6th European Women in Mathematics Summer School
 Institut Mittag-Leffler, 23-27 June 2014

Apollonian Circle Packings

IML official website

Sponsored by the EWM, EMS and IML in partnership with the
with additional funding from the National Science Foundation and the Number Theory Foundation.



Aim and description

An ancient theorem of Apollonius of Perga, circa 200BC, proves that for any three mutually tangent circles or lines there are precisely two other circles or lines which are tangent to all three. Thus given four mutually tangent circles, one of them internally tangent to the other three, there is a unique way to inscribe a circle into each of the curvilinear triangles that appear between the circles; iterating this procedure indefinitely gives a construction called an Apollonian circle packing.

One can study an Apollonian circle packing (ACP) from many different angles. Such packings are certainly of interest in classical geometry. But ACP's encode fascinating information of an entirely different flavor: an infinite family of so-called integer ACP's encodes beautiful and mysterious number theoretic properties. Many of these properties can be observed or conjectured experimentally, and the statements of conjectures or theorems in this area can be pleasingly simple. But these properties turn out to be fantastically difficult to prove. ACP's are intimately related to questions on orbits of thin groups, and the summer school will adopt this modern point of view, which has led to recent breakthroughs by employing methods involving geometric group theory, equidistribution, expanders, and the affine sieve. Two wonderful references are the following survey articles written by the two lecturers: Counting problems in Apollonian packings by Elena Fuchs and Apollonian circle packings: dynamics and number theory by Hee Oh.

The aim of the European Women in Mathematics summer school is to provide a stimulating intellectual environment for female and male PhD students and postdocs from different countries and different mathematical disciplines to learn new mathematics (outside the scope of their own research) and to meet new colleagues. We hope that these contacts will help the forming and development of a network for PhD students and between PhD students and established mathematicians.


During the summer school, the featured lectures by Hee Oh and Elena Fuchs will take place in the mornings. In the afternoons, pairs of participants will give review lectures on background material relevant to the morning lectures, and then the participants will coalesce into small groups to work on problems. Three months in advance of the summer school the organizers will group the participants into pairs and assign a topic to each pair; each pair will then write an expository manuscript on that topic and prepare a chalkboard presentation to give at the summer school. A collection of all the expository manuscripts will be given to each participant upon arrival at the summer school. The atmosphere of the historic Institut Mittag-Leffler will facilitate the focused scientific program, and will also provide ample opportunities for casual discussions about subjects such as work/life balance and math careers.




Apollonian circle packings: dynamics and number theory (H. Oh) pdf
Counting problems in Apollonian packings (E. Fuchs) pdf
List of references for the lecture series on arithmetic of ACPs (E. Fuchs) pdf


The Institut Mittag-Leffler will provide accommodation, breakfasts, lunches and a conference dinner. There is some funding for travel.


  • Claire Burrin
  • Pritha Chakraborty
  • Sneha Chaubey
  • Giacomo Cherubini
  • Anne-Maria Ernvall-Hytonen
  • Amy Feaver
  • Yongqi Feng
  • Anna Haensch
  • Min Lee
  • Poj Lertchoosakul
  • Benny Loeffel
  • Beth Malmskog
  • Jana Medkova
  • Wenyu Pan
  • Anke Pohl
  • Damaris Schindler
  • Kate Stange
  • Anitha Thilaisundaram
  • Xin Zhang



  • Alina Bucur (UCSD)
  • Pirita Paajanen (The Wellcome Trust Sanger Institute)
  • Lillian Pierce (Duke University and Hausdorff Center for Mathematics)

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