REU Participant Details

This week, I had sadly not joined the REU program yet, because I was studying abroad in Switzerland and my semester hadn't ended yet. However, I did have a chance to brush up on my hyperbolic geometry knowledge, which I will need for the REU.

This week, I joined the REU program and met my mentor, Professor Feng Luo. We discussed the project and I learned about circle packing on surfaces. I began to understand the background of the project and gave a presentation on the topic for intro presentation day!

I also started reading some notes written by Professor Luo about circle packing which culminated in the proof of the Köbe-Andreev-Thurston theorem. Here is the statement of the theorem:

Theorem 1.1 (Köbe-Andreev-Thurston) Let (S, T) be a simplicially-triangulation of 2-sphere. Then there exists a circle packing P on S², unique up to Mobius transformations, whose nerve is isomorphic to the 1-skeleton of T.

This week, I worked to understand and prove the Discrete Schwartz Lemma which is a key result in theory of circle packings. The lemma is a extension of the classical Schwartz lemma from complex analysis to the discrete setting of circle packings. The statement of the lemma is as follows:

Lemma 1.1 (Discrete Schwartz Lemma) Let (S, T, Φ) be a closed connected simplicially-triangulated surface, where Φ : E(T) → [0, π/2]. Let V(T) = V₁ ⊔ V₂ such that V₁ ≠ ∅ and in the case Euclidean background, V₂ ≠ ∅. Suppose R and r are two circle packing radius assignments on (S, T) with either Euclidean or hyperbolic background such that R|_V2 ≥ r|_V2 and their curvatures k_R and k_r satisfy k_R|_V1 ≥ k_R|_V1. Then R ≥ r and for any two vertices v₁ and v₂:

d_R(v₁, v₂) ≥ d_r(v₁, v₂)

Furthermore, I worked to learn more about circle packings through reading Stephanson's book on circle packings.

This week, I explored the proof of Discrete Schwartz Lemma, specifically its use of Perron's Method. Where to show R ≥ r, we show they both lie in a set X = { x ∈ ℝ^V_>0 | x|_V2 ≥ r|_V2, k_x|_V1 ≥ k_r|_V1 }, which is closed under minimums and is convex. We then show that r is the infimum of X thus R ≥ r. I wanted to see if the "opposite set" would still work to prove the theorem, so I defined Y = { x ∈ ℝ^V_>0 | x|_V2 ≤ R|_V2, k_x|_V1 ≤ k_R|_V1 } and showed that it is closed under maximums and convex. But an issue pops up where there may exist a sequence (x_n)_n∈ℕ ∈ X such that ∃v ∈ V₁ such that:

lim_n→∞ x_n(v) = ∞

, and the argument would not work. The reason I am doing this is because for the cases where circle packings are no longer tangent, the set of possible radii assignments may no longer be ℝ^V_>0 and won't be closed under minimums. But the set may still be closed under maximums, so I wanted to see if the "opposite set" would work.

Additionally, I am also working through Stephanson's book on circle packings, and proved the theorem that for any combinatorial closed disk, there exists a unique (up to Mobius transformations) maximal hyperbolic circle packing on 𝔻 with the same combinatorial structure.

I began the week starting to explore the generalizations of Discrete Schwartz Lemma, particularly in the context of circle packings with inversive distances. I was specifically experimenting with a triangle with a single vertex in the center which are connected to all three vertices of the triangle, where the boundary edges have a inversive distance of 1 (ie the circles corresponding to the boundary vertices are each mutually tangent to each other). I tried to understand the function F : ℝ² × (0, ∞) → [−1, ∞)³ where you give the coordinate of the center vertex and the radius of the center circle, and it outputs the inversive distances of the edge connecting the center vertex to each of the three boundary vertices. I showed that the det(∂F) ≠ 0 for all points in the open disk contained by the circle intersecting the 3 vertices of the triangle, and thus the function is a local diffeomorphism. I then tried to show that the function is a global diffeomorphism, but I was unable to do so. I also tried to see if the function was injective, but I was unable to show that either. I will continue to explore this next week.

