# Snapshots of the history of mathematics, spring 2015

### What is going on?

Some extra exercises for compensating for missing hours have been uploaded: check the "Exam" section below.

Here some of the essays written by the students of the course (published with their permission):

Zenith Purisha, Galileo Galilei

Matias Von Bell, Highlights from the History of Graph Theory

Francesca Corni, The Essence of Mathematics Lies in its Freedom

Oluwatosin Ishmeal, On Fermat Last Theorem

Clifford Gilmore, William Rowan Hamilton: Mathematician, Poet and Vandal

### Some lecture slides

Most details presented and especially almost all of the mathematics at the blackboard are not shown in these slides, they just give a partial sceleton of the material.

### Lecturer

### Description of the course

The course is not a 'standard history of mathematics course'. Instead, it focuses on some specifically chosen themes or turning points in the development of mathematics. In addition, the topics do not aim at all to cover full areas or the complete time evolution of some historic period in mathematics. Naturally such selection also reflects the lecturers own interests in the history of mathematics.

Examples of topics that will be presented in the lectures:

- Solving polynomial equations - fundamental theorem of algebra and prehistory of Galois theory
- Work of Emmy Noether and Sophia Germain
- History of spectral theory
- Fourier series and development of rigor in analysis in the 1800's
- Road to modern probability
- Gauss, Weil, Grothendieck, Deligne: long path to modern level of abstraction
- Along the way, life of some interesting math personalities will be described

The course is mainly aimed to master students and graduate students. During many of the lectures the __main aim is to try to present some bits of the mathematics involved.__ Hence knowledge of basic analysis courses, probability and algebra is needed to follow many the lectures. Level of mathematics in the lectures may vary considerably from lecture to lecture.

One can pass the course by attending most lectures, writing an essay on a given topic and giving a presentation of it in the exercise class run and supervised by Paola Elefante. Their timetable will be decided later on.

The topics for the student presentations will be decided during the first weeks of the course. Own suggestions are welcome!

### Scope

6 sp.

### Type

Advanced studies

### Prerequisites

Some basic knowledge of analysis courses, probability and algebra are needed to follow all the lectures, see the course description above.

### Lectures

Weeks 4-9 and 12-18 (with some exceptions/changes mentioned later on) Tuesday 10-12 and Wednesday 10-12 in room C123.

Occasionally the Wednesday lecture will start at 9 (it will be communicated in this webpage).

Easter Holiday 2.-8.4.

### Bibliography

All written math since antiquity... but, in particular, here are some interesting sources:

Biographies of mathematicians.

Biographies of women in science.

C. B. Boyer, *A History of Mathematics*, Wiley, 1968.

V. J. Katz, K. H. Parshall, *Taming the Unknown: a history of algebra from antiquity to the early twentieth century*, Princeton University Press, 2014.

R. Netz, W. Noel, *The Archimedes Codex*, Da Capo Press, 2007.

H. W. Lenstra Jr., *Solving the Pell Equation* (link).

G. Cardano, *The book of my life*, NYRB Classics (Cartano's autobiography).

H. Edwards, Galois theory. Springer.

K. Plofker, *Mathematics in India: 500 BCE–1800 CE*, Princeton University Press, 2009.

V. J. Katz. *The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook*. Princeton University Press, 2007.

R. Cooke, *The Mathematics of Sonya Kovalevskaya*, Springer-Verlag, 1984.

S. Kovalevskaya, *A Russian Childhood*, Springer-Verlag, 1978 edition.

J. J. Watkins, *Number Theory: a Historical Approach*, Princeton Press, 2013.

H. Edwards, *Fermat's Last Theorem*: A Genetic Introduction to Algebraic Number Theory, Springer.

V. I. Arnold, *Huygens & Barrow, Newton & Hooke*, Birkhäuser, 1990.

Jason Bardi, *Calculus Wars*, High Stakes 2006

Isaac Newton, *Mathematical Principles of natural Philosophy*, (translated by Abdrew Motte, and revised by Florian Cajori, Univ. of Calfornia Press, 1934 .

Hans Niels Jahnke (ed.), *A History of Analysis*, Am. Math. Soc, 2003.

### Registration

Did you forget to register? What to do?

### Exam

The course can be passed by completing the following:

- regular attendance (at least 80% =
*you can skip up to 5 lectures*) (see below for exceptions) - writing an essay
- giving a presentation on the essay

The final examination (essay and presentation) can be carried out in pairs (of your choice) or on your own. The essay shall contain some mathematics and some history. As a general indication, the essay should be minimum 15 page long, 5 of which should be non-math.

If you work in pairs, the presentation is required to be 2 academic hour long (45+45 minutes), otherwise it can be either 2 or 1 academic hour long. The presentation can be held at some Friday time slot (see table below) or as replacement for some lecture (agree with Eero Saksman).

Please remember that essay and presentation must be in English language.

