What is mathematical analysis and/or discrete mathematics used for?

[mitcho]

I am starting a maths major and I will going to go into pure maths. I am going to specialize in either analysis or discrete maths. I understand that mathematical analysis has a very strong connection to calculus and that discrete mathematics is used mainly in the cryptography and security industry. What other areas are either of these two studies used in? Also, what area of mathematics would be used to predict trends in things like the stockmarket for example?
Any help would be appreciated.
Thanks.

 

[discrete*]

I am starting a maths major and I will going to go into pure maths. I am going to specialize in either analysis or discrete maths. I understand that mathematical analysis has a very strong connection to calculus and that discrete mathematics is used mainly in the cryptography and security industry. What other areas are either of these two studies used in? Also, what area of mathematics would be used to predict trends in things like the stockmarket for example?
Any help would be appreciated.
Thanks.

I'm curious; how does you know what you're going to specialize in if you don't even know what the fields are?

Analysis does have connections to Calculus. In the course of your bachelor's degree you'll probably take a course (or two) called Advanced Calculus. This is typically Calculus presented in a rigorous manner and usually functions as a bridge course to Real Analysis and the Theory of Functions of a Complex Variable (AKA: Complex Analysis).

That being said, Analysis is the mathematical study of theories of integration, differentiation, measure, functions etc. Informally, it's the mathematics of change. Analysis tends to break complex things into smaller parts and analyze those constituent parts.

Discrete Mathematics is a large field of mathematics that studies discrete models. What this means is that Discrete Math seeks to study noncontinuous objects. I wish I had a better way to describe it, because if you're not yet in some advanced courses, you probably don't know what I'm talking about. We can define Discrete Math, rather loosely, as the study of countable sets.

Discrete Mathematics has deep connections to Computer Science, and most (if not all) CS majors take the course. Discrete Math has some very interesting subfields. A few off the top of my head are Number Theory, Set Theory & Logic, Probability, Combinatorics...

There are many members here at the PF that can answer your last question much better than I can, but I'll give you what I can. To answer questions in finance, I believe Stochastic Calculus is very important. Anything that studies nonlinear functions would most likely have a use.

Both a very broad fields and very interesting subjects in their own right. You should get some coursework under your belt before you start thinking about specializing in any field. Good luck.

 

[mitcho]

Hey and thank you for the reply.
I am unsure at this stage what to specialize in for that reason; I don't yet understand what the two are. The reason I have decided on either one of these is because I have a keen interest in pure mathematics rather than applied simply because of its 'beauty' and I was under the impression that these fields are the two main fields in pure maths. Given the information that you have, do you know of any do you know any industries that would employ a discrete mathematician (apart from cryptography) or a mathematician specializing in analysis as I am worried about employment prospects for a pure mathematician outside of research.
Thanks.

 

[discrete*]

Hey and thank you for the reply.
I am unsure at this stage what to specialize in for that reason; I don't yet understand what the two are. The reason I have decided on either one of these is because I have a keen interest in pure mathematics rather than applied simply because of its 'beauty' and I was under the impression that these fields are the two main fields in pure maths. Given the information that you have, do you know of any do you know any industries that would employ a discrete mathematician (apart from cryptography) or a mathematician specializing in analysis as I am worried about employment prospects for a pure mathematician outside of research.
Thanks.

It really depends on what you study and what you take to doing research in while at university. For example, if you study and do research in Theoretical Computer Science (a subject under Discrete Mathematics) than you might be able to work for companies like Microsoft, IBM, Apple, etc... Now, you may be able to get a job with these companies with any specialization, but I would think that they'd look for persons with an interest in the theory of computing.

The techniques of Analysis are widely applied. Curiously, you've said that you want to work in pure maths, but then work in industry. Pure mathematics typically are not worked on by industry professionals. Perhaps if you get a government research position or something, but other than that, industry is looking for mathematicians that can apply their skills to real world problems.

I think you have a while before you have to worry about this type of stuff. Focus on doing well in all your maths and see what you like best along the way. When I first started I thought I'd be interested in Geometry, then I took a course on the Logic and Set Theory and found my niche so to speak. I never thought I'd study the Foundations, but it was a natural progression for me to choose that as a track for my masters, and soon my Ph.D, and I couldn't be happier. Take your time, and let the chips fall where they will.

 

[Matrix0]

Mathematical analysis is a branch of mathematics that deals with the study of continuous functions and their properties. It is often referred to as calculus, as it builds upon the concepts and techniques of calculus to analyze and understand functions in a more rigorous way. It is used in a wide range of fields, including physics, engineering, economics, and finance.

In terms of pure mathematics, analysis is a fundamental tool for understanding and proving theorems in various areas such as number theory, geometry, and topology. It is also used in the development of other branches of mathematics, such as complex analysis and functional analysis.

On the other hand, discrete mathematics deals with discrete structures and objects, such as graphs, networks, and combinatorial objects. It has applications in various fields, including computer science, operations research, and cryptography. In the cryptography and security industry, discrete mathematics is used to develop algorithms and protocols for secure communication and data protection.

In addition to these areas, analysis and discrete mathematics are also used in various other fields, such as biology, chemistry, and social sciences. They provide powerful tools for modeling and analyzing complex systems and phenomena.

To predict trends in the stock market, one would typically use mathematical modeling and statistics. These areas of mathematics involve the use of data analysis, probability theory, and regression techniques to identify patterns and make predictions about future trends. Time series analysis, in particular, is a branch of mathematics that is specifically focused on analyzing and predicting trends in data over time.

In summary, both mathematical analysis and discrete mathematics have a wide range of applications in various fields, making them important and valuable areas of study for a mathematician. As you continue your studies in pure mathematics, you will gain a deeper understanding of these subjects and their potential uses in different industries and research areas.