**The practice of mathematics** - a series of popular
lectures at the Institute during the academic year 1999-2000.

The talks were recorded at the Institute,
and were put into RealVideo G2 format with the help of Dave Morrison.
They can be found at
`http://www.math.duke.edu/langlands/`.

(Photographs taken by C. J. Mozzochi in one of the lectures)
**The trace formula and its applications**
(an introduction to the work of James Arthur)

First appeared in the *Canadian Math. Bull.* **XX** (2001).

**The report**

Jim Arthur at the Institute, April 2001

(Click **here** for a photograph
of Jim lecturing.)

**Laudatio**
(on the occasion of an award to Günter Harder)

Langlands' comment:*
The following brief discourse was delivered in
Erlangen in October, 2004, on the occasion of
the award of the Karl Georg Christian von
Staudt-Preis to Günter Harder. It does
not do justice to his many contributions
to mathematics, but does attempt to express my great
admiration of him and my great respect for
the passion and the tenacity with which he
continues to reflect on what seem to me some
of the central problems of the modern
theory of numbers.
*

*
At one or two points in the text there are
references to diagrams. The diagrams are
hardly necessary for a mathematically
experienced reader and are not included.*

**The article**

**Review of ***SL*_{2}(**R**) by Serge Lang

First appeared in *
B. A. M. S.* **82** (1976).

**The review**

**Review of ***The theory of Eisenstein systems* by M. Scott Osborne and Garth Warner

First appeared in *
B. A. M. S.* **9** (1983).

**The review**

**Review of Harish-Chandra's ***Collected Works*

First appeared in *
Bull. London Math. Soc.* **17** (1985)

**The review**

**Review of ***Elliptic Curves* by Anthony Knapp

First appeared in *B. A. M. S.* **30** (1994).

**The review**

Portrait engraved by van Schooten the younger, editor and translator
of the Latin edition of *La géometrie.*

Descartes said of it, "La barbe & les habits ne ressemblent aucunement."

(From the Rosenwald Collection at the Institute in Princeton)
**Euclid's windows and our mirrors** - review
of *Euclid's Window* by Leonard Mlodinow

First appeared in the *Notices* of the
American Mathematical Society, May 2002.

**The review**

**Benim tanidigim Cahit Arf**
(Recollections of a year in Turkey with Cahit Arf)

**The memoir**

**Langlands' comment:** This note contains a few recollections of a year
I spent in Turkey in 1967/68, where my office was adjacent to
that of Cahit Arf, known, among other things, for the Hasse-Arf
theorem and the Arf invariant. It was he who referred me --
as I was first attempting to define local factors for
Artin L-functions -- to the paper of Hasse published in the Acta
Salmanticensia. Hasse's paper was my first introduction to the
methods that had already been introduced for calculating and
comparing the -factors.
The note was published in the winter number of Ali Nesin's
Matematik Dünyasi for the year 2004.

**Descartes ile Fermat** (in Turkish)

**Langlands' comment:**
The article is an exercise in the reading of mathematics
from earlier times. An explanation of Descartes's solution
of the problem of Pappus as included in the appendix "La
géométrie" to "Discours de la méthode" and an explanation
of a solution to another form of the same problem by
Fermat, described briefly in a letter included in his collected
works, are taken as an occasion to compare the mathematical
styles of the two men and to observe their mutual debt to
Apollonius as well as the differences in their depth of
understanding of his work. The purpose is not scientific
or historic; it is simply to encourage the private reading
of classical mathematics by those who like me have no special
knowledge of the history of mathematics.

This article appeared on pages 54-61 of the
mathematical magazine Matematik Dünyasi, No. 2, 2005.
The magazine (as mentioned above, edited by Ali Nesin), is
published in Istanbul and has a wide circulation among mathematics students
and teachers in Turkey.

**The article**

A review of Haruzo Hida's
**p-adic automorphic forms on Shimura varieties**

To appear in the *Bulletin of the American Mathematical Society*,
autumn 2006.

**Langlands' comment:**
This review comes with a supplement (footnote) that contains the
comments of several leading specialists and will be much more useful
to the potential reader of the book, whether a novice or a specialist,
than the review itself.

Just after the review itself comes a collection of remarks about
a preliminary version of it by
Langlands as well as by Michael Harris, Ching-Li Chai,
Richard Taylor, J. Tilouine, Ralph Greenberg, and Laurent Clozel.

**The article**

**A supplement**