Alex Elzenaar
I am a PhD student at the MaxPlanckInstitut für Mathematik in den Naturwissenschaften in Leipzig under the supervision of Prof. Bernd Sturmfels. I usually work in between the areas of geometric group theory, geometric topology, and metric geometry. I am particularly interested in modern classical geometry (for instance as studied by Coxeter and Thurston) and relationships with other branches of mathematics (knot theory, number theory, algebraic geometry, ...).
I am also interested in visualisation of mathematical objects (including mathematical art) and the study of mathematical writing.
A polished
black sedan was a good subject, especially if parked at the intersection of a treebordered street and
one of those heavyish spring skies whose bloated grey clouds and amoebashaped blotches of blue
seem more physical than the reticent elms and evasive pavement. Now break the body of the car into
separate curves and panels; then put it together in terms of reflections. These will be different for each
part: the top will display inverted trees with blurred branches growing like roots into a washily
photographed sky, with a whalelike building swimming byan architectural afterthought; one side of
the hood will be coated with a band of rich celestial cobalt; a most delicate pattern of black twigs will
be mirrored in the outside surface of the rear window; and a remarkable desert view, a distended
horizon, with a remote house here and a lone tree there, will stretch along the bumper. (Nabokov, Pnin, Chapter 4, Section 5.)
Recently my work has looked at structures in algebraic geometry which capture 'knottedness' of 3manifolds. In the process of doing this I produced this nice picture. Joint work is forthcoming in which this will be explained (and also some more interesting stuff will be done).
Email address: elzenaar@mis.mpg.de
Here is my Curriculum Vitae.
Many of the things which I am interested in are very easy to start thinking about, even with no technical background. Some videos which I like include: Coxeter discusses the math behind Escher's circle limit (Harold Coxeter) Not Knot (The Geometry Center) Python for mathematical visualization (David Dumas) Mathematics as Metaphor (Curtis McMullen) Doodling in Math: Spirals, Fibonacci, and Being a Plant (Vi Hart) How to write mathematics badly (JeanPierre Serre) NonEuclidean virtual reality (Vi Hart, Andrea Hawksley, Sabetta Matsumoto, and Henry Segerman) Every StrictlyConvex Deltahedron (Michael Stevens) The Suggestive Power of Pictures (Caroline Series) Knots Don't Cancel (John Conway) How to make mathematical candy (JeanLuc Thiffeault) Maths Your Own Kaleidoscopic Shapes! (Matt Parker)
Some other geometric things which interest me: some sculptures around Pōneke/Wellington exhibition of the work of Hilma af Klint at the City Gallery in 20212022 Robin White: Making of That Vase Energy Work: Kathy Barry/Sarah SmutsKennedy (at Vic's Te Pātaka Toi Adam Art Gallery) "Rita" by Quentin Angus Patrick Pound at City Gallery Wellington Len Lye: A Colour Box, Colour Cry, Kaleidoscope
Here is a list of opinions about books. My books are catalogued on LibraryThing. I will update the
bibliography that was here when I have time.
Publications and preprints
 Proceedings article: Alex Elzenaar, Gaven Martin, and Jeroen Schillewaert. "Concrete one complex dimensional moduli spaces of hyperbolic manifolds and orbifolds". In: 202122 MATRIX annals. Ed. by David R. Wood, Jan de Gier, Cheryl E. Prager, and Terrence Tao. MATRIX Book Series 5. Springer, to appear. arXiv:2204.11422 [math.GT]. A version with minor corrections: PDF.
 Preprint: Alex Elzenaar, Gaven Martin, and Jeroen Schillewaert. "The combinatorics of the Farey words and their traces." April 2022. arXiv:2204.08076 [math.GT].
A version with minor corrections: PDF.
 Preprint: Alex Elzenaar, Gaven Martin, and Jeroen Schillewaert. "Approximations of the Riley slice." November 2021. arXiv:2111.03230 [math.GT]. A version with minor corrections: PDF. Accepted by Expo. Math., to appear.
