Sometimes you decide that you need a specific theoretical result. For example, you may need a closed form formula for a given quantity. Yet, you barely know how to begin.

Maybe, you end up looping: you keep on revisiting the same ideas, again and again. You don’t seem to make any progress at all!

What do you do? Don’t be hasty to conclude that you can’t do theory. Mathematicians are just ordinary people.

Some advice that has worked for me:

• Be patient. Work incrementally.
• At first, don’t try to prove conjectures. Begin by trying to disprove them! It is often much easier to disprove something than to prove it. And most conjectures end up being false.
• Find the simplest non-trivial variation on your problem, and work from there instead. What you will learn from solving a closely related problem might teach you how to break the real problem. A related strategy is to work on specific examples.
• Be paranoid. Check every single fact or claim twice. And then check it again.
• Run computer simulations to check your mathematics, or to suggest new conjectures.
• Draw pictures. Use your visual cortex.
• Try to abstract out the problem: find a more general problem. Sometimes, specific problems appear more difficult than general ones because you are thrown off by irrelevant details.
• Be thorough. When trying to solve a difficult theoretical problem, be neat. For example, work as if you are trying to explain your problem to someone else.

I decided to copy Daniel Tunkelang’s idea and maintain a list of some people who read my blog. This is not meant to be an ego-boosting or name-dropping project. My goal is to prove that blogging is a serious business. Blogging is part of my day job! Indeed, this list proves that professional networking results from my blog.

Like most dads who have professional careers, I have little time for games. Nevertheless, I spend some time playing war games. Here are some I like:

• Civilization: Call to Power was one of my favorites.
• I still play to Master of Orion II even though the game is nearly 15 years old.
• The Age of Wonders series is great too.
• Wesnoth is not a bad game, and it is free.

All these games have taught me some basic principles:

• Retreat instead of losing a battle. When life gets difficult, cut your losses and withdraw at once. This is the most important lesson of all. The best way to win is to avoid battles that you will lose. (In life, don’t fight when the odds are against you. Come back another day.)
• Focus on one or two fronts. Never multitask widly even if you have far superior numbers. You can actually lose the war because you won too many small battles! With exhausted troops all over the map, you can become vulnerable to a focused offensive. (In life, do not fight all battles at once. Try to get one or two solid victories instead.)

While pure theory is wasteful, bringing applied research to academia is equally wasteful!

Applied research seeks to produce applied results. These results are good or bad depending on their real-world application. But in academia, success is defined by peer review. Peers do not test real-world applications! Moreover, researchers have an approximative idea of real-world problems.

Hence, I cringe every time I hear about a new academic applied project. Professors are rarely good at solving real problems! A better option is to use the world around us as a stimuli for research ideas.

To put it bluntly, you cannot live in an ivory tower and pretend to tell others what to do. Instead, use your tower to have a bird’s eye view on the world! Study the world! But be humble about what you can do to transform it!

Honesty is charisma. Whenever you are honest, even if you are controversial, many people will respond positively to you.