||

10 Nov | Holonomy | ||||

17 Nov | Positivity in complex and algebraic geometry | ||||

24 Nov | Introduction to moduli problems | ||||

1 Dec | Toric II: fans and examples | ||||

8 Dec | Tropical curves | ||||

15 Dec | 27 lines | ||||

Spring term | |||||

12 | 12 Jan | Classifying spaces, equivariant cohomology, localisation | Segal | Benjamin Aslan | Yin Li |

13 | 26 Jan | The ordinary double point | Thomas | Petru Constantinescu | Gregorio Baldi |

14 | 9 Feb | Deformation theory | Thomas | Samuel Stark | Giulia Gugiatti |

15 | 23 Feb | Line bundles and intro to higher dimensional geometry | Cascini | Ashwin Iyengar | Tim King |

16 | 9 Mar | Koszul Duality | Ed Segal | Sarah Nowell | Fabian Lehmann |