Recent talks I have given
Here is a list of talks I have given this year (2022). Older slides can be found below in their respective sections.
 17 to 20 January 2023: Apocrypha and ephemera on the boundaries of moduli space now in minicourse form (Uni. of Auckland). See below!
 7 December 2022: Apocrypha and ephemera on the boundaries of moduli space at MPI, Lecture notes (these are full of errors, use at your own risk).
 10 October 2022: Uniformisation, equivariance, and vanishing: three kinds of functions hanging around your Riemann surface, at MPI, Lecture notes. Subsequent to
this, Remarks on computing the algebraic curve corresponding to a Riemann surface.
 21 September 2022: What is a Kleinian group?, a talk aimed at undergraduates and beginning postgraduate students in the Australian Postgraduate Algebra Colloquium, slides, recording.
 3 August 2022: Reproducibility in Computer Algebra (MPI MIS), handouts for practical activity (event coorganised with Christiane Görgen and Lars Kastner)
 15 July 2022: On the MathRepo page "Farey Polynomials", in the MathRepo: Data for and from your Research event (MPI MIS), slides
 24 May 2022: Projective varieties over \(\mathbb{C}\), in the Lorentzian polynomials day which I organised, slides
 4 May 2022: Pictures of hyperbolic spaces, in the Discrete Mathematics and Geometry Seminar (TU Berlin), slides
 27 April 2022: Strange circles: The Riley slice of quasiFuchsian space, in the Seminar on Nonlinear Algebra (MPI MIS), slides
 17 March 2022: Strange circles: The Riley slice of quasiFuchsian space, in Pedram Hekmati's seminar on moduli spaces (Uni. of Auckland), slides.
Apocrypha and ephemera on the boundaries of moduli space
Henry Moore: Bronze Form (1988). In situ, Wellington Botanic Garden ki Paekākā.
I will teach a minicourse at UoA from
16 1720 Jan 2023. The goal (which will not be achieved, but we will get some way towards it) is to explain the rough structure of the following equivalent
objects: (i) the moduli space of Schottky groups; and (ii) the space of hyperbolic 3manifolds with visual boundary a compact Riemann surface (handlebody). In the process we will learn some of
the Birman theory of braid groups, some knot theory, some of the quasiconformal deformation theory of Kleinian groups, and a lot of geometric topology. The only prerequisite is comfort thinking
about quotients in metric spaces and some algebraic topology, but we will go fast and so you should expect to lose track of some details fairly quickly. To try to fix this I will also be handing
out problem sheets (which will include some "basic" problems, some research level problems, and some computational problems). Each lecture should also be fairly selfcontained. People who attended
the graduate seminar/course I taught in Sem 1 of 2021 will find things easier but it is not necessary at all for you to have followed that. The talk in
Nelson
by Benson Farb is also very good preparation.
I anticipate 5 lectures, at 2PM every day in 303.257 (this schedule is only guaranteed for the first talk, I think some of the more enthusiastic people will want more time to discuss the ideas and
so we will wing it as we go). The lectures will be:
 A crash course in Kleinian groups problems
 Sociology problems
 Bgroups and other degeneracies problems
 Braids, links, and mapping class groups
 ???
I have some
rough notes but they are very incomplete and full of silly mistakes, and I will hand out more polished versions with
different mistakes next week.
Background reading
The length of this list is an indication of width not depth.
