Math 8001: Giving a Good Math Talk November 30, 2012 Any current - - PowerPoint PPT Presentation

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 = {(x1, x2, ...xn) ∈ Rn | x2

1 + x2 2 + · · · + x2 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 Rn. 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

• f Rn. (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. highest density = volume of sphere volume of smallest cell = (4/3)π (1 − .293) · 82 = 4.189 5.656 ∼ = 74.05% 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 Problem1 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.

1See “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 15 10 $1.44 McGCh 400 42 31 8 15 15 8 $2.29 Fries 220 26 3 15 2 $0.77 SMcMf 345 27 15 4 20 15 $1.29 1Milk 110 12 9 10 4 30 $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

• ffer, 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:

Comments from Early Viewers

Before the video went viral, commenters had some knowledge of mathematics. The successful student: The struggling student:

Comments from Early Viewers

Before the video went viral, commenters had some knowledge of mathematics. The successful student: The struggling student: The student studying a different chapter:

Tuesday, November 20, 2007 @ 11:14pm

Tuesday, November 20, 2007 @ 11:14pm

“YouTube, meet Mathematics!”

When the general YouTube viewing public encountered high level mathematics, the tone of the comments changed.

“YouTube, meet Mathematics!”

When the general YouTube viewing public encountered high level mathematics, the tone of the comments changed.

“YouTube, meet Mathematics!”

When the general YouTube viewing public encountered high level mathematics, the tone of the comments changed.

“YouTube, meet Mathematics!”

When the general YouTube viewing public encountered high level mathematics, the tone of the comments changed.

“YouTube, meet Mathematics!”

When the general YouTube viewing public encountered high level mathematics, the tone of the comments changed.

The self-confident:

The self-confident: The deep thinker:

The self-confident: The deep thinker: The poet:

The self-confident: The deep thinker: The poet: The observant:

The 13-year old:

The 13-year old: The conversation:

The 13-year old: The conversation: The pyschedelic:

Inflation...

“M¨

• bius Transformations Revealed is a wonderful video clarifying a

deep topic.”

• Edward Tufte

“On YouTube a beautiful video explaining the geometric interpretation of these transformations has been viewed one million six hundred thousand times so far–more people than can be reached by even ten thousand mathematics lecturers.”

• Terence Tao

Deflation...

For comparison, Lady Gaga has gar- nered 196 million hits [on YouTube]. Hence Jon’s video is worth about 1%

• f a gaga.
• Mathematics Department Newslet-

ter, Augustana College (SD)

Reality Check

It was a nice moment to have mathematics featured prominently

• n YouTube. However....

ATC Clip

Möbius Transformations Revealed

Douglas N. Arnold and Jonathan Rogness

M

öbius Transformations Revealed is a short film that illustrates a beautiful correspondence between Möbius transformations and mo- tions of the sphere. The video received an Honorable Mention in the 2007 Science and Engineering Visualization Challenge, cosponsored by the National Science Foundation and Science magazine. It subsequently received extensive coverage from both traditional media

• utlets

and

• nline

blogs. Edward Tufte, the world’s leading expert on the visual display of information, came across the video and reported

• n his blog “Möbius Transformations Revealed

is a wonderful video clarifying a deep topic… amazing work…” But the film has also attracted a far less expert audience. As of this writing, it has been viewed nearly 1.5 million times on the video-sharing website YouTube and is rated as the number three top favorite video of all time in YouTube’s educational category. Over 11,000 viewers have declared it among their favorites, which makes it one of the YouTube top favorites of all time. From the more than 4,000 written comments left by YouTube viewers it is clear that many of them had little mathematical background, and some were quite young. To view Möbius Transformations Revealed, visit the website http://umn.edu/˜arnold/moebius/. In this article we discuss some of the technical details behind the video and offer a “behind the Douglas N. Arnold is McKnight Presidential Professor of Mathematics at the University of Minnesota. His email ad- dress is arnold@umn.edu. Jonathan Rogness is assistant professor of mathemat- ics at the University of Minnesota. His email address is rogness@umn.edu. scenes” look at its production. We begin with a brief overview of the visualization of functions

• f a complex variable, especially the technique

used throughout the video, which we refer to as homotopic image mapping. This is followed by a discussion of Möbius transformations and the specific theorem illustrated in the video. We con- clude by describing aspects of the movie that are generally unnoticed by the public but can be appreciated by mathematicians. Visualization of Functions Among the most insightful tools that mathematics has developed is the representation of a function

• f a real variable by its graph. In fact, historically,

graphs of functions appeared before the notion

• f function itself. A graph of the inclinations of

planets as a function of time appears already in a tenth century manuscript [1], and in the fourteenth century Nicolas Oresme published a graphical method for displaying data that leads to graphs that appear quite familiar (see Figure 1). By the late seventeenth and early eighteenth century, when the notion of function was devel-

• ped by Leibniz, John Bernoulli, Euler, and others,

graphs appeared in their works that would not be out of place in today’s calculus texts. Who today would attempt to teach the trigonometric functions, without drawing a graph? The situation is quite different for a function of a complex variable. The graph is then a surface in four-dimensional space, and not so easily drawn. Many texts in complex analysis are without a single depiction of a function. Nor is it unusual for aver- age students to complete a course in the subject with little idea of what even simple functions, say trigonometric functions, “look like”. (Fortunately Figure 3. A colored rectangle and its image mapped via cos z.

Summary

Once you’ve finished writing your talk, ask yourself:

◮ Is it too long? ◮ Is it too technical? ◮ Will it connect to the audience?

Finally

Have a (relevant!) “kicker” at the end of the talk, i.e.

◮ A memorable image ◮ A cool video ◮ A surprising example or conclusion