A CIMPA school on
Lattices, Heights and Diophantine Approximation
Urgench State University, Urgench, Uzbekistan
June 24th - July 5th 2024

Khiva, hometown of Al-Khwārizmī a few kilometers away from Urgench

The school aims at introducing the students to a few lines of research in number theory revolving around the concepts of lattices, heights and diophantine approximation. Part of the school is devoted to presenting the recent proof by Maryna Viazovska that the densest sphere packing in dimension 8 is the one given by the E_8 lattice. In 2022 she was awarded the Fields medal. To this end we introduce the students to lattices, sphere packings and modular forms. We will prove the Cohn-Elkies bound and finally we will study Viazovska's construction of the function which optimizes such bound and proves that the E_8 lattice is the densest packing in dimension 8. In the other part of the program we will introduce the machinery of heights, which are standard tools of Diophantine geometry used to measure arithmetic complexity of objects. We will then demonstrate the use of height functions in Diophantine approximation and Diophantine geometry discussing results and conjectures such as the Mordell-Weil theorem, Faltings' theorem, Siegel's lemma, Cassels' theorem, Lehmer's conjecture, and many others. On the Diophantine approximation side, we will discuss the central themes of equidistribution and approximation of reals by algebraic numbers. We expect the participants to gain familiarity with these central and vibrant areas of mathematics that have been at the forefront of mathematical research for over a hundred years. Our program is specifically designed in a way that assumes a rather modest background, but with a concentrated and focused approach takes the audience to some of the modern-day research directions.

  • Elisa Lorenzo Garcia, external coordinator (Université de Neuchâtel & Université de Rennes 1)
  • Zafar Ibragimov, local coordinator (Urgench State University)

    Scientific Committee
  • Lenny Fukshansky (Claremont McKenna College)
  • Elisa Lorenzo Garcia (Université de Neuchâtel & Université de Rennes 1)
  • Kate Petersen (University of Minnesota Duluth)
  • Valerio Talamanca (Università Roma Tre)

    Organizing Committee
  • Bakhrom Abdullaev (Urgench State University)
  • Alimardon Atamuratov (V.I.Romanovsky Institute of mathematics)
  • Aygul Babajanova (V.I.Romanovsky Institute of mathematics)
  • Nargiza Boboyorova (Urgench State University)
  • Zafar Ibragimov, (Urgench State University)
  • Jumanazar Khujamov (Urgench State University)
  • Gayrat Urazboev (Urgench State University)
  • Mokhira Vaisova (Urgench State University)