 Alex Elzenaar, Gaven Martin, and Jeroen Schillewaert, Concrete one complex dimensional moduli spaces of hyperbolic manifolds and orbifolds
 Bernard Maskit, Kleinian groups (Springer)
 Jessica Purcell, Hyperbolic knot theory (AMS)
 William Thurston, Threedimensional geometry and topology (Princeton) and The geometry and topology of threemanifolds
 Michael Kapovich, Hyperbolic Manifolds and Discrete Groups (Birkhaüser)
 Benson Farb, Dan Margalit, A Primer on Mapping Class Groups (Princeton)
 Joan Birman, Braids, Links, and Mapping Class Groups (Princeton)
 David Mumford, Caroline Series, and David Wright, Indra's pearls (Cambridge)
MSc. thesis: Deformation spaces of Kleinian groups
My
Master of Science thesis was completed in 202122 in the Department of Mathematics at the
University of Auckland, under the supervision
of Dist. Prof. Gaven Martin (NZ Institute of Advanced Study, Massey University) and Dr. Jeroen Schillewaert.
Abstract. It has been known since at least the time of Poincaré that isometries of 3dimensional hyperbolic space \( \mathbb{H}^3 \) can be represented by \( 2 \times 2\) matrices over the complex numbers: the
matrices represent fractional linear transformations on the sphere at infinity, and hyperbolic space is rigid enough that every hyperbolic motion is determined by such an action at infinity.
A discrete subgroup of \( \mathrm{PSL}(2,\mathbb{C}) \) is called a Kleinian group; the quotient of \( \mathbb{H}^3 \) by
the action of such a group is an orbifold, and its boundary at infinity is a (possibly empty or disconnected) Riemann surface.
The Riley slice is the moduli space of Kleinian groups generated by a pair of parabolic elements which are free on those generators and whose corresponding surface is supported on a
4punctured sphere; Robert Riley introduced this object in the 1970s while studying 2bridge knot groups. The Riley slice is naturally embedded in \( \mathbb{C} \) and so is particularly amenable to
study since one can draw pictures of it. Linda Keen and Caroline Series studied this embedding in the early 1990s via a family of polynomials
which gave a foliation (local product decomposition) of the slice. We will discuss the KeenSeries theory and extend it to allow torsion elements as generators. We also discuss some new results
of a combinatorial flavour and some applications. We aim for the exposition to be accessible to beginning graduate students, despite the high bar for entry to this subject in terms of prerequisite material.
If this sounds interesting, you might want to start with our expository article
Concrete one complex dimensional moduli spaces of hyperbolic manifolds and orbifolds
which we wrote to give historical and mathematical background: we aimed for this to be accessible to beginning graduate students with only a little complex analysis and topology knowledge.
My
thesis gives more detailed background to the study of quasiconformal deformation spaces and the relevant knot theory and lowdimensional topology, with references to graduate
textbooks and the original literature, before discussing our new results. For the new results please cite our papers which are based in part on the thesis rather than the thesis itself, as this was joint work:
 Slides for talks and other interesting links 
Here is a picture of the exterior of the Riley slice:

 Talk on computation and visualisation in hyperbolic geometry, in the Discrete Mathematics and Geometry Seminar at TU Berlin, 4 May 2022: presentation slides.
 The Riley slice as a moduli space, in the MPI MiS Seminar on Nonlinear Algebra,
27 April 2022: presentation slides.
 Talk in Pedram Hekmati's seminar on moduli spaces, 17 March 2022: presentation slides.
 Waiheke workshop, 6 December 2021: presentation slides on the Farey polynomials.
 Contributed talk for the MATRIX workshop on groups and geometries
discussing recent work done jointly with Gaven Martin and Jeroen Schillewaert, 2 December 2021: presentation slides.
 Halfyear Postgraduate Project Presentations talk, July 2021: Presentation slides.
 Talk for the UoA Dept. of Mathematics Student Research Conference, June 2021: Extended abstract; presentation slides.
 Very rough lecture notes for the informal graduate seminar I taught on Kleinian groups in Semester 1, 2021: PDF 1,
PDF 2, further reading list, and postmortem.
 Riley slice computational package (GitHub).
 Some animations, and some limit sets.


