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# Ben Green's lectures on nilsequences

Go back to the program

https://motivichomotopy.hatenablog.jp/entry/2020/10/19/171757

http://people.maths.ox.ac.uk/greenbj/videos.html

### Lecture 1

1:18:00 Explains the von Neumann theorem for Gowers uniformity norms

### Lecture 2

He mentions GI(s)

https://youtu.be/27Ljd_M--Qw?t=230
And its converse directly after that.

t=660
Introduces the Gowers norms

t=4000
Rigorous statement of GI(s)

### Lecture 3

This lecture is about the converse to GI(s). Along the way he explains useful notions and techniques about polynomial sequences in nilmanifolds.

The lecture starts with recap of the statement of GI(s)

https://youtu.be/u4inkf0VJAU?t=28

The notion of polynomial sequences revisited

https://youtu.be/u4inkf0VJAU?t=900

He says he can send typed notes

https://youtu.be/u4inkf0VJAU?t=1260

Derivatives of nilsequences $\Delta _h \chi$

https://youtu.be/u4inkf0VJAU?t=1289

which enables one to use induction on $s$ in some situations. The final statement of this part is stated at:

https://youtu.be/u4inkf0VJAU?t=3110

Be careful that the derivative $\partial _h p$ of a polynomial sequence $p\colon \mathbb Z \to G$ and the derivative $\Delta _h \chi$ of a nilsequence $\chi \colon \mathbb Z \xrightarrow p G \xrightarrow{\varphi } \mathbb C$ are two different things!

Proof of converse to GI(s) begins
https://youtu.be/u4inkf0VJAU?t=3347

At 1:14:00 (t=4440) the proof is done.

### Lecture 4

Equidistribution of polynomial sequences.

https://youtu.be/yFdoQwDBMRE?t=28

First he talks about the general motivation.

Then he introduces the equidistribution phenomenon in the $\mathbb R/ \mathbb Z$ case.

https://youtu.be/yFdoQwDBMRE?t=345

Theorem of Leon Green

https://youtu.be/yFdoQwDBMRE?t=494

which reduces some problems to the abelianized case $G/[G,G]$.

Definition of "δ-equidistributed"

https://youtu.be/yFdoQwDBMRE?t=889

and the equidistribution theorem.

Announcement of the main result

https://youtu.be/yFdoQwDBMRE?t=1245

Explanation of the symbols in the statement lasts until 34:00 (t=2040).

##### Idea of the proof of the equidistribution theorem

https://youtu.be/yFdoQwDBMRE?t=2110

The van der Corput lemma is part of the machinery which enables induction on $s$.

Its proof lasts until 44:00 (t=2640).

Next he explain's Weyl's idea

https://youtu.be/yFdoQwDBMRE?t=2673

which employs Fourier analysis to get vertical frequency.

He uses the induction technique from Lecture 3

https://youtu.be/yFdoQwDBMRE?t=3254

and spend some minutes to explain how it works.

Idea of proof finished at 1:07:20.

"Why do I care about equidistribution of polynomial sequences on G/Γ?"

https://youtu.be/yFdoQwDBMRE?t=4200

including the decomposition theorem

https://youtu.be/yFdoQwDBMRE?t=4753

Why one has to consider polynomial sequences even if one is primarily interested in linear sequences $n\mapsto g^n x$

https://youtu.be/yFdoQwDBMRE?t=5163

Lecture 5