A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. I am interested in mathematics only as a creative art – Godfrey Harold Hardy

Recently a friend who is theoretical computer scientist was commenting to another friend (who has a degree in arts and design) about how much mathematics has in common with arts and design. He explained how some one doing mathematics creatively tries to find intricate and useful structure which may help in problem solving. He stressed how a creative combination of small ideas leads to the ‘final piece of art’. I was amused when our artistic friend gave an incredulous look. This is a common reaction. It can be frustrating to explain to others that mathematical activity is creative in its essence and not mechanical like arithmetic.

Perhaps, Sir Michael Atiyah does of a better job at explaining at least one aspect of this issue:

The art in good mathematics, and mathematics is an art, is to identify and tackle problems that are both interesting and solvable. Proof is the end product of a long interaction between creative imagination and critical reasoning. Without proof the program remains incomplete, but without the imaginative input it never gets started.

One can see here an analogy with the work of the creative artist in other fields: writer, painter, composer, or architect. The vision comes first, it develops into an idea that gets tentatively sketched out, and finally comes the long technical process of erecting the work of art. But the technique and the vision have to remain in touch, each modifying the other according to its own rules.

There is a tendency among some young mathematicians to think that using fancy and sophisticated language means that what they are doing is deep. Nevertheless, modern tools are powerful when they are understood properly and when they are combined with new ideas. Those working in certain fields (number theory, for example) who do not put in the time and substantial effort needed to learn these tools are putting themselves at a great disadvantage. Not to learn the tools is like trying to demolish a building with just a chisel. Even if you are very adept at using the chisel, somebody with a bulldozer will have a huge advantage and will not need to be nearly as skilful as you.

This point is further highlighted by great papers. Authors of great papers worked extra hard to simplify the arguments and make the paper more readable and presentable.

I was notified of a newly launched AMS Graduate Student Blog which is a blog by and for math graduate students. The blog is being handled by Frank Morgan, AMS vice-president, and professor of mathematics at Williams College. The Graduate Student Editorial Board members are Asher Auel, Adam Boocher, Diana Davis, Daniel Erman, Fernando Galaz, Brian Katz, Alex Levin, Kathryn Lindsey, Andrew Obus, David Shea Vela-Vick, Clay Shonkwiler, Annalies Z. Vuong, and Tom Wright.

The blog promises to cover many of the issues which are faced by PhD students from other fields and other geographical regions. Some of the issues already covered in the blog include organizing a reading seminar, how to give a good mathematics talk, advice for beginning teaching assistants, navigating seminars and finding an advisor. One entry which caught my eye was on the hardest part of writing a mathematics paper. It highlights the challenge of writing a good introduction:

My experience in writing the paper from that summer has given me an appreciation for solid introductions in research papers.  Reading a paper, it is easy to lose sight of the big picture when struggling through the technical details.  A glance at a well-written introduction can serve as a much-needed reminder of the context and outline of the argument.  And even if many technical details in a paper prove elusive, the introduction can help frame one’s discussion with someone who would be able to help understand the paper.

“The economic role of science will be of even more importance than before,”

“Some say that now is not the time to invest, but the bottom line is that the downturn is no time to slow down our investment in science. We will not allow science to become a victim of the recession, but rather focus on developing it as a key element of our path to recovery.”

The pledge came at Oxford University in a lecture on science education policy. The occasion was the 2009 Romanes Lecture, an event hosted annually by the University of Oxford since 1892. Previous speakers include Winston Churchill, Karl Popper and Iris Murdoch. The pledge includes the increase in science education and new opportunities for science graduates. The BBC page says that government intention is to double the number of students taking “triple science”, which includes biology, chemistry and physics. One hopes that there is as much ambition to encourage mathematics which drives all the sciences.

With the general doom and gloom, this is a really positive sign especially for UK Science with implications to increase in the science budget. One hopes this leads to increase in higher education funding for computer science and mathematics!

