Mathematics: Math in the Making - Linear Algebra Research Project

Fields of Mathematics:
Graph Theory
Topology
Number Theory
Game Theory
Different types of Geometries (Non-Euclidean, Projective...)

Mathematical Applications

Optimization

Actuarial Science

Operations Research

Cryptography

Machine Learning /
Artificial intelligence

Fractals

Feuds in Mathematics

Newton vs. Leibniz
Bernoulli vs. Bernoulli

Epistemology
Absolutists/Platonists
vs.
Fallibilists/Constructivists

Fascinating Conundrums:
The Millennium Prize Problems / Clay Mathematics Institute
sample article:
"Who Wants To Be a Millionaire?"
by Steve Kennedy
American Scientist 2003
Book review which discusses role of prizes in mathematical community.

International Mathematical Union
Fields Medal - chronological listing
 

Objective:   Take a closer look at a topic in Mathematics to learn more about:

Fields,  Applications, and Personalities

Background reading:   "Manifold Destiny:  A legendary problem and the battle over who solved it" by Sylvia Nasar and David Gruber. Annals of Mathematics. New Yorker, August 28, 2006.  The article opens with a fascinating overview (truncated here):

Mathematics, more than many other fields, depends on collaboration. Most problems require the insights of several mathematicians in order to be solved, and the profession has evolved a standard for crediting individual contributions that is as stringent as the rules governing math itself. As Perelman put it, “If everyone is honest, it is natural to share ideas.” ... “Politics, power, and control have no legitimate role in our community, and they threaten the integrity of our field,” Phillip Griffiths said. (44)

When searching an area of mathematics, address the following:

• When did it begin?
• Was the field need driven or did they field come first and needs found later?  
• Who are some key mathematicians involved and what are their contributions?  
• What are some key foundational theorems that perhaps we can understand?  
• How has the field evolved over the years?  
• How is it used today?

Sample search:  "Topology innovation"

When researching the genesis of mathematical creation consider the nature of mathematical knowledge :

• How does inspiration become a fundamental concept? How does a theory become a theorem?
• Are mathematical advances discovered or invented?
• Is mathematical knowledge a function of logic or intuition?  (long running feud - compare Alfred North Whitehead and Bertrand Russell (Principia Mathematica 1913) to Henri Poincaré ("Mathematical Creation" 1908)
• How does mathematics evolve because of access to a common pool of knowledge?
• How do new forms of mathematical knowledge grow?
• What is the surveyability of proof?

Questions about mathematical feuds:

• Who gets credit for an original contribution to mathematical theory? ... the mathematician who had the inspiration?  or the ones who flesh out the gaps?  As Nassar  and Gruber point out, "Occasionally, the difference between a mathematical gap and a gap in exposition can be hard to discern."  (52)
• How does the role and purpose of proof in mathematics impact mathematical disputes?

When searching, it is important to play with your keywords.

• Alternative terms for "feud"  could be controversy, dispute, rivalry, conflict
• Frame your search in the context of intellectual property.    Other terms might be academic credit, professional reputation, even "glory" or ownership.

Suggestion - search Jstor with Google Scholar.