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On an early paper of Maryam Mirzakhani

William J. Martin

TL;DR

This memorial piece surveys Mirzakhani's early mathematical life and the seeds of her later breakthroughs. It highlights the 1996 BICA paper that introduced the planar graph $M$, proving a $3$-colorable graph can fail $4$-choosability, via a $42$-cycle $P$ and hub structure, yielding a $63$-vertex counterexample to Jensen's conjecture. The essay recounts her formative mentorship network (Akbari, Beheshti Zavareh, Mahmoodian) and early publications on $5$-cycle decompositions that catalyzed her trajectory. It ties these beginnings to her later work on hyperbolic geometry and moduli spaces, illustrating how combinatorial ideas informed her geometric and probabilistic methods and helping explain the significance of her Fields Medal.

Abstract

Maryam Mirzakhani, the first female (and first Iranian) Fields Medalist, passed away on July 14, 2017 at the age of 40. This short note remembers her 1996 article in the Bulletin of the Institute of Combinatorics and its Applications and her early years as a mathematician.

On an early paper of Maryam Mirzakhani

TL;DR

This memorial piece surveys Mirzakhani's early mathematical life and the seeds of her later breakthroughs. It highlights the 1996 BICA paper that introduced the planar graph , proving a -colorable graph can fail -choosability, via a -cycle and hub structure, yielding a -vertex counterexample to Jensen's conjecture. The essay recounts her formative mentorship network (Akbari, Beheshti Zavareh, Mahmoodian) and early publications on -cycle decompositions that catalyzed her trajectory. It ties these beginnings to her later work on hyperbolic geometry and moduli spaces, illustrating how combinatorial ideas informed her geometric and probabilistic methods and helping explain the significance of her Fields Medal.

Abstract

Maryam Mirzakhani, the first female (and first Iranian) Fields Medalist, passed away on July 14, 2017 at the age of 40. This short note remembers her 1996 article in the Bulletin of the Institute of Combinatorics and its Applications and her early years as a mathematician.

Paper Structure

This paper contains 3 sections, 1 equation, 3 figures.

Table of Contents

  1. We all got the news
  2. As for so many, it started with graph coloring
  3. A community of talented students and their mentors

Figures (3)

  • Figure 1: The Mirzakhani graph $M$ was discovered in 1996 and first appeared in the journal BICA as a proof that there exist planar 3-colorable graphs which are not 4-choosable.
  • Figure 2: Subgraphs of the Mirzakhani graph. All lists assigned to the vertices of the first graph are 4-element subsets of $\{1,2,3,4,5\}$; e.g., $L^1=\{2,3,4,5\}$. Mirzakhani concisely explained why any proper list-coloring of the first graph must use color 1 at some vertex on the outer face. This is seen by looking at each subgraph isomorphic to the second graph and assuming color 1 is unavailable.
  • Figure 3: Maryam Mirzakhani (front row, right) at the Mathematical Olympiad in Yazd in 1995. Roya Beheshti is also in the front row, at the left. Photo provided by Ebad Mahmoodian.