I am currently a fourth year PhD student exploring the geometry and topology of 3-manifolds under the supervision of Dr Saul Schleimer. I am mostly interested in problems related to veering triangulations.
You can get in contact with me via .
Information about my teaching experience can be found here.
Recently I have posted the following preprint
There I give algorithms to compute the taut and veering polynomials of veering triangulations. These invariants were introduced by Landry, Minsky and Taylor. Using their work I also give algorithm to compute the Teichmüller polynomial of any fibred face of the Thurston norm ball of any hyperbolic 3-manifold.
All algorithms have been implemented by me, Saul Schleimer and Henry Segerman. The source codes are available here.
Now I’m finishing a preprint about the relation between the taut polynomial of a veering triangulation and the Alexander polynomial of the underlying manifold. The results extend McMullen's Theorem 7.1 relating the Teichmüller polynomial with the Alexander polynomial.
A BIT OF HISTORY
Before coming to Warwick, I obtained BSc and MSc degrees in mathematics from the University of Gdańsk, Poland.
My mathematical interests were centred around mapping class groups.
The aim of my master’s thesis was to investigate the existence and possible degrees of roots of Dehn twists, crosscap slides and crosscap transpositions in the mapping class groups of closed nonorientable surfaces. (It is known that a finite number of Dehn twists and a single crosscap slide or a crosscap transposition generate the whole group).
The content of my master’s thesis is summarised in the following two papers, written jointly with my former supervisor Michał Stukow.
- A. Parlak, M. Stukow. Roots of crosscap slides and crosscap transpositions.
Periodica Mathematica Hungarica (2017), Vol. 75, Issue 2, pp 413 – 419.
- A. Parlak, M. Stukow. Roots of Dehn twists on nonorientable surfaces.
Journal of Knot Theory and Its Ramifications, Vol. 28, No. 12, 1950077 (2019).
Over the last few years I gave some talks:
- Veering triangulations, the Teichmüller polynomial and the Alexander polynomial
• Algebra/Topology Seminar, University of Copenhagen, January 2021.
• Junior Topology and Group Theory Seminar, University of Oxford, November 2020.
- The taut polynomial and the Alexander polynomial
• UCR Topology Seminar, University of California – Riverside, November 2020.
• Topology Seminar, Oklahoma State University, November 2020.
- Bristol Junior Geometry Seminar, University of Bristol, October 2019.
Fibrations of 3-manifolds over the circle and their corresponding veering triangulations
- Junior Geometry and Topology Seminar, University of Warwick; May 2019.
Fibrations of a 3-manifold carried by the same veering triangulation
- Postgraduate Seminar, University of Warwick; February 2019.
Pseudo-Anosov homeomorphisms of surfaces
- Junior Geometry and Topology Seminar, University of Warwick; January 2018.
Roots of Dehn twists
- Young Topologists Meeting 2017; Stockholm, July 2017.
Roots in the mapping class group of a nonorientable surface
- The 19th International Workshop for Young Mathematicians - Algebraic Geometry; Kraków, September 2016.
Automorphisms Groups of Hyperelliptic Riemann Surfaces
- XIII Workshop for Mathematical Students' Associations; Hel, May 2016.
Classification of finite subgroups of the mapping class group of a punctured sphere
- 18th Andrzej Jankowski Memorial Lecture Mini Conference; Gdańsk, May 2016.
Roots of crosscap slides and crosscap transpositions
- The 18th International Workshop for Young Mathematicians - Algebraic and Differential Topology; Kraków, September 2015.
Some Remarks on Quandles and Their Applications