Welcome

Summer Haag

PhD Student in Mathematics · University of Colorado Boulder

About Me My Research
About Me

Summer Haag

I am currently a fourth year mathematics PhD student at the University of Colorado Boulder. I enjoy questions that are simple to explain but have complex solutions and explorations, especially focusing on the visual explorations. I'm interested in number theory, specifically Apollonian circle packings, Kleinian groups, continued fractions, and quadratic forms. These topics relate to many areas like hyperbolic geometry, arithmetic groups, and lattices. My advisor is Katherine E. Stange.

B.A. in Mathematics, University of Georgia, May 2022.

Institution
University of Colorado Boulder
Department
Department of Mathematics
Office
MATH 340
Email
summer.haag@colorado.edu
Email CV Resume
Summer Haag
Research

My Work

I am interested in algebraic number theory, focusing on illustrating mathematics, computations, and connecting topics to get different perspectives. My current research is focused on Apollonian circle packings.

2024

The local-global Conjecture for Apollonian circle packings is false

Summer Haag, Clyde Kertzer, James Rickards, Katherine E. Stange

In a primitive integral Apollonian circle packing, the curvatures that appear must fall into one of six or eight residue classes modulo 24. The local-global conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski closure), and are a result of quadratic and quartic reciprocity, reminiscent of a Brauer-Manin obstruction. Based on computational evidence, we formulate a new conjecture.

Annals of Mathematics ↗ arXiv ↗ Quanta Magazine ↗
2021

Computational study of non-unitary partitions

A.P. Akande, Tyler Genao, Summer Haag, Maurice Hendon, Neelima Pulagam, Robert Schnider, Andrew V. Sills

Following Cayley, MacMahon, and Sylvester, we define a non-unitary partition as an integer partition with no part equal to one, denoted ν(n). Building on prior work, we refine and prove earlier conjectures about the growth of ν(n). Key results include proving that p(n) ~ ν(n)√(n/ζ(2)) as n approaches infinity, and establishing Ramanujan-like congruences such as p(5n) ≡ ν(5n) (mod 5).

arXiv ↗
Teaching

Courses & Mentorship

I enjoy teaching in interactive ways with active learning and exploratory persepctives.

Fall

Precalculus Instructor

2025
Spring

Precalculus TA

2025
Fall

Calculus II Instructor

2024
Summer

Calculus II Instructor

2024
Spring

Calculus III TA

2024
Fall

Calculus II Instructor

2023
Spring

Calculus II TA

2023
Fall

Calculus I TA

2022
Presentations

Talks & Presentations

March 2026

The Parameter Space of Apollonian Circle Packings ↗

Aix Marseille University

Marseille, France
May 2025

The Local to Global conjecture for Apollonian Circle Packings

University of Houston

Houston, Texas, USA
June 2024

The Local to Global conjecture for Apollonian Circle Packings ↗

University of Connecticut Number Theory 2024

Storrs, Connecticut, USA