# Balázs Strenner

Hale Visiting Assistant Professor

Georgia Institute of Technology

School of Mathematics

Office: Skiles 019

# Research

## Papers

- Fast Nielsen-Thurston Classification (with D. Margalit and Ö. Yurttas), in preparation. (animated ppt slides) (pdf slides) (poster)
- Fibrations of 3-manifolds and asymptotic translation length in the arc complex (arXiv) (pdf slides)
- Minimal Penner dilatations on nonorientable surfaces (with L. Liechti), to appear in Journal of Topology and Analysis (arXiv)
- Minimal pseudo-Anosov stretch factors on nonorientable surfaces (with L. Liechti), to appear in AGT (arXiv)
- The Arnoux-Yoccoz mapping classes via Penner's construction (with L. Liechti) (arXiv)
- Algebraic degrees of pseudo-Anosov stretch factors,
*GAFA*27 (2017), no. 6, 1497-1539. (arXiv) - Galois conjugates of pseudo-Anosov stretch factors are dense in the
complex plane,
*IMRN*(2017), rnx221. (arXiv) - Lifts of pseudo-Anosov homeomorphisms of nonorientable surfaces have
vanishing SAF invariant,
*Math. Research Letters*25 (2018), no. 2, 677-685. (arXiv). - Pseudo-Anosov mapping classes not arising from Penner's
construction (with H. Shin),
*Geometry & Topology*19 (2015), no. 6, 3645-3656. (arXiv) -
How large dimension guarantees a given angle? (with V.
Harangi, T. Keleti, G. Kiss, P. Maga, A.
Máthé, P. Mattila),
*Monatshefte für Mathematik*, 2013, Volume 171, Issue 2, pp 169-187. (arXiv) - n-pont halmazok a síkban (in English: "n-point sets in the plane"), Master's thesis, 2010. (PDF)

## Talks

- Colloquium talk on stretch factors. pdf slides
- Number-theoretic aspects of surface homeomorphisms. pdf notes

## Undergraduate mentoring

### Change of Dehn-Thurston train tracks under elementary moves of pants decompositions

REU at Georgia Tech with Ian Katz (Georgia Tech, first year graduate), Yandi Wu (Berkeley, undergraduate) and Yihan Zhou (Georgia Tech, undergraduate). (2017 Summer)

### Tightening curves on surfaces

Undergraduate project with Georgia Tech undergraduates with Shreyas Casturi, Vignesh Raman, Kyle Xiao (freshmen) and Jonathan Chen (sophomore). (2017 Fall)

The project was about curves on surfaces. One can encode a curve on a surface as a path connecting various points on a surface. Some paths are clearly not efficient: if we go from A to B, then B to C, we could have simply gone from A to C. On higher genus surfaces, paths can be inefficient in more complicated ways, but one can make a list of what the inefficient paths can look like and how they can be made more efficient. The students were designing a way to encode paths as a sequence of letters and numbers, and wrote a Python-implementation of the curve tightening process, which looks for inefficient subpaths of a path and replaces them with their efficient counterparts. This implementation is a contribution to Macaw.

### Transition matrices of real, periodic, quadratic polynomials

Summer undergraduate research project with Logan White (Chicago), Jacob Schulkin (Michigan) and Agniva Roy (Georgia Tech). (2018 Summer)

A post-critically finite polynomial, as a branched covering from the complex plane to itself, acts on its Hubbard tree in a way that vertices map to vertices and edges go to edge paths. Corresponding to this graph map is a transition matrix. The largest eigenvalue of this transition matrix is a notion of stretch factor for the polynomial. The project was centered around the problem of understanding what numbers arise as stretch factors of polynomials this way.

# Teaching

Notes for my Geometric structures of surfaces course taught in 2019 Spring at Georgia Tech. Most of the material is on Mirzakhani's thesis. There are also a few pages on billiard in polygons. Poetry by students of the class.

## Linear algebra demos

# Programs

## Donut (formerly Macaw)

Program for fast computations in mapping class groups, based on joint work with Dan Margalit and Öyku Yurttas. One of its main features is that it works for closed surfaces in addition to punctured ones. When completed, it will be able to determine the Nielsen-Thurston type (finite order, reducible, pseudo-Anosov) of mapping classes very efficiently, in quadratic time in the word length. It will also compute other pieces of data such as stretch factors, invariant foliations, reducing curves, invariant train tracks, veering triangulations and translation surfaces. Some of the currently implemented features can be found on the project website. Macaw was written Python. Later the whole codebase has been rewritten in Julia, resulting in Donut. Thanks to Julia, Donut is often 10-20x faster than Macaw on the same tasks. Donut's code is also significantly better organized than Macaw and has some extra functionality. Unfortunately both projects are unfinished and it is unlikely that I will have time to finish them. If anyone is interested in working on Donut, I am happy to help.

## Escher

Python script for creating a tiling of the hyperbolic plane with a specified image.

## probability-models

Program for creating pictures of simple probability experiments. I wrote this while teaching an introductory probability class for math teachers.