Fran Herr

Talks

I love giving talks and thinking about how to communicate mathematics! Here is a list of talks I have given in recent history.

• Crocheted Surfaces Bounded by Knots

  March 2026; (invited) Colorado College

  Abstract: Studying knots, one can often find themselves in a tangled mess. It is rather hard to differentiate two knots: even holding them and moving them around will often return little success. Mathematicians study knots using invariants— numbers, polynomials, or other algebraic objects. Many of these invariants are related to surfaces with a given knot as a boundary. Bounding a knot with a surface gives it structure, rigidity, makes it easier to hold… and gives rise to beautiful art pieces. With crochet, these complicated surfaces are made tangible, and we gain an appreciation for the beauty and complexity within the wide world of knots and links. In this talk, we will learn about the role of knots and links in modern low-dimensional topology. We will then learn how to crochet surfaces which bound knots and have a chance to make such a surface! (No crochet experience is necessary.)

• Leafwise Branched Coverings of Foliated 3-Manifolds

  May 2025; GTA: Philidelphia 2025; slides

  Abstract: A foliation of a 3-manifold is a decomposition into a disjoint collection of surfaces so that it locally looks like a product R^2 × R. A well-loved example is a 3-manifold fibering over the circle (a mapping torus) where the fibers form the surfaces in the foliation. Such a foliation is taut: there is a closed circle transversely intersecting all the fibers. In a 2021 paper, Calegari gives a combinatorial proof that a taut foliation is equivalent to the existence of a leafwise branched covering. We examine the relationship between the monodromy of a mapping torus and the necessary complexity of this leafwise branched cover. We are interested in mapping tori of both finite and infinite type surfaces.

• Modular Multiplication and Dancing Planets

  May 2025; U Chicago Math Club; slides

  Abstract: Modular multiplication tables are popular objects from recreational mathematics that display fascinating and varied designs. Take m points equally spaced around a circle, choose an integer multiplier a, and draw a chord from point p to a*p mod m. What pattern will appear as the envelope of the chords? We will investigate this question using dynamics, topology, coding, and crafting!

• One circle to rule them all...

  Feb 2025; Farb and Friends student seminar

  Abstract: ... One circle to find them, One circle to bring them all and in the darkness bind them.
  Given a two-dimensional foliation of a 3-manifold with hyperbolic leaves, the fundamental group π_1(M) has a natural action on a space of many circles. These are the circles at infinity of the leaves in the foliation of the universal cover of M. A universal circle binds these circles together and gives an action on a single circle instead. In this example-heavy talk, we will learn how to construct a universal circle for taut foliations and see how universal circles relate to modern research directions in 3-manifold topology.

• Realizing a Maximally Symmetric Surface

  Oct 2024; U Chicago Math Club

  Abstract: For a genus g surface with g at least 2, the Hurwitz Theorem places a strange upper bound on the number of possible symmetries: 84(g - 1). We will see why this theorem is true by learning about orbifolds and branched covering maps. Then we will study the Klein quartic, a maximally symmetric genus 3 surface. No prior knowledge of algebraic topology or hyperbolic geometry will be assumed, but we will see how these subjects play with each other to understand symmetry.