You can agree your chosen topic and book a time slot with Paola Elefante. The deadline to send the essay is **10.5.2015**. Please send it via email in pdf format to Paola Elefante.

For the essay, you can choose any topic of your interest, but if you lack ideas, here are some suggestions:

- Sophia Kovalevskaya
- Hilbert and his existence proof for invariants
- Evolution of axioms of geometry:
- 1) Euclid etc. (role in the history)
- 2) History of non-Euclidian geometry
- 3) Hilbert axioms for geometry and program for axiomatizing the whole of mathematics
- Gödel (
**taken**) - Lexell
- the Lindelöf school of function theory (Nevanlinna, Myrberg, Ahlfors etc)
- Emergence of probability theory (Pascal, Fermat)
- Prehistory of the Gaussian distribution
- Mathematical history of x-ray tomography
- Galileo Galilei (
**taken**) - Alice Roth
Paul Erdös

To avoid repetitions of presentations and with lectures, please agree the choice of topic with the lecturer.

About attendance: if you *really really really* have need to skip more than 20% of lectures, you can cover them by doing some exercises given by the lecturer. You can find them here (in constant update: check the date of the document): extra exercises. Please agree with Eero Saksman or Paola Elefante, in case.

### Exercises

Friday h 10-12 can be used for office hours (contact Paola Elefante beforehand) or as time slots for the final presentations.

Please contact Paola Elefante to book a time slot. Remind is first come, first serve.

#### Presentations calendar

Speaker | Title | Day | Time | Place | Instructor |
---|---|---|---|---|---|

Unknown User (axhartma@helsinki.fi) (***) | Leibniz and analysis notation | Friday 6.3 | 10-12 | C122 | Paola Elefante |

Unknown User (samelson@helsinki.fi) (1 hour) | A Brief History of Western Mathematical Notation | Friday 13.3 | 10-11 | C122 | Paola Elefante |

Unknown User (ishmeal@helsinki.fi), Unknown User (gilmore@helsinki.fi) | Fermat's last theorem | William Rowan Hamilton: Mathematician, Poet and Vandal (**) | Tuesday 17.3 | 10-12 | C123 | Paola Elefante |

Anne Isabel Gaudreau (§), Unknown User (sjdaniel@helsinki.fi) | Unexpected repercussions of knot theory | Women in Mathematics | Friday 20.3 | 10-12 | C122 | Paola Elefante |

Friday 27.3 | 10-12 | C122 | Paola Elefante | ||

Unknown User (mavo@helsinki.fi), Atte Walden | The history of graph theory | History of approximation | Friday 10.4 | 10-12 | C122 | Paola Elefante |

Unknown User (nidia@helsinki.fi) and Fiyinfoluwa Soyoye | The History of Cryptography | Friday 17.4 | 10-12 | C122 | Paola Elefante |

Unknown User (purisha@helsinki.fi) (1 hour), Unknown User (corni@helsinki.fi) (1 hour) | The First Physicist, Galileo Galilei | The essence of mathematics lies in its freedom (§§) | Friday 24.4 | 10-12 | C122 | Paola Elefante |

Unknown User (soultani@helsinki.fi) | Bourbaki | Wednesday 29.4 | 9-10 | C123 | Paola Elefante |

Janne Junnila and Jesse Jääsaari | The Rising Sea (°°) | Friday 8.5 | 10-12 | C122 | Paola Elefante |

Yijie Jiang | The history of Pi | Wednesday 13.5 | 10-11 | C124 | Paola Elefante |

(**) Clifford Gilmore - *William Rowan Hamilton: Mathematician, Poet and Vandal*

To mark St Patrick's Day we will hear about the remarkable life of Ireland's greatest mathematician, William Rowan Hamilton (1805-1865). Hamilton was a child prodigy whose contribution to physics and mathematics transformed algebra and laid the foundations for, among others, quantum mechanics and computer graphics.

He is also famous for his "eureka" moment, when he used a penknife to inscribe the quaternion formula into a bridge on Dublin's Royal Canal.

Did you miss this talk? Here are some pictures ( taken by Glen).

(***) Find the slides here.

(§) Anne Isabel Gaudreau - *Unexpected repercussions of knot theory*

(§§) Francesca Corni - *The essence of mathematics lies in its freedom*

A little history about infinite, through Cantor's main results about the discover of different levels of infinite. During his researches Cantor found out an apparently harmless question that will have lead him to mental illness. Godel and other great mathematicians then tried to answer to the same charming problem, without success.

(°°) Janne Junnila and Jesse Jääsaari - *The Rising Sea*

In this talk we will go through the developments of algebraic geometry, starting from the analytic geometry of Descartes and Fermat and then moving in a rapid pace towards the today's state of art.

The main points are the increase of abstraction in mathematics during the 20th century and showing how the classical more concrete concepts were translated into this modern framework developed by Grothendieck et al.

In the end we will also shortly discuss an application to Weil conjectures, maybe the biggest result in number theory in the 20th century along with Fermat's last theorem.