Next Seminar:  July 3, 2025 at ISTA

Location: Mondi 2 (Main building)

Time:  July 3, 13.00 - 18.00

Schedule: 

13:00-14:00:  Martin Widmer (TU Graz)

Title: Universal quadratic forms over infinite extensions  

Abstract: By Lagrange's Theorem the sum of four squares is a universal quadratic form, i.e., represents all (totally) positive integers. This is almost never  the case when we replace the rational integers by the integers of a totally real number field. However, there is always some universal quadratic form. 

The situation is fundamentally different for infinite totally real extensions. Daans, Kala and Hang Man have shown that in this case the Northcott property is an obstruction to the existence of a universal quadratic form, and  asked whether it is the only obstruction. 

We show that most (in a suitable topological sense) totally real extensions do not have a universal quadratic form or the Northcott  property. The main tool is a new obstruction to the existence of a universal quadratic form.  I will discuss the result and some ideas of the proof.

This is joint work with Nicolas Daans, Vitezslav Kala, Siu Hang Man, and Pavlo Yatsyna.

14:00-14:30: Tea break

14:30-15:30: Lena Wurzinger (ISTA)

Title: Towards lattices of fixed determinant

Abstract: I will report on ongoing work regarding counting (primitive) lattices in Zn of fixed determinant. In particular, I will derive asymptotics for the number of rank-2 lattices in  Zn of fixed determinant and n > 8, focusing on the case of 4 dividing n and the determinant being squarefree. 

 15:30-16:30: Break

16:00-17:00: Chris Lutsko (University of Houston)

Title: Apollonian Circle Packings  

Abstract: The Apollonian circle packing is a well-known configuration of circles generated by a discrete subgroup, whose radii satisfy a quadratic form calculated by Descartes. Thus, the Apollonian counting problem, where one counts circles in a given configuration, is equivalent to counting orbit points of a group living on a variety. In this talk I'll explain how to use spectral information to get an effective asymptotic estimate for the count. This work is joint with Alex Kontorovich and I will show a result joint with Dubi Kelmer and Alex Kontorovich. 

17:30: Dinner at Redlinger Hütte