Lorentzian polynomials and algebraic geometry on matroids
If \( X \) is a sufficiently nice variety, the Chow group \( A^*(X) \) provides a homology theory on \( X \); in fact, it admits a ring structure coming from the intersection product. It turns out
that such a theory can be made to work on more general spaces, for example one can define a Chow ring for matroids; then the various Hodgetype results (Poincaré duality, the hard Lefschetz theorem,
and the HodgeRiemann relations) carry over. Various nice polynomials can be defined with respect to this generalised Hodge theory and the associated cones of 'ample divisors' (which turn out
to be submodular functions); these are the
Lorentzian polynomials of Brändén and Huh.
A Day of Geometry and Lorentzian Polynomials
At the end of May 2022 there was a
seminar at the Institut MittagLeffler on the work of Branden, Huh, Katz, and various
other people on Lorentzian polynomials and the geometry of matroids; before this event on
Tuesday 24 May, I organised a very informal Zoom workshop on some of the geometric background
material.
Abstract.
Even if you do not know what Lorentzian polynomials are, you may have heard of Minkowski volume polynomials, the polynomials of the form \( \mathrm{vol}(x_1 K_1 + \cdots + x_n K_n) \)
where \( K_1,\ldots,K_n \) are convex bodies—and these are somehow the "canonical examples" of Lorentzian polynomials. The goal of the workshop is to give many different examples of Lorentzian
polynomials arising in geometry. The talks will be very informal, nontechnical, and have many pictures.
The final schedule was as follows (all times are CET). Many of the speakers have kindly allowed me to share their slides and/or lecture notes.
 9.30am—Matroids and chromatic polynomials (Tobias Boege, MPI MiS): Slides
 10:15am—Varieties over C and embeddings into projective space via elliptic curves (Lukas Zobernig, The University of Auckland): Slides
 11:00am—Hyperbolic polynomials (Hisha Nguyen, V.N. Karazin Kharkiv National University)
 Break (hopefully the morning talks are finished by 11:45, or 12 at the latest if we run over time).
 1:30pm—Convex geometry & mixed volumes (Mara Belotti, TU Berlin): Slides
 2:15pm—Projective varieties over \( \mathbb{C} \) (Alex Elzenaar, MPI MiS): Slides
Some background material
Spherical designs
A spherical \((3,3)\)design in \( \mathbb{R}^3 \) of 16 vectors.
Spherical \((t,t)\)designs are arrangements of points on the sphere (possibly with weights) which are spaced 'far apart from each other': they are finite sets in \( \mathbb{R}^d \)
such that the integral over the sphere of each homogeneous polynomial of degree \(2t\) in \( d \) variables is equal to its average value on the set. There are generalisations
of this definition to subsets of \( \mathbb{C}^d \) and \( \mathbb{H}^d \) (the \(d\)fold product of the Hamiltonian quaternion algebra, not hyperbolic \(d\)space!).
Optimal designs and neardesigns
Shayne Waldron and I have a paper in preparation:
Putatively optimal projective spherical designs with little apparent symmetry,
computing various spherical designs in order to find those of minimal order; a large set of designs and neardesigns are archived on on Zenodo
at
DOI:10.5281/zenodo.6443357. You can look at the code used to generate these
on
GitHub; it uses the
Manopt optimisation toolbox. This work was was funded in
part by a University of Auckland Summer Research Scholarship (201920). You can view the
final report for the scholarship.
Spherical designs and sums of squares
BSc(Hons) dissertation: Toric Varieties
I completed my BSc(Hons) dissertation in 2020 under the supervision of Dr. Jeroen Schillewaert.
College (secondary school) notes on mathematics and physics
In 2017–2020 I was a private tutor to various students in mathematics, physics, chemistry, and geography at a variety of levels from NCEA Level 1 (approx. 15 years old) to
NZ Scholarship (the premier NZ college examinations for highachieving students). I produced a wide range of teaching materials which have been tested on several different students
and seem relatively successful on the whole. These materials can be found on a
separate website. I no longer tutor privately
and these notes are not updated any more, but I am happy to give advice or comment on possible errors.