Math 8001: Giving a Good Math Talk November 30, 2012
Any current issues in your own teaching?
The Problem Most colloquia are bad. They are too technical, and aimed at too specialized an audience. Consequently, most mathematicians skip colloquia in areas not in their general field (unless the speaker is famous: mathematicians are very class-conscious). So when a conscientious speaker actually listens to the routinely ignored advice to prepare a lecture “accessible to graduate students,” he or she looks out on the audience and sees only experts in the field, and feels stupid for preparing an elementary lecture. — John McCarthy (Article posted on 8001 Homepage)
Why Does it Happen? I have heard terrible colloquia from senior mathematicians, who have been giving bad talks for 30 years. I can only conclude that they do not realize their talks are bad. Why? Because afterwards, people come up politely and say “Nice talk,” thinking it is a harmless white lie. It is not: it means that the next unfortunate audience will have to sit through a bad talk, the speaker obliviously thinking that he or she is doing a great job. — John McCarthy (Same Article)
Today’s advice is not focused on research seminars or even colloquia, but rather your future speaking endeavors: ◮ Job talks, especially at smaller schools. ◮ General math talks to mathematicians in other fields. ◮ Outreach talks to high school students. ◮ Public lectures
Your Goal You want people to learn something... ...and want to learn more!
Basics The mechanics and public presentation skills we discussed with respect to lecturing all still apply. See the Math 8001 webpage for links to relevant articles.
A Word About PowerPoint Remember the Rule of 33 i.e. “Less is More.”
What is the Sphere Packing Problem? How can we pack spheres as efficiently as possible in 3D space? • In 1611, Kepler conjectured that the most efficient way was the same that grocers have used throughout history: lay out the first layer, then put the second layer in the dimples of the first, and so on. • This seems intuitive, but it took mathematicians 387 years to prove it, a situation that some called “scandalous” and others “ridiculous.” • Claude Ambrose Rogers wrote in 1958, “Many mathematicians believe, and all physicists know, that Kepler’s Conjecture is true.” Sphere Packing – p.3/28
A Formal Statement of the Problem Define the unit sphere in n -dimensional space to be the set of all points which have a distance of 1 from the origin: S = { ( x 1 , x 2 , ...x n ) ∈ R n | x 2 1 + x 2 2 + · · · + x 2 n = 1 } Certainly we can move spheres around so that they are centered at a different point as well. A sphere packing is some collection of spheres which fill up R n . The spheres are not allowed to intersect, although they can touch (“kiss,” from billiards). The (global) density of a packing is the volume contained in the spheres divided by the total volume of R n . (Wait a second....) More precisely, we can use a limit. Look at the ratio of the volume covered in an n -dimensional box of length k ; then let k → ∞ and hope the ratio approaches a number! Sphere Packing – p.4/28
The 3D Case continued... This fact is all we need to prove that the FCC (or HCP , if you prefer) is the densest lattice packing. volume of sphere (1 − . 293) · 8 2 = 4 . 189 (4 / 3) π highest density = volume of smallest cell = = 74 . 05% ∼ 5 . 656 That’s the density of the FCC packing, and so in the midst of a book review, Gauss polished off the “easy” part of the problem. The general case would take another 167 years, although there were plenty of failed attempts in the meantime. Most of these involved looking at the 3D Voronoi Cells and trying to minimize their volume. At most, people managed to lower the theoretical upper bound for the density to the high 70s. Sphere Packing – p.18/28
Hsiangs Proof...? Very quickly, problems developed. • Early preprints included egregious errors, such as the following: “If several objects do not fit into a given area, then they do not fit into a smaller area.” Most (all?) of these errors were removed from the final paper, but the damage was done. • His paper was published in the International Journal of Mathematics , which was published by his own department. The period of time from the original submission to a submission of the revised version was less than 16 weeks ! This raised concerns about the refereeing process, or lack thereof. • Hsiang repeatedly used phrases such as, “it is easy to see” and “the general case follows using the same method” in cases where the experts in the field did not find it obvious at all. • In Math Reviews , Gabor Fejes-Toth (son of the Fejes-Toth Sphere Packing – p.21/28
Basic Advice First and foremost: ◮ Know your audience! (Especially if you are unfamiliar with the setting...) ◮ Know what’s expected! A rigorous talk? An entertaining talk? An inspiring talk?
