Chapter 4:
Presentation Technique

Structure of the Lecture

Theoretical Part

Practical Part

Structure of the Lecture Notes

Importance of Written and Oral Presentations

Types of Mathematical Documents

Monographs

Articles in Journals

Articles in Collections of Articles

Articles in Conference Proceedings

Technical Reports of Research Institutions

Papers vs. Publications

Quality Control: Anonymous Refereeing

An Algorithm for Fast Refereeing of Conference/Journal Papers (due to Bruno Buchberger)

(Typical) Referee Report Form

Writing Mathematical Papers

Save the time of the reader !!!

Analysis of Readership

Specification of Goals

Now, clear goals have to be specified and structured. A topic for a paper is not yet a goal! It is helpful to always specify goals in the form of operational goals, i.e. by describing the operations (actions) the reader should be able to perform after having read the paper. Operational goals offer an easy way to check the quality of one's own paper: If readers from the intended group of readers are not capable of performing the operations specified as the operational goals then the paper does not meet the goals.
Clear operational goals give the reader a clear motivation to read the paper, namely the desire to acquire the skills that are described by the goals of the paper.
After the goals have been specified they need to be structured. It may be that one goal is a subgoal of another, one goal can be a prerequisite of another, or two goals may be independent. A hierarchical structure of goals, subgoals, subsubgoals, etc. often structures the actual paper naturally into chapters, sections, subsections, etc.

Collecting Material

The necessary material needs to be collected. In this phase, the literature needs to be studied, i.e. one has to look for relevant books, articles, reports, etc. that contain relevant information for writing the paper. The use of available literature consists of basically four steps:

An Algorithm for Finding Relevant Literature Items

Retrieving Relevant Literature Items

Processing Literature Items

Documenting Used Literature

@ARTICLE {Paue92,
  AUTHOR={Pauer,G.},
  TITLE={{On Lucky Ideals for Gröbner Basis Computations}},
  JOURNAL={JSC},
  YEAR={1992},
  VOLUME={14},
  NUMBER={5},
  PAGES={471--482}}

Structuring Papers

The Title Block

When choosing a title for a paper it is often a good idea to address “the three coordinates of mathematical space”, namely the problem, the data domain, and the method (the algorithm, the theorem). This will help the user to quickly decide about the relevance of the paper. If a coordinate is missing in the title then the reader should expect the paper to give an overview over several or all instances on that axis or that this aspect is maybe not addressed at all in the paper. Examples of titles in this style:

Abstract

Introduction

Finally, spend some words on how the remainder of the paper is structured (“reading instructions”) so that readers can maybe skip certain parts of the paper. For long papers, a table of contents may be included in the introduction.

Exact Presentation for the Users

It is often advisable to give an easy example for the problem at the beginning of this part. If the problem given in the example cannot be solved using existing techniques or the existing solutions show severe disadvantages, then this is usually a strong motivation for the reader to study the paper. Do not forget to come back to the solution of this introductory example later at the end of this part, where further (even more difficult) examples may be presented.
The choice of good examples is very important for the success of a paper. Try to consider the following guidelines:

Detailed Presentation for the Insiders

The exact presentation for insiders is what for software is typically called “reference manual”.

Conclusion

Appendices

Structuring Text

Chapters, Sections, and Subsections

Paragraphs, Sentences, and other Means for Structuring Text

As a general rule, try to use short sentences. It is often advantageous to use indentation within sentences in order to reflect the structure. The indented textblocks are often accompanied by certain symbols (e.g. •, ◦, ◻, etc.) or numbers at the beginning in order to indicate some enumeration–like structure. Consider the following example: “The operation of substitution has three arguments: an expression e in which the substitution takes place, a (free) variable x for which a term should be substituted, and a term t which is substituted for the variable.”. Using indentation, this sentence could be written in one of the following styles:

Important short parts within sentences can be emphasized using italics or bold face. Another possibility to structure text is the use of footnotes. However, footnotes tend to destroy the flow of reading, therfore it is often better to restructure the sentences such that footnotes are not anymore necessary.

Text and Formulae

Numbers and labels are used for refering to the respective formal part, i.e. one may later refer to a certain theorem by “… by Theorem 1 …”. One might consider to use “semantic labels” such as “Theorem (Limit of Continuous Function)”. One may then refer to this theorem by “… by the theorem on the limit of continuous functions …”. Often a combination of decimal numbering with semantic labels such as “Theorem 5.28 (Limit of Continuous Function)” may be the most helpful for the reader. One would refer to this theorem then typically by “… by Theorem 5.28 on the limit of continuous functions …”.
If it seems unclear where a formal part ends, then use some marker to clearly separate formal from informal parts. The end of a proof is usually marked by “□” or “q.e.d”, but also for long definitions it might often be useful the explicitely mark where they end.
Both within formal and informal text there might occur mathematical formulae. Usually, important formulae are written on a line for themselves (displayed formulae), whereas less important formulae are written within the running text (inline formulae). Within a formula, mathematical symbols denoting constants or variables are written italics, e.g. f[x] > a, except for known function constants such as “sin”, “exp”, “log”, etc. In most of the cases formulae do not stand on their own but they are embedded in natural language text. In such situations, be careful that the formula acts as a correct part of the sentence! In particular, punctuation (“.” and “,”) must be used appropriately, even if it is a displayed formula, e.g. “in this case, we have

f[
          x] < log[
            a]  .”

Formal parts should not be embedded into running text. Don't write “We will later need

After reading several mathematical texts you will get some feeling on what the habits in the community are. Much is, of course, personal taste but much of the commonly used styles has developed over the years and there are good reasons why certain things are written in a certain style. Try to develop your personal style! Try to copy the good things that you see from other authors! Try to avoid the bad things that you detect in other authors' writing!

Technical Matters

Giving Mathematical Talks

Analysis of Audience

Specification of Goals

Preparation of the Talk

The Presentation

Analysis of Success