Category: Meta

Higher reciprocity laws

Today I wanted to share with you a write-up of a talk I have given to other PROMYS Europe 2018 counsellors about two weeks ago. The topic of the talk was higher reciprocity laws, which is a general term used to describe a great variety of results which, in one way or another, can be seen as generalizations of the classical law of quadratic reciprocity. I have tried to give a somewhat motivated account of how each and every one of those generalizes the previous ones and, ultimately, how all of them are just special cases of one theorem (or of a single, even more general conjecture!)

This is a topic I have been fascinated by for years now and I have been seeking an opportunity/excuse to read more into it. Around a year ago I have given a 30-minute talk on this topic during a student conference I was attending, but due to time constraints I couldn’t get far into the topic nor did I get deep into it myself. My talk during PROMYS Europe was different – I would like to thank all counsellors for having patience to sit through my 2.5 hour long (!) talk, despite my promises I will keep it reasonably short. I have greatly enjoyed reading up on the topic and then preparing to present it to others.

I must admit I am quite proud of the write-up for this talk, which is why I would like to share it with everyone. Except for some minor fixes, this is a version which ended up in our yearbook, but I hope some people find some benefit in this, perhaps even find this topic as interesting as I do!

Direct download link to the PDF

Starting a new project – Solutions to exercises in Marcus’ book (+some info)

The book “Number Fields” by D. Marcus is a very well-known introductory book on algebraic number theory. Its most memorable aspect is, without a doubt, the great number of exercises it contains. They vary from short(ish) computational exercises, through various technical results used later in the book, to series of exercises aimed to establish (sometimes very deep) results in number theory. They are structured in a way which allows even an unexperienced reader be able to solve most, if not all, exercises even on their first reading, thanks to (often very elaborate) hints provided.

However, even then a reader might want to refer some external source in order to see how the exercise can be solved, because otherwise it might be difficult to proceed any further (I myself would appreciate such a source at times). And, as they say, if you want something done right, do that yourself.

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Restarting the blog

Alright, it’s been almost 4 years since I have created this blog and I still haven’t posted anything apart from the introduction post. This wasn’t caused by the lack of time or anything like that, but rather by me being lazy and not motivated. But today I’ve decided to get around and get something done on this blog.

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Introduction

Hi, my name is Wojtek, I am middle school student in Poland. I created this blog inspired by Adam  P. Goucher’s blog – Complex Projective 4-space. Like in his blog, I want to write about math stuff. I won’t write here about basics, so I don’t mind if you don’t understand everything 🙂

Because of personal difficulties, Posts will appear very irregularly, but I’ll do my best. Hope to see any feedback soon.

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