An Apollonian circle packing is an arrangement of tangent circles in the plane. Packings where all curvatures are integral has been a rich topic of study in the last 20 years. Instead of fixing a packing and considering the possible curvatures in the packing, we will invert the problem: fix a curvature, and consider all packings that have a circle of this curvature. One natural way to study this set is to ask: "What is the distribution of the minimal curvatures of these packings?" This problem leads to a staircase-like distribution, which we name the "Apollonian Staircase." By exploring a connection to binary quadratic forms and hyperbolic geometry, we will describe and prove some properties of the staircase.
The Apollonian Staircase
Nov. 01, 2022 1:25pm (Zoom)
Grad Algebra/Logic
Bernardo Rossi (University Linz, Austria)
X
Universal algebraic geometry, first introduced by B. Plotkin, extends some basic notions of classical algebraic geometry to universal algebra. For a clone on a set , a subset of is called algebraic if it is the solution set of a system of equations from the -ary part of . A basic fact in classical algebraic geometry is that the union of two algebraic sets is algebraic. This is no longer true in the present setting, and clones with this special property are called equationally additive. We prove that on a finite set with at least three elements there is a continuum number of equationally additive constantive clones. We characterize the Mal'cev algebras whose clone of polynomial functions is equationally additive in terms of properties of the binary term condition commutator, and equationally additive E-minimal algebras in terms of their TCT-type. Joint work with E. Aichinger and M. Behrisch.
On when the union of two algebraic sets is algebraic
Nov. 01, 2022 2:30pm (MATH 3…
Lie Theory
Manabu Hagiwara (Chiba University (Japan))
X
This talk introduces the topics of coding theory and its nontrivial relationship with root systems and perfect codes, with examples. The talk will focus on introductory aspects. Mainly two topics will be discussed: one is the connection between the combinatorial structures in representation theory and deletion perfect codes. The other is the connection between the sphere-packing problem and bit-flipping perfect codes.
The Apollonian Staircase