Fundamentals of Numerical Analysis and Symbolic Computation: Lecture on Writing Mathematical Text
-------------------------------------------------------------------------------------------------

11.12.2020

Below is an anonymised protocol of the discussion:
some statements were made by me, some by you,
and there was not always a consense

Basic rules:
- correct syntax of math formulas
- correct English grammar
- define objects before using (or immediately after first mention)
- precise statements in theorems and lemmas

Any writing depends on (explicit or implicit) assumptions on the reader. It could be
- the tutor/professor
- a mathematician working in the same area
- a mathematician working in a different area
- a student
Trick for writing if you do not know your reader: invent one! For instance,
yourself two years ago.

Possible assumptions on the way of reading:
- reader reads carefully line by line
or
- reader is only interested in a part of the text
Comment: writing a paper so that the reader may be able to skip a part,
without getting lost, is a real challenge! (modularize, give entry points)
- reader thinks about statements and is willing to fill in proof details
- reader already knows the content, he/she only reads for checking

About phrases like "It is easy to see that.." and omissions:
- common suspicion: is it just a cheap excuse for laziness?
- Problem: sometimes the reader does not find it easy to fill in the details.
To avoid this problem, it would be good to give hints, outlines, and/or specific
references of theorems that are needed in order to fill in the detail
- Leaving details to be filled in is okay if the assumption is that the reader
may be willing to do this (it is actually good!), but problems should be avoided

About introductions:
- often gives a small survey of related literature
sometimes this survey is historical
- statements of main results
- states what exactly is new (in research papers / Phd theses)
- points out the scientific value/importance of the main results
- purpose of an introduction: helps the reader to decide if he/she should read the rest
comment: many existing introductions contain lot of text not serving this purpose!
comment: try to avoid disappointment if the reader decides to read
and does not what he/she expects
put limitations of the result in the intro
- in research papers: may contain unrelated citations enforced by reviewers

About abstracts:
- brief summary of main result
- good place to put searchable keywords (but keywords can also be put extra, sometimes)
- an opportunity to advertise your paper - the abstract is more accessible than the full paper

Main text:
- engaging part: the first part after the intro, where we assume that reader has already decided
to read, and where it becomes complicated, is critical: the reader is maybe not yet "hooked"
and may change his/her decision because of small misunderstandings
So: check this part carefully and try to avoid possible misunderstanndings
- avoid disppointments: do not introduce concepts that you do not use later

About conclusions:
- open problems, open questions raised by the results
- possible continuations of research
- not every paper does have a conclusion.
- summary of results, mostly repetitive (is this useful?)
- limitations of the results (can sometimes be better explained afterwards)

--

About talks (especially differences between papers and talks):
- main difference: speaker is responsible for the timing
very often too fast!
- audience better known as readers
- more options for choosing other media for communication
- not good:
- tables in talks - information in tables is not suitable for realtime processing
- complicated sentences, full versions of theorems
- repeated complaints about the shortage of time
- good:
- slide give skeleton, spoken text gives more details
- leave space for questions (not only after talk, but also during talk)
some questions may need a longer answer, and this may put the main point of the talk at risk
in seminars it is easier to address this type of questions, in conferences difficult
- take questions serious: if there was a misunderstanding, try to guess what the misunderstanding is
(it may not help to just answer the question and then proceed)


=======
Homework Assignment:

Part 1: deadline 31.1.2021
--------------------------
Write a text, 1-5 pages, on a mathematical result or insight that you believe is not known
to the other participants of this course.
This could be a result by yourself, but it does not need to be.
The readers will be the other participants.
Try to write self-contained: references only allowed if easily accessable (e.g. wikipedia).

Part 2: read the texts of the other participants
------------------------------------------------
(this part will not be graded)