Advice on preparing academic presentations. Brent Doiron Mathematics Department University of Pittsburgh

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1 Advice on preparing academic presentations Brent Doiron Mathematics Department University of Pittsburgh

2 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 (ecept their very narrow acre). Not only do they know nothing besides their narrow epertise, 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 eperts 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 Narrow specialists and tacticians would soon be superseded by computers.

3 Problem in academic upbringing 1. We first eperience learning mathematics in a lecture setting. 2. We first eperience presenting mathematics in a lecture setting. 3. Academic talks are not lectures. In a lecture the audience is later epected 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.

4 What you should epect 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)

5 What you should NOT epect 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.

6 Eercise 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.

7 Talk 1: The Fundamental Theorem of Calculus Brent Doiron Mathematics Department University of Pittsburgh

8 Thrm. Let f be a continuous, real valued function defined on the closed interval [a, b]. Let F be the function for all in [a, b] by Z F () = a f(t)dt Then F is continuous on [a, b], di erentiable on the open interval (a, b) and for all in (a, b). F 0 () =f()

9 Proof For any two numbers 1 and 1 + in [a, b] wehave this gives F ( 1 )= F ( 1 + ) = Z 1 a Z 1 + F ( 1 + ) F ( 1 )= a f(t)dt f(t)dt Z f(t)dt The mean value theorem for integration gives Z f(t)dt = f(c) c 2 [ 1, 1 + ]

10 Proof Combining equations gives Taking limits F ( 1 + ) F ( 1 ) = f(c) F 0 () = lim!0 f(c) The number c is in the interval [ 1, 1 + theorem we have lim!0 c = 1,yieldingtheresult F 0 ( 1 )=f( 1 ) ] sobythesqueeze

11 Corollary For the definite integral we have Z b a f(t)dt = F (b) F (a)

12 Eample Z 0 sin(t)dt = cos( ) [ cos(0)] = 1 [ 1] = 2

13 Talk 2: The Fundamental Theorem of Calculus Brent Doiron Mathematics Department University of Pittsburgh

14 Tangent to a curve Area under a curve Goal: Show that tangent and area are fundamentally related

15 Thrm. Let f be a continuous, real valued function defined on the closed interval [a, b]. Let F be the function for all in [a, b] by Z F () = a f(t)dt Then F is continuous on [a, b], di erentiable on the open interval (a, b) and for all in (a, b). F 0 () =f()

16 Proof Consider a curve f() and let the area under the curve be F (). f() F () +

17 Proof f() F () } Area f() +

18 Proof f() F () } Area f() + F ( + ) =F ()+f() + error

19 Proof f() F () } Area f() + F ( + ) =F ()+f() + error Rearranging F ( + ) F () = f()+ error

20 Proof f() F () } Area f() + F ( + ) F () = f()+ error lim!0 F ( + ) F () = F 0 ()

21 Proof f() F () } Area f() + F ( + ) F () = f()+ error O( 2 ) O( ) error O( )! 0 as! 0

22 Proof f() F () } Area f() + F ( + ) F () = f()+ error Taking lim!0 on both sides gives F 0 () =f()

23 Corollary For the definite integral we have Z b a f(t)dt = F (b) F (a)

24 Eample Z 0 sin(t)dt = cos( ) [ cos(0)] = 1 [ 1] = 2 sin() Area 0 2

25 - Conclusion Clear relation between the area under a curve and tangents to the curve F 0 () =f() Can be etended to integration and differentiation in R n Can be etended to apply to piecewise continuous f()