1 What Is Mathematical Research? Xiaoya Zha Department of Mathematical Sciences (a topic that is too big for me)

2 A math game

3 (1) To begin with Interests Fun (at least, no pain) Proud (sometimes) Show off (you will be happy for the rest of the day if you are the only person in your class who solved the problem) (most important) a lot of questions

4 Many people received the following in 2003 “Guess you folks know this one already … but since I had not seen this particular one before, I just thought I would share it. I wonder … is it true that 2003 is the only year that it will work? First of all, pick the number of times a week that you would like to have dinner out. (try for more than once but less than 10). Multiply this number by 2. Add 5. Multiply it by 50. If you have already had your birthday this year add If you haven’t, add Now subtract the four-digit year you were born. You should have a three digit number. The first digit of this was your original number (i.e., how many times you want to have eat out each week). The next numbers are …. YOUR AGE (they say THIS IS THEONLY YEAR (2003) IT WILL EVER WORK, SO SPREAD IT AROUND WHILE IT LASTS. ”

5 What do you do after reading this ? Does it work? (try a few times, and it works) Ask questions! 1.How does it work? What is the mathematical reasoning? 2.Is it true that “this is the only year (2003) it will ever work”? 3.Do we need to “multiply by 2”? If not, what other numbers can we multiply? 4.Do we need to “add 5”? If not, what other numbers may we add? 5.Does this game always work for any person with any age? 6.What other changes may we make to modify this game? Create your own game

6 (2) After “Show Off” (more serious stuff) Modify existing theorems, theories, systems Weaken the hypothesis and strengthen the result Develop new theorems, theories and systems Attack open problems and conjectures

7 Theorem: If A then B Keep the following questions in your mind when reading the theorem If A - then B (weakening the hypothesis) If A - then B - (weakening the hypothesis and weakening the results) If A then B + (strengthening the result) If A + then B + (stronger hypothesis and stronger results)

8 IfA then B, then C (extend the result) IfX then Y (analogue in other system) If A then B (analogue in bigger system) If α then B (different hypothesis)

9 (3) Some “necessary” conditions Curiosity Constant interests Mathematical Intuition Be able to picture your math objects Do not have to be too abstract Use examples to understand concepts Work on small cases to find patterns, direction of proof, etc Solid Mathematics Background Well-Motivated

10 (4) Some Suggestions Avoid hard questions/conjectures at the beginning Try some open-ended questions so that partial results can be obtained Write down every small things you have observed, obtained (even though they don’t look useful, they may be good later) Don’t expect success in your first attempt Need practice a lot to get feelings and experiences

11 Try to submit something to undergraduate research journals and some low-rating journals (not necessarily refereed by experts) Finish your first project to certain level (may not be publishable), you will be encouraged and more enjoyable If you give up too soon (unless it is an un-reasonable problem), you will be discouraged Open your mind, talk to your teachers, and get started