Advice on preparing academic presentations

Brent Doiron Mathematics Department University of Pittsburgh

A quote from Doron Zeilberger

“I just came back from attending the 1052nd AMS (sectional) meeting at Penn State, last weekend, and realized that the Kingdom of Mathematics is dead. Instead we have a disjoint union of narrow specialties, and people who know everything about nothing, and nothing about anything (except their very narrow acre). Not only do they know nothing besides their narrow expertise, they don't care!”

His reasoning:

“You can't really blame the audience for not showing up [to plenary lectures], since they were probably burnt out from countless previous invited talks where they didn't understand a word, or from reading the very technical abstracts of the current talks. Most speakers have no clue how to give a general talk. They start out, very nicely, with ancient history, and motivation, for the first five minutes, but then they start racing into technical lingo that I doubt even the experts can fully follow.”

His plea:

“For the good of future mathematics we need generalists and strategians who can see the big picture. Narrow specialists and tacticians would soon be superseded by computers.”

Problem in academic upbringing

1. We first experience learning mathematics in a lecture setting.

2. We first experience presenting mathematics in a lecture setting.

3. Academic talks are not lectures.

In a lecture the audience is later expected to reproduce results. Details are critical.

In an academic talk the audience is hoping to gain a general intuition for your field and the contributions that you have made to it. Details obscure.

What you should expect from your audience

1. A genuine interest in the topic of your talk.

2. The ability to follow trains of thought that last < 5mins.

3. Working knowledge of basic mathematics (undergraduate)

What you should NOT expect from your audience

1. That they know more about your problem than you do.

2. That they hate being told things that are “obvious”.

3. Working knowledge of your field of mathematics.

4. That their initial passion for your problem matches yours.

Exercise

I will present two talks on the Fundamental Theorem of Calculus as an academic talk. The first will be bad - and we will discuss why. The second will be better - and we will discuss why.

Brent Doiron Mathematics Department University of Pittsburgh

Talk 1: The Fundamental Theorem of Calculus

Fundamental theorem of calculus

Thrm.

F (x) =

Zx

a

f(t)dt

F

0(x) = f(x)

Let f be a continuous, real valued function defined on the closed

interval [a, b]. Let F be the function for all x in [a, b] by

Then F is continuous on [a, b], di↵erentiable on the open interval (a, b)

and

for all x in (a, b).

Fundamental theorem of calculus

Proof

For any two numbers x1 and x1 +�x in [a, b] we have

F (x1) =

Zx1

a

f(t)dt

F (x1 +�x) =

Zx1+�x

a

f(t)dt

this gives

F (x1 +�x)� F (x1) =

Zx1+�x

x1

f(t)dt

The mean value theorem for integration gives

Zx1+�x

x1

f(t)dt = f(c)�x

c 2 [x1, x1 +�x]

Fundamental theorem of calculus

Proof

Combining equations gives

F (x1 +�x)� F (x1)

�x

= f(c)

Taking limits

F

0(x) = lim�x!0

f(c)

The number c is in the interval [x1, x1 +�x] so by the squeeze

theorem we have lim

�x!0c = x1 , yielding the result

F

0(x1) = f(x1)

Fundamental theorem of calculus

Corollary

Z b

af(t)dt = F (b)� F (a)

For the definite integral we have

Fundamental theorem of calculus

Example

Z ⇡

0sin(t)dt = � cos(⇡)� [� cos(0)] = 1� [�1] = 2

Brent Doiron Mathematics Department University of Pittsburgh

Talk 2: The Fundamental Theorem of Calculus

Fundamental theorem of calculus

Tangent to a curve

Area under a curve

Goal: Show that tangent and area are fundamentally related

Fundamental theorem of calculus

Thrm.

F (x) =

Zx

a

f(t)dt

F

0(x) = f(x)

Let f be a continuous, real valued function defined on the closed

interval [a, b]. Let F be the function for all x in [a, b] by

Then F is continuous on [a, b], di↵erentiable on the open interval (a, b)

and

for all x in (a, b).

Fundamental theorem of calculus

Proof

f(x)

F (x)

x x+�x

Consider a curve f(x) and let the area under the curve be F (x).

Fundamental theorem of calculus

Proof

f(x)

F (x)

x x+�x

f(x)�x

Area}

Fundamental theorem of calculus

Proof

f(x)

F (x)

x x+�x

f(x)�x

Area}

F (x+�x) = F (x) + f(x)�x+ error

Fundamental theorem of calculus

Proof

f(x)

F (x)

x x+�x

f(x)�x

Area}

F (x+�x) = F (x) + f(x)�x+ error

F (x+�x)� F (x)

�x

= f(x) +

error

�x

Rearranging

Fundamental theorem of calculus

Proof

f(x)

F (x)

x x+�x

f(x)�x

Area}

F (x+�x)� F (x)

�x

= f(x) +

error

�x

lim�x!0

F (x+�x)� F (x)

�x

= F

0(x)

Fundamental theorem of calculus

Proof

f(x)

F (x)

x x+�x

f(x)�x

Area}

F (x+�x)� F (x)

�x

= f(x) +

error

�x

error

�x

⇠ O(�x) ! 0 as �x ! 0

O(�x

2) O(�x)

�x

Fundamental theorem of calculus

Proof

f(x)

F (x)

x x+�x

f(x)�x

Area}

F (x+�x)� F (x)

�x

= f(x) +

error

�x

Taking lim

�x!0on both sides gives

F

0(x) = f(x)

Fundamental theorem of calculus

Corollary

Z b

af(t)dt = F (b)� F (a)

For the definite integral we have

Fundamental theorem of calculus

Example

Z ⇡

0sin(t)dt = � cos(⇡)� [� cos(0)] = 1� [�1] = 2

0 ⇡ 2⇡

sin(x)

x

Area

Fundamental theorem of calculus - Conclusion

Clear relation between the area under a curve and tangents to the curve

F

0(x) = f(x)

Can be extended to integration and differentiation in Rn

Can be extended to apply to piecewise continuous f(x)