Getting frustrated with this approach, I tried to see if I can approach the problem computationally. I wrote a program to find counter examples to the Discrete Schwartz Lemma in the case of inversive distances where the inversive distances greater than 1. I wrote a program to generate random triangulations of a equilateral triangle, and then assign random inversive distances to the edges. I then checked if the Discrete Schwartz Lemma holds for the triangulation where V₁ are the interior vertices and V₂ are the boundary vertices, and if it does not, I output the triangulation. I was able to find a counter example where the lemma does not hold, but I am still trying to understand why it does not hold.

I presented the counter example I found last week to Professor Luo, and he suggested that I try to find a counter example where the inversive distances are less than 1. I tried to do this by generating random triangulations of a equilateral triangle, and then assigning random inversive distances to the edges. However, I was unable to find a counter example where the inversive distances are less than 1. I will continue to explore this next week.

I was also exploring the theorem on Rigidity of Circle Packings, which states that if two circle packings have the same combinatorial structure and the same inversive distances, then they are equivalent up to a scaling for euclidean geometry and are completely equal in hyperbolic geometry. The theorem is proven for the case where the circle packings have inversive distances are greater than or equal to 0, but I am trying to see if the theorem holds for the case where the inversive distances greater than −1 (corresponding to the case where at least 1 circle intersect at angles greater than 90 degrees and less than 180 degrees).

I continued to explore Rigidity of Circle Packings for the case where at least 1 pair of circles intersect at angles greater than 90 degrees and less than 180 degrees. The issue with this case is that the inversive distances can be negative, thus the triangle inequality may not hold for all possible radii. I tried to pivot to find counter examples for this case using computational methods, but the issue is that I have assert equality between the discrete curvatures of the two circle packings, which is hard to check computationally. I tried to find a way to check if two circle packings have the same discrete curvatures, but I was unable to do so, because the equations for the discrete curvatures are not easy to compute.

This week was presentation week so most of the time was spent preparing for the presentation. But I was able to make some progress on the Discrete Schwarz Lemma for Circle Packings. I was able to prove that the discrete Schwarz lemma holds for circle packings with more general intersection angles.

Let (S, T, Φ) be a weighted simplicially-triangulated surface. Φ : E → [0, π) is a weight function on the edges of the triangulation that describes the intersection angles of the circles in the circle packing.

Figure 1: Intersection Angle in Circle Packings

For simplicity of notation, forall triangles △ vᵢvⱼvₖ let Φₐ = Φ(vᵦv꜀) where {a, b, c} = {i, j, k}. Then γₐᵇ'꜀ = cos(Φₐ) + cos(Φᵦ) cos(Φ꜀).

Theorem 1.1 Discrete Schwarz Lemma for Circle Packings

{ Let (S, T, Φ) be a closed connected weighted simplicially-triangulated surface where for any topological triangle △ vᵢvⱼvₖ ∈ F we have γᵢʲ'ᵏ ≥ 0, γⱼⁱ'ᵏ ≥ 0, γₖⁱ'ʲ ≥ 0. Let V = V₁ ⊔ V₂ such that V₁ ≠ ∅ and in the case Euclidean background, V₂ ≠ ∅. Suppose R and r are two circle packing radius assignments on (S, T) with either euclidean or hyperbolic background such that R|ᵥ₂ ≥ r|ᵥ₂ and their curvatures k_R and k_r satisfy K_R|ᵥ₁ ≥ K_r|ᵥ₁. Then R ≥ r. }

This theorem encompasses the known case for circle packings with intersection angles less than or equal to 90 degrees, as well as a subset of the case of intersection angles greater than 90 degrees. The proof is similar to the proof of the discrete Schwarz lemma for circle packings with intersection angles less than or equal to 90 degrees, but it requires a more careful analysis of the intersection angles and the curvature conditions.

Last week of the program! I can't believe how fast it has gone by. I spent most of the week writing the final report. I also spent some time reading about the Discrete Uniformization Theorem after a talk with Dr. Zhenghao Rao.