Rationality, cooperation, fairness and conflict are fundamental concepts. It is no surprise then, that game theory has come a long way. Since the seminal work of von Neumann and Morgenstern, game theory has made its presence felt not only in economics and political science but also in computer science and operations research. The field further caught the public imagination when Russell Crowe portrayed Nobel Laureate in Economics and game theorist John Nash in the film ‘A Beautiful Mind (2001).

Michael Maschler (July 22, 1927 – July 20, 2008) was a professor at the Center for the Study of Rationality at the Hebrew University of Jerusalem. He was a long standing collaborator of Nobel Prize Winner Robert Aumann. His important contributions include devising cooperative solutions such as the bargaining set and kernel which are stable ways to allocate resources in a cooperative setting. Maschler is famous for his work on repeated games with incomplete information. He also researched on the applications of cooperative game theory to games on networks and different aspects of voting systems.

David Gale (December 13, 1921 – March 7, 2008) was another well known game theorist who passed away in 2008. Gale completed his PhD from Princeton and taught at Brown and Berkeley. Gale’s most famous contribution is the stable marriage algorithm which he came up with in the 1960s with another game theorist Lloyd S. Shapley. For any equal number of men and women, the algorithm ensures that every one gets married and also that the marriage is stable i.e., no man and woman who are not married to each other have an incentive to break their marriages. Gale also played a role in developing the theory of linear programming and other areas of mathematics.

The short answer is that graphs can be used to reason symbolically about any kind of pairwise relationship between any kind of entity, and that we like to think about pairwise relationships because unary relationships aren’t powerful enough and k-way relationships for k>2 add extra complication without adding any real power.

We have graphs representing people and social networks connecting them (online friendships, sexual contacts, parenthood, coauthorship, etc). We have graphs representing subroutines in a computer program and caller-callee relations between them. We have graphs representing web pages and html links between them. We have graphs representing proteins in your body and the chemical interactions they participate in. Etc etc.

OxDE also gives a nice explanation of why abstracting problems is useful in mathematics and computer science:

Graphs are powerful because the same kinds of problems and algorithms turn out to be important in many of these different applications. So by taking away the application-specific features of all of those different problems and turning them into something as abstract as a graph, we only have to solve these problems once instead of repeatedly solving the same problems in different disguises.

To quantify the many facets of the 200 jobs included in our report, we determined and reviewed various critical aspects of all of the jobs, categorizing them into five “Core Criteria;” that is, the general categories that are inherent to every job: Environment, Income, Outlook, Stress and Physical Demands.

Although the financial and other aspects of mathematics related jobs are easy to see for most people, it is not always easy to communicate the fun aspect of mathematical sciences. This is one misconception that is difficult to change because the only experience that many people have had with mathematics is arithmetic. This gives the impression that mathematics is concerned with number crunching and not with creative problem solving, precise thinking, smart reasoning and logic.

The Wall Street Journal has further coverage of the survey, wherein a mathematician called Ms. Courter is profiled:

“It’s a lot more than just some boring subject that everybody has to take in school,” says Ms. Courter, a research mathematician at mental images Inc., a maker of 3D-visualization software in San Francisco. “It’s the science of problem-solving…Problem-solving involves a lot of thinking…I find that calming.”

