Each lecture hopefully requires only 1 hour.

Part 1: Introduction - 3 lectures - [L]

Part 2: Generalities of Nilsequences - 2 or 3 lectures - [Q]

Part 3: Equidistribution - 2 lectures - [Book] and [P]

Part 4: MN(s) - 1 lecture - [M]

Part 5: GI(s) - ?? lectures - [GTZ]

Part 6: Back to linear equations - ?? lectures - [L]

[L] = Linear equations in primes, [Q] = The quantitative behaviour of polynomial sequences, [Book] = Higher Order Fourier Analysis, [M] = The Mobius function is strongly orthogonal to nilsequences, [GTZ] = Green-Tao-Ziegler.

### Part 1: Main results of linear equations in primes

In this paper, the main results are proven assuming conditions called GI(s) and MN(s). These two conditions were later confirmed by Green-Tao and Green-Tao-Ziegler.

References: the linear equation paper (98 pages) by Green and Tao

#### Lecture 1

-§§ 1, 4 of the paper.

-Introduce enough notions to explain how the results announced in Introduction of the linear equation paper are related to each other.

-There are linear algebra tricks involved; though they are elementary in nature, it would be nice to cover them to make sure we are on the same page.

#### Lecture 2

- Statement of the conjectures GI(s) and MN(s) from §8 of the linear equation paper.

- To state this, introduce the Gowers norm (linear equation paper Appendix B) and the notion of nilsequences following the linear equation paper §8. This latest Bourbaki Seminar article (2020) also gives a compact summary.

#### Lecture 3

Try your best to explain how the main result can be proven under GI(s) and MN(s). At this stage, a rough roadmap would suffice.

### Part 2: Generalities on simply connected nilpotent Lie groups and polynomial sequences

Such Lie groups and sequences appear in the statements of GI(s) and MN(s).

References: Green's videos (a YouTube playlist; the list in Green's page is also useful http://people.maths.ox.ac.uk/greenbj/videos.html ),

the polynomial orbits paper (76 pages) by Green and Tao.

How to get prepared:

Watch video 2 and 3 of Green's 6 lectures.

Read §§ 1, 2 and 6 of the polynomial orbits paper.

#### Lecture 1

Explain the following notions perhaps with illustrating examples:

##### - simply connected nilpotent Lie groups/algebras.

The category of connected nilpotent Lie groups is equivalent to the category of nilpotent Lie algebras via the functor $G\mapsto \mathfrak g := \mathrm{Lie}(G) $. (The essential part of this equivalence is called *Lie's third theorem*. See also this post of mine (in Japanese).)

##### - filtered groups in the sense of Green-Tao

See Green's video 2/6

https://youtu.be/27Ljd_M--Qw?t=1285

and Introduction to the polynomial orbits paper.

##### - Lattices and Malcev bases of simply connected nilpotent Lie groups

See Green's video 2/6 44:00-47:00

https://youtu.be/27Ljd_M--Qw?t=2626

Green's video 2/6 56:30--1:03:00

https://youtu.be/27Ljd_M--Qw?t=3390

and §2 of the polynomial orbits paper.

#### Lecture 2

##### - define the notion of polynomial maps $H\to G $ between filtered groups and in particular polynomial sequences $\mathbb Z \to G $; equivalent formulations of these notions.

See Green's video 3/6 11:00-21:00

https://youtu.be/u4inkf0VJAU?t=669

and §6 of the polynomial orbits paper.

Section 1.6 of Tao's *Higher Order Fourier Analysis* is also useful.

#### Lecture 3 (optional)

##### - Present a proof of the converse to GI(s) (the easier implication).

Since it is not logically needed, **this lecture may be skipped**. However, the proof in Green's video 3/6 56:00

https://youtu.be/u4inkf0VJAU?t=3379

contains nice arguments using induction on $s$. This proof also shows how the notion of general polynomial sequences is useful when one is only interested in the linear sequence $n\mapsto g^n x$.

You might also want to refer to Proposition 8.2 of the paper "Linear equations in primes," which refers to Proposition 12.6 of the paper "An inverse theorem for the Gowers $U^3(G)$ norm."

https://doi.org/10.1017/S0013091505000325

### Part 3: Equidistribution of polynomial sequences

This seems to be an important ingredient for the proof of both conjectures.

#### Optional activity

One could read pp. 2--27 of Tao's book "Higher order Fourier analysis" to familiarize themselves with the equidistribution phenomenon of sequences defined by polynomials $f\in \mathbb R [X]$: \[ \mathbb Z \xrightarrow{n\mapsto f(n) } \mathbb R \longrightarrow \mathbb R /\mathbb Z \] which is classical. We could as well have a homework session discussing this classical case.

#### How to get prepared:

Watch this video of Green's (= Lecture 4/6). It explains a rough idea of what the topic is about. See this post

https://motivichomotopy.hatenablog.jp/entry/2021/03/02/123330

for what is done in this video.

Have a quick look through *The quantitative behaviour of polynomial orbits on nilmanifolds* by Green and Tao. In doing so, you could read the introduction first and take a comprehension test here.

Note that this paper contains errors in §8. See this erratum (the arXiv version is more detailed: https://arxiv.org/abs/1311.6170 ). In Erratum, the authors describe how to establish the results in the parallel context of multivariate polynomial sequences $\mathbb Z ^t \to G $.

#### Lecture 1

##### - main results.

Introduce enough materials from the paper so that you can state its main results (Theorems 2.9 and 1.19) and explain how you can (or cannot) appreciate their beauty/importance.

##### - from Theorem 2.9 to Theorem 1.19.

Make a comment on how Theorem 2.9 can be used to prove Theorem 1.19.

We ** don't** have to know the proof of Theorem 2.9. It's too technical.

#### Lecture 2

**- Explain some technical contents of the paper **

**https://doi.org/10.4007/annals.2012.175.2.2**

**and the Erratum. https://arxiv.org/abs/1311.6170**

In particular, **Lemma 2.1** from Erratum, **Proposition 2.3** from Erratum, **Proposition 8.4** from the paper and **Theorem 8.6** from the paper (whose proof is completed in Erratum) turn out to be useful in the proof of **MN(s)**. The case $N_1=\cdots =N_t=N$ suffices. See also this post of mine. https://motivichomotopy.hatenablog.jp/entry/2021/01/12/005343

##### - Make some comments on proofs of some of them, or discuss examples, or draw pictures.

### Part 4: The condition MN(s)

This was proven in the paper titled "The Mobius function is strongly orthogonal to nilsequences" (26 pages) by Green-Tao. It was published directly after the polynomial orbits paper in the same volume.

Good news is that the main content of this paper is completed within the first 13 pages.

### Part 5: The codition GI(s)

Main reference is this paper by Green-Tao-Ziegler.(142 pages...)

You might also want to take a look at this Bourbaki exposition by Bloom on the work of Manners on a new proof of GI(s) with better numerical implications. https://arxiv.org/abs/2009.01774

### Part 6: Back to linear equations

Follow the path from GI(s) and MN(s) to the main result more closely than done in Part 1.