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Mathematics Institute

Anna Parlak

I submitted my thesis, titled Veering triangulations and polynomial invariants of three-manifolds, in April 2021.

Information below is outdated. Please see my personal webpageLink opens in a new window.

Between 2017-2021 I was a PhD student exploring the geometry and topology of 3-manifolds under the supervision of Dr Saul SchleimerLink opens in a new window. I am mostly interested in problems related to veering triangulations.

Information about my teaching experience can be found here.

Recently I have posted the following preprints

Computation of the taut, the veering and the Teichmüller polynomialsLink opens in a new window

The taut polynomial and the Alexander polynomialLink opens in a new window

In the first one I give algorithms to compute the taut and veering polynomials of veering triangulations. These invariants were introduced by Landry, Minsky and TaylorLink opens in a new window. 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 hereLink opens in a new window.

In the second preprint I compare the taut polynomial of a veering triangulation with the Alexander polynomial of the underlying manifold. I also consider Dehn fillings of veering triangulations to relate the image of the taut polynomial under the Dehn filling and the Alexander polynomial of the Dehn-filled manifold.

I also explain how in the fibred case the close relationship between the Alexander polynomial and the Teichmüller polynomial is controlled by orientable fibred classes.

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).

This was mainly a generalisation/extension of the results previously obtained by Margalit & SchleimerLink opens in a new window, McCullough & RajeevsarathyLink opens in a new window and MondenLink opens in a new window in the orientable case.

The content of my master’s thesis is summarised in the following two papers, written jointly with my former supervisor Michał StukowLink opens in a new window.

  1. A. Parlak, M. Stukow. Roots of crosscap slides and crosscap transpositionsLink opens in a new window.
    Periodica Mathematica Hungarica (2017), Vol. 75, Issue 2, pp 413 – 419.
    arXiv:1601.06096 [math.GT]Link opens in a new window
  2. A. Parlak, M. Stukow. Roots of Dehn twists on nonorientable surfacesLink opens in a new window.
    Journal of Knot Theory and Its Ramifications, Vol. 28, No. 12, 1950077 (2019).
    arXiv:1701.00531 [math.GT]Link opens in a new window
My bachelor's thesis concerned simple knot invariants.
I also have a BSc degree in biotechnology.
Over the last few years I gave some talks:
  1. Veering triangulations and polynomial invariants of 3-manifolds 
    • Topology and Geometric Group Theory SeminarLink opens in a new window, Cornell University, April 2021.
    • Topology SeminarLink opens in a new window, University of Texas at Austin, March 2021.
    • Topology SeminarLink opens in a new window, University of Oxford, February 2021.
  2. Veering triangulations, the Teichmüller polynomial and the Alexander polynomial
    •     Algebra/Topology SeminarLink opens in a new window, University of Copenhagen, January 2021.
    • Junior Topology and Group Theory SeminarLink opens in a new window, University of Oxford, November 2020.
  3. The taut polynomial and the Alexander polynomial
    • UCR Topology SeminarLink opens in a new window, University of California – Riverside, November 2020.
    • Topology SeminarLink opens in a new window, Oklahoma State University, November 2020.
  4. Bristol Junior Geometry SeminarLink opens in a new window, University of Bristol, October 2019.
    Fibrations of 3-manifolds over the circle and their corresponding veering triangulations 
  5. Junior Geometry and Topology Seminar, University of WarwickLink opens in a new window; May 2019.
    Fibrations of a 3-manifold carried by the same veering triangulation
  6. Postgraduate Seminar, University of WarwickLink opens in a new window; February 2019.
    Pseudo-Anosov homeomorphisms of surfaces
  7. Junior Geometry and Topology Seminar, University of WarwickLink opens in a new window; January 2018.
    Roots of Dehn twists
  8. Young Topologists Meeting 2017Link opens in a new window; Stockholm, July 2017.
    Roots in the mapping class group of a nonorientable surface
  9. The 19th International Workshop for Young Mathematicians - Algebraic GeometryLink opens in a new window; Kraków, September 2016.
    Automorphisms Groups of Hyperelliptic Riemann Surfaces
  10. XIII Workshop for Mathematical Students' Associations; Hel, May 2016.
    Classification of finite subgroups of the mapping class group of a punctured sphere
  11. 18th Andrzej Jankowski Memorial Lecture Mini ConferenceLink opens in a new window; Gdańsk, May 2016.
    Roots of crosscap slides and crosscap transpositions
  12. The 18th International Workshop for Young Mathematicians - Algebraic and Differential TopologyLink opens in a new window; Kraków, September 2015.
    Some Remarks on Quandles and Their Applications

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