Topics in Mathematics for a 15 minute demonstration

Question

I need to appear for an interview for the post of Assistant Professor in Mathematics in an undergraduate college.

My Backgorund : I have studied topics like Algebra comprising of Group Theory,Ring Theory and Field Theory. I have also studied Linear Algebra and Matrices,Complex Analysis,Functional Analysis. The above topics are of interest to me and I study these topics in my PhD.

Background of College:The college also teaches undergraduate students in the courses mentioned above and also teaches some applied mathematics as a part of Bachelors in Science Course.The syllabus of the college in the website at least says this .

I received a call letter from the college that I need to give a demonstration about some topic of interest of mine for 15 minutes.

The problem is I have never given such a demonstration before and I really dont know how to handle this.In fact I have only given talks about my PhD topics for my PhD and I am pretty sure that I cant say those things for a 15 minute demonstration.

I understand that there are numerous topics on which such a demonstration can be made but I am not sure what will fit into a 15 minutes demonstration.

Can someone please help me on how to select some topics and how to manage them in 15 minutes?

NOTE:There is no powerpoint presentation involved.It should be done in Blackboard.

Any help will be appreciated.

Answer 1

Maybe just pick a topic/area you're interested in and search Mathematics Stack Exchange for questions/answers on this topic/area until you find something that works for you. What I often did in the 1990s and early 2000s for job talks that I gave (roughly 5 to 8 during each of five different search years) was to go to a university library and browse through books (and sometimes look through the tables of contents in bound volumes of MAA journals and Math. Gazette) until I found something that seemed appropriate. Of course, I also sometimes just used things based on my own interests (such as large numbers 1, 2, 3) or something based on (or inspired by) a supplementary handout I'd written for students (done this several times, one example being this with 9-year later comments here).

In particular, the following comments extracted from this might be useful in how to think about what's appropriate, although it's important to note that I had 60 minutes for this particular talk that I gave in March 2001.

I decided to change a job interview (a tenure-track math position) presentation the evening before I was to leave town because I decided that what I had originally planned to talk about might not be sufficiently focused to pull off very easily. [Square/cube law in biology and engineering, applications of scaling principles in mathematics (the Pythagorean theorem can be proved in this manner, for instance), how the Cantor set and other "fractals" scale, and using big 'Oh' and little 'Oh' to describe growth rates (those already discussed, along with linear approximation errors, trapezoid and Simpson rule errors, etc.).] The audience would consist of upper level mathematics students and I could assume a background through sophomore level multivariable calculus. The topic needed to be something relating to undergraduate level real analysis. Although I could assume a bit more mathematical maturity than the typical student who had just finished multivariable calculus (the students were, after all, upper level math majors), I shouldn't assume that they would have already taken a real analysis or advanced calculus course. I decided to discuss the rate at which the harmonic series diverges. By grouping the terms in groups of $2,$ $4,$ $8,\; \ldots,$ in two different ways, you can obtain a divergence rate that is within a factor of $2$ of log-base-$2$ of the number of terms added. By looking at appropriate upper and lower Riemann sums for $y = 1/x,$ you can show that the divergence is asymptotic to log-base-$e$ of the number of terms added. Indeed, you're able to show that the sum of the first $n$ terms lies between $\ln(n) + 1/n$ and $\ln(n) + 1,$ a much stronger result. By considering appropriate triangular regions that get ignored during the lower Riemann sum calculation, you can improve this by showing that the sum lies between $\ln(n) + 1/2 + 1/(2n)$ and $\ln(n) + 1.$ At this point I would show (a bounded monotone sequence argument) that the difference between the sum and $\ln n$ actually approaches a limit---Euler's constant $\gamma,$ and I would then show some of the many ways Euler's constant $\gamma$ comes up in math.

Incidentally, this was before Julian Havil's well known 2003 book Gamma: Exploring Euler's Constant was published, so the stuff I talked about was not as well-trodden for a talk such as this as it probably would be now.

Answer 2

1. I think you should tell us what the audience is. Just the selection committee or more general? Is the intent to be more of a research talk or a teaching demonstration or general interest talk? How prestigious is the school? HPYS? UCal? CalState? Juco? Is this a teaching position or a tenure track research position? (I doubt it since you are not doing an hour talk, but let us know.) I'm not being nosy but my point is that these are "variables" that affect the "solution".

2. Length: you should be able to do a talk in 5 minutes, covering the world or all day covering a small part of your Ph.D. Note: this, of course, is easier said than done. But just do the sweat and figure it out. It is a life skill.

2.5. And after you write your talk, then practice it 3 times. And then cut it more. It ALWAYS TAKES LONGER in real life. And practice it several times! Skill in public speaking for neophytes is highly correlated (rsq > 0.9) to practice amount. Yes, it is "cheating" in that this is not your extemperaneous skill. Just double down and do the cheat. It will be worth it regardless if you get the job and of all variables (to include topic selection and writing, I would say it is the single most important variable...highest partial derivative.)

3. My sneaky advice to getting hired is find something mildly topical (elections, fracking, Facebook social graph, global warming, TV viewing, etc.) and make some talk that plays off of it but shows something of interest of yours. Show your basic topic and how it affects something. People love that stuff. You don't have to go overboard, but get a hook. Even mathematicians earn money, make love, read the newspaper, fight with their kids, etc. Human interest sells. Used to see great articles in The American Scientist. Hint, hint.

(Feel free to convert to comment. -cookiemouse)