Planning Your Talk ◮ Don’t overestimate your audience. ◮ Cover half of what you want to. ◮ Don’t get bogged down in technical details. That might mean: ◮ No proofs (!), or ◮ Sketches of proofs for 1-2 basic results ◮ You don’t have to be an world renown expert in the field. ◮ The three most important things in your talk are...?
Examples, Example, Examples Examples will stimulate your audience’s interest in the mathematics. Find a way to connect your subject to the math they already know or their everyday lives. Every field has great examples. Find them!
The McDonalds Diet Optimization Problem 1 The problem: how can you reach your recommended daily nutritional allowances at McDonald’s for the least amount of money? This is an optimization problem, because we’re trying to minimize the cost function. We also have constraints , because we need to meet certain requirements: 2000 calories, 55 grams of protein, 100% of our RDA of Vitamin C, and so on. 1 See “Examples” at www.ampl.com.
Item Cal Carb Pro A C Ca Fe Cost Min 2000 350 55 100 100 100 100 - QPChz 510 34 28 15 6 30 20 $1.84 McLnD 370 35 24 15 10 20 20 $2.19 BigMc 500 42 25 6 2 25 20 $1.84 Filet 370 38 14 2 0 15 10 $1.44 McGCh 400 42 31 8 15 15 8 $2.29 Fries 220 26 3 0 15 0 2 $0.77 SMcMf 345 27 15 4 0 20 15 $1.29 1Milk 110 12 9 10 4 30 0 $0.60 OJ 80 20 1 2 120 2 2 $0.72
The computer gives us the following solution. ◮ 4.39 Quarter Pounders with Cheese ◮ 6.15 Small Fries ◮ 3.42 Milks Total Cost : $14.85. (Total Calories: 3965)
The computer gives us the following solution. ◮ 4.39 Quarter Pounders with Cheese ◮ 6.15 Small Fries ◮ 3.42 Milks Total Cost : $14.85. (Total Calories: 3965) The computer has given us insight. We left out an important constraint: we want integer solutions!
The new, more practical solution, looks like this. ◮ 4 Quarter Pounders with Cheese
The new, more practical solution, looks like this. ◮ 4 Quarter Pounders with Cheese ◮ 5 Small Fries
The new, more practical solution, looks like this. ◮ 4 Quarter Pounders with Cheese ◮ 5 Small Fries ◮ 4 Milks
The new, more practical solution, looks like this. ◮ 4 Quarter Pounders with Cheese ◮ 5 Small Fries ◮ 4 Milks ◮ 1 Fillet-O-Fish Total Cost : $15.05. (Total Calories: 3950)
If we give the computer all 63 items on the McDonald’s menu, we get the following solution. ◮ 2.06 Cheeseburgers ◮ 4.12 Sweet’n’Sour Sauces ◮ 16.2 Honeys ◮ 0.04 Chunky Chicken Salads ◮ 2.27 Cheerios ◮ 1.78 Milks ◮ 0.41 Orange Juices Total Cost : $5.36. (Total Calories: 2018)
In fact, if we give the computer everything McDonald’s has to offer, we get an even sillier solution. ◮ 55 packets of Bacon Bits. ◮ 50 packets of Honey ◮ 50 packets of Barbecue Sauce. Total Cost : $0.00.
The computer has given us the insight to add three new constraints: 1. We should ask for integer solutions. 2. You can’t get certain items (such as Sweet’n’Sour Sauce or Honey) without the accompanying meal. 3. Variety would be good – we should limit ourselves to no more than two of the same item. We can also add other constraints, such as “one drink per meal.”
The “Final” Solution (organized into three meals) Meal 1 Meal 2 Meal 3 Cheerios Cheeseburger 2 Hamburgers English Muffin Side Salad Chocolate Shake Cinn Raisin Danish Croutons Orange Juice HiC Orange (large)
Conclusion During the rest of your studies – and career – remember that calculators and computers can be valuable tools, but they are only tools. They can give you insights into a problem, but you still need to do the thinking.
Public Lectures With nonspecialists, don’t be afraid to be light, fluffy and entertaining. Also (from Joe Gallian): don’t be modest if mathematics catches the public eye. Play it up!
Comments from Early Viewers Before the video went viral, commenters had some knowledge of mathematics.
Comments from Early Viewers Before the video went viral, commenters had some knowledge of mathematics. The successful student:
Comments from Early Viewers Before the video went viral, commenters had some knowledge of mathematics. The successful student: The struggling student:
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