Ask HN: Which areas of math are practical to programming/algorithms and why?

At my previous job ( https://www.rsa.com/ ), where I worked for 7 years, I've often had to use various mathematical techniques to do my work as a software engineer.

Here are some concrete examples from my career.

* Combinatorics and probability theory - Entropy analysis of authentication schemes, algorithms for enumeration and analysis of related combinations and permutations, etc.

* Calculus and probability theory - Implementation of bloom filter for space-efficient indexing, tests and analysis of false-positive rates, etc.

* Statistics - Adaptive authentication schemes, performance measurements and predictions, network event correlation, etc.

* Discrete mathematics and algorithms - Tree traversals, graph search algorithms, etc. were useful in a variety of situations, e.g. data retrieval in tree-based distributed databases.

* Asymptotic analysis, Big O notation, etc. - Data deduplication in distributed databases.

* Modular arithmetic - I didn't really implement any cryptography algorithms; I only used them. But the understanding of modular arithmetic was essential to understanding some of the limitations of cryptographic algorithms based on modular arithmetic, e.g. why the message size cannot exceed the size of the modulus, as well as for software engineering tasks like decoding certificates found in network traffic.

* Formal language theory, DFAs, etc. - Parser generators for network event parsing, SQL engine development and related query optimizations, etc.

So these are 7 examples from 7 years. There may be more examples but I cannot remember them right now.

I have felt from my personal experience that although most of the work does not involve mathematics, once in a while an interesting problem comes up that can be solved efficiently with the knowledge of mathematics. It is not easy to predict when such a problem would come and it is not always easy to recognize whether the problem at hand can be solved or analyzed efficiently by known mathematical techniques. So it is good to have someone in the team or be the person who has a good breadth of knowledge in mathematics, so that when such problems do come up, they are addressed efficiently. Fortunately for me, I love mathematics and there were people around me who also loved mathematics, so we got a lot of interesting work done.

First I would recommend Cal Newport's blog about using time wisely. http://calnewport.com/blog/2008/11/25/case-study-how-i-got-t... Plenty more great info in the archive: http://calnewport.com/blog/archive/ Obviously the best way would be to go through your school's course syllabus for whatever classes you want to take and look at the material you will be doing, but these are good for a general preparedness:

Axler - Precalculus 2nd version http://precalculus.axler.net/ He works through every odd exercise solution, in full. Also gives you a good intro to sets and series, summation notation, binomial theorem, all this will come up later in discrete math. He assumes the reader knows nothing about Trig as well.

Gilbert Strang's videos "Highlights of Calculus" https://ocw.mit.edu/resources/res-18-005-highlights-of-calcu... just to get an overview of what calculus really is since I assume your university will throw you into an applied single variable class first year. MIT Open Courseware has their calculus (18.01/18.02) course lectures up if you want to watch them too to get an idea of what you'll run into.

"Elements of Mathematics: From Euclid to Gödel" gives quite a good explanation of Mathematics as a whole, like why we learn elementary math the way we do and how it applies to more advanced concepts, explains Rings/Fields and has a really good introduction to Logic. I wish this book existed 5 years ago when I started http://press.princeton.edu/titles/10697.html

"An Introduction to Mathematical Reasoning" by Eccles is short and excellent. You can also download the lecture notes here from CMU's course on proofs http://math.cmu.edu/~svasey/concepts-summer-1-2014/ and try some of the homework if you want but you probably won't see any of this until second year judging by most university calendars and recommended program course list I've seen.

CLRS https://en.wikipedia.org/wiki/Introduction_to_Algorithms which has a great first chapter "Foundations" that compliments Knuth's 'Mathematical Preliminaries' chapter in TAOCP vol 1. Knuth's book you can really drill yourself in these concepts though with the many, many exercises writing proofs. The skills I learned doing these exercises paid in full years later when I started more advanced math and even at work.

Topology is useful in robotics and control to define the set of achievable states in a system and their connectivity.

False. Decisions like this can be made at a higher level than the department. The mandatory courses in my programme were applied univariate calculus and an introduction to statistical modelling (with examples drawn from biology), imposed by the faculty of science on all science majors - all the while the CS professors were screaming for an introduction to logic and discrete maths to be added to the curriculum.

I will definitely take that advice to heart. I literally just hit the submit button on my first conference article.