1. Good mathematical problem-solving (e.g. a major breakthrough on an important mathematical problem);
2. Good mathematical technique (e.g. a masterful use of existing methods, or the development of new tools)
3. Good mathematical theory (e.g. a conceptual framework or choice of notation which systematically unifies and generalises an existing body of results);
4. Good mathematical insight (e.g. a major conceptual simplification, or the realisation of a unifying principle, heuristic, analogy, or theme);
5. Good mathematical discovery (e.g. the revelation of an unexpected and intriguing new mathematical phenomenon, connection, or counterexample);
6. Good mathematical application (e.g. to important problems in physics, engineering, computer science, statistics, etc., or from one field of mathematics to another);
7. Good mathematical exposition (e.g. a detailed and informative survey on a timely mathematical topic, or a clear and well-motivated argument);
8. Good mathematical pedagogy (e.g. a lecture or writing style which enables others to learn and do mathematics more effectively, or contributions to mathematical education);
9. Good mathematical vision (e.g. a long-range and fruitful program or set of conjectures);
10. Good mathematical taste (e.g. a research goal which is inherently interesting and impacts important topics, themes, or questions);
11. Good mathematical public relations (e.g. an effective showcasing of a mathematical achievement to non-mathematicians, or from one field of mathematics to another);
12. Good meta-mathematics (e.g. advances in the foundations, philosophy, history, scholarship, or practice of mathematics);
13. Rigorous mathematics (with all details correctly and carefully given in full);
14. Beautiful mathematics (e.g. the amazing identities of Ramanujan; results which are easy (and pretty) to state but not to prove);
15. Elegant mathematics (e.g. Paul Erdos’ concept of “proofs from the Book”; achieving a difficult result with a minimum of effort);
16. Creative mathematics (e.g. a radically new and original technique, viewpoint, or species of result);
17. Useful mathematics (e.g. a lemma or method which will be used repeatedly in future work on the subject);
18. Strong mathematics (e.g. a sharp result that matches the known counterexamples, or a result which deduces an unexpectedly strong conclusion from a seemingly weak hypothesis);
19. Deep mathematics (e.g. a result which is manifestly non-trivial, for instance by capturing a subtle phenomenon beyond the reach of more elementary tools);
20. Intuitive mathematics (e.g. an argument which is natural and easily visualisable);
21. Definitive mathematics (e.g. a classification of all objects of a certain type; the final word on a mathematical topic)

Blum refers to the joke that ‘a PhD knows more and more about less and less
until he knows everything about nothing.’ He explains that although, it may seem that one is working on a minute area, if one masters that minute area, it helps one see the broader universe in a new light.

There are different kinds of research in mathematics. One could try to prove something true or expected to be true. One could also search for what is true or disprove what you expect to be true. Blum’s suggestion is that while trying to prove something true, it can be helpful to try disproving it. This can give you a useful insight. The conclusion is that one should keep one’s mind open and be prepared for the unexpected.

For thesis writing, Blum writes that it is better to get the thesis checked and proofread by one’s peers. Not every supervisor is good at or keen on proofreading. The advice article also contains the following quotation:

“First have something to say,
Second say it,
Third stop when you have said it,
and
Finally give it an accurate title.”
JOHN SHAW BILLINGS [1838-1913]

Blum’s last words are the most inspiring:

One does not have to be brilliant, a genius, to be special.
To do something better than anyone/everyone else. To be
UNMATCHED,
One has only to choose an END
any END
that MATTERS
that INSPIRES
YOU
And then DO IT.

• Blum advises to not use books like scrolls and read everything from the start to finish. In mathematical sciences especially, it is better to identify and read relevant material and try to understand something in depth.
• He also advises to write down whatever one reads. This is definitely useful when reading complex material.
• Another piece of advice is to have the belief that there must be some easy way to overturn a hard problem. This helps one to look for innovative solutions and remain optimistic.
• In case someone solves a problem before you, one should take it as an opportunity to analyze how one could have approached the problem better. Moreover, one should read the other person’s solution carefully to learn the method.
• Blum emphasizes the importance of contradiction and paradoxes in mathematical reasoning.
• There is also a suggestion to list all possible ways to solve a problem, by, for example, looking at small examples in order to get a handle on the problem.
• Blum writes that thesis advisors could be of two types. One is someone who gives you a problem from his or her own research area. This might be solvable or very hard but it is advisable to look for a non-standard answer. The second type of supervisor may encourage or let you work in an area that the supervisor is familiar with, which means that you will have to do a lot of literature research. Both types of supervisors have their pros and cons.
• The most important advice in the whole article is that whatever one does, one must enjoy doing it. This is critical in maintaining sufficient enthusiasm to solve problems.