Topics in complex analysis
MATH-327
Some advice on reading mathematical texts
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Description
This page contains some personal advice about reading mathematical texts (scripts, books, papers, etc.) you may find helpful for this course and beyond.
Page content
The following advice is a very condensed version of a subset of Manfred Lehn's guide "Wie halte ich einen Seminarvortrag?" (in German, viz. "How do I conduct a seminar presentation?"; although the title is a bit off-topic, the text contains some good hints which are applicable to reading scientific texts).
You can read a mathematical text in two different modes.
Bird's eye view. What are the rough guidelines? What is the content of the text? What are the central terms and definitions, what are the central results and theorems? What are the rough proof strategies? What is all that done for?
Worm's eye view. How are things done in detail? How does a proof work? Why do you need the hypotheses of a theorem? What happens when these are dropped?
Frequently, you have to switch between these two modes. First, you have to gain an overview over what you are aiming for in, say, a textbook chapter. Otherwise you will be stuck at the first technical lemma that comes along. At first reading, you should skip the proofs and concentrate on the results. Sometime there will be a point at which you don't understand the theorems any more, because you haven't developed a feeling for the introduced terms. Then it is time to take a closer look at the proofs you skipped first. If you have understood the relevant technical details, you should step back again and figure out the overall context, etc.
In a modified form, this procedure also applies to your approach to understanding single theorems or examples. For instance, when confronted with a new theorem, you may ask the following questions before, during, or after the study of its proof. What are simple cases of the theorem (e.g. special cases)? What are simple counterexamples in which certain hypotheses of the theorem are not satisfied? Do the theorem or its proof connect to previously known concepts? Is there a characteristic example which shows all essential phenomena?
This "circular" approach to working into the course material, especially the textbook, should be understood as an inspiration, not as an enforcement to change your (possibly successful) learning strategy. Usually, it is an effective way of reading and understanding also more complicated mathematical literature.
You can read a mathematical text in two different modes.
Bird's eye view. What are the rough guidelines? What is the content of the text? What are the central terms and definitions, what are the central results and theorems? What are the rough proof strategies? What is all that done for?
Worm's eye view. How are things done in detail? How does a proof work? Why do you need the hypotheses of a theorem? What happens when these are dropped?
Frequently, you have to switch between these two modes. First, you have to gain an overview over what you are aiming for in, say, a textbook chapter. Otherwise you will be stuck at the first technical lemma that comes along. At first reading, you should skip the proofs and concentrate on the results. Sometime there will be a point at which you don't understand the theorems any more, because you haven't developed a feeling for the introduced terms. Then it is time to take a closer look at the proofs you skipped first. If you have understood the relevant technical details, you should step back again and figure out the overall context, etc.
In a modified form, this procedure also applies to your approach to understanding single theorems or examples. For instance, when confronted with a new theorem, you may ask the following questions before, during, or after the study of its proof. What are simple cases of the theorem (e.g. special cases)? What are simple counterexamples in which certain hypotheses of the theorem are not satisfied? Do the theorem or its proof connect to previously known concepts? Is there a characteristic example which shows all essential phenomena?
This "circular" approach to working into the course material, especially the textbook, should be understood as an inspiration, not as an enforcement to change your (possibly successful) learning strategy. Usually, it is an effective way of reading and understanding also more complicated mathematical literature.