Large cardinals and combinatorial set theory

August 28, 2018

Paul Larson and I are organizing a special session at the Fall Central Sectional Meeting 2018 in Ann Arbor, on Large cardinals and combinatorial set theory. The session will take place Saturday October 20 and Sunday October 21. See here for the schedule and additional details. Paul and I are trying to organize dinner for Saturday (it is work in progress).

I transcribe the schedule below:

  • Saturday October 20, 2018, 8:30 a.m.-11:20 a.m.

Room 2336, Mason Hall


  • Saturday October 20, 2018, 2:00 p.m.-4:20 p.m.

Room 2336, Mason Hall


  • Sunday October 21, 2018, 8:00 a.m.-10:20 a.m.

Room 2336, Mason Hall


  • Sunday October 21, 2018, 1:00 p.m.-3:50 p.m.

Room 2336, Mason Hall


Riemann on Riemann sums

November 16, 2013

Though Riemann sums had been considered earlier, at least in particular cases (for example, by Cauchy), the general version we consider today was introduced by Riemann, when studying problems related to trigonometric series, in his paper Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe. This was his Habilitationsschrift, from 1854, published posthumously in 1868.

Riemann’s papers (in German) have been made available by the Electronic Library of Mathematics, see here. The text in question appears in section 4, Ueber den Begriff eines bestimmten Integrals und den Umfang seiner Gültigkeit. The translation below is as in

  • A source book in classical analysis. Edited by Garrett Birkhoff. With the assistance of Uta Merzbach. Harvard University Press, Cambridge, Mass., 1973. MR0469612 (57 #9395).

Also zuerst: Was hat man unter \displaystyle \int_a^b f(x) \, dx zu verstehen?

Um dieses festzusetzen, nehmen wir zwischen a und b der Grösse nach auf einander folgend, eine Reihe von Werthen x_1, x_2,\ldots, x_{n-1} an und bezeichnen der Kürze wegen x_1 - a durch \delta_1, x_2 - x_1 durch \delta_2,\ldots, b - x_{n-1} durch \delta_n und durch \varepsilon einen positiven ächten Bruch.  Es wird alsdann der Werth der Summe

\displaystyle S = \delta_1 f(a + \varepsilon_1 \delta_1) + \delta_2 f(x_1 + \varepsilon_2 \delta_2) + \delta_3 f(x_2 + \varepsilon_3 \delta_3) +\cdots \displaystyle +\delta_n f(x_{n-1} +\varepsilon_n \delta_n)

von der Wahl der Intervalle \delta und der Grössen \varepsilon abhängen.  Hat sie nun die Eigenschaft, wie auch \delta und \varepsilon gewählt werden mögen, sich einer festen Grenze A unendlich zu nähern, sobald sämmtliche \delta unendlich klein werden, so heisst dieser Werth \displaystyle \int_a^b f(x) \, dx.

In Birkhoff’s book:

First of all: What is to be understood by \displaystyle \int_a^b f(x)\,dx?

In order to establish this, we take the sequence of values x_1,x_2,\ldots, x_{n-1} lying between a and b and ordered by size, and, for brevity, denote x_1 - a by \delta_1, x_2 - x_1 by \delta_2,\ldots, b - x_{n-1} by \delta_n, and proper positive fractions by \varepsilon_i. Then the value of the sum

\displaystyle S = \delta_1 f(a + \varepsilon_1 \delta_1) + \delta_2 f(x_1 + \varepsilon_2 \delta_2) + \delta_3 f(x_2 + \varepsilon_3 \delta_3) +\cdots \displaystyle +\delta_n f(x_{n-1} +\varepsilon_n \delta_n)

will depend on the choice of the intervals \delta_i and the quantities \varepsilon_i. If it has the property that, however the \delta_i and the \varepsilon_i may be chosen, it tends to a fixed limit A as soon as all the \delta_i become infinitely small, then this value is called \displaystyle \int_a^b f(x) \, dx.

(Of  course, in modern presentations, we use \Delta_i instead of \delta_i, and say that the \delta_i approach 0 rather than become infinitely small. In fact, we tend to call the collection of data x_1,\dots,x_{n-1}, \varepsilon_1,\dots,\varepsilon_n a tagged partition of {}[a,b], and call the maximum of the x_{i+1}-x_i the mesh or norm of the partition.)


November 15, 2013

This link should take you to the YouTube channel of Algo-rythmics, or see their website.

Different sorting algorithms (bubble sort, insertion sort, quicksort, selection sort, shell sort) illustrated through folk dance.


November 5, 2013


I recognize I owe much to Messrs. Bernoulli’s insights, above all to the young, currently a professor in Groningue. I did unceremoniously use their discoveries, as well as those of Mr. Leibniz. For this reason I consent that they claim as much credit as they please, and will content myself with what they will agree to leave me.

L’Hôpital, in the preface (page xiv) of his Analyse des Infiniment Petits pour l’Intelligence des Lignes Courbes (1696), the first calculus textbook, published anonymously. (A posthumous second edition, from 1716, identifies L’Hôpital as the author.)

L'Hospital-2 Read the rest of this entry »

Office Hours

October 28, 2013

This week, I will not be able to hold office hours on Thursday (my son does not have school on Hallowe’en). Instead, they will be on Wednesday, 3-4:30 pm.

2013-08-11 11.41.23

170 – Extra credit

September 20, 2013

Choose a mathematician whose work is somehow related to calculus. The connection may be loose (of course, Newton or Leibniz qualify, but so does John Nash). To avoid repetitions, email me your choice, and I’ll let you know whether your choice has not yet been claimed.

Write (well, type) an essay on their life and mathematical work. It may be short. Make sure to follow reasonable standards of style when citing references, etc. Due December 2 (after Thanksgiving break).

170 – Hippasus of Metapontum

September 16, 2013

Erroll Morris ran a series of essays on the New York Times a couple of years ago, on the topic of incommesurability. The whole series is highly recommended. You may particularly enjoy reading Part III, on Hippassus of Metapontum, the mythical Pythagorean who proved the irrationality of square root of two, and was killed by the other followers of the cult as a result.

The legend is told in many places; Morris lists a few in his essay. I first found it as an appendix to Carl Sagan‘s book version of Cosmos.