Alex Elzenaar

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 Hodge-type results (Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann 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 there is a seminar at the Institut Mittag-Leffler 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, non-technical, 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 – Toric varieties & their projective embeddings (Alex Elzenaar, MPI MiS): Slides

Interesting links on the background material

Petter Brändén, Jonathan Leake: Lorentzian polynomials on cones and the Heron-Rota-Welsh conjecture
Petter Brändén, June Huh: Lorentzian polynomials (direct multivariable-calculus proof of the theory without mentioning Hodge theory). See also this lecture by June Huh
Matthew Baker: Hodge theory in combinatorics (an expository paper)
Karim Adiprasito, June Huh, Eric Katz: Hodge Theory for Combinatorial Geometries (definition of the Hodge ring for arbitrary matroids)

MSc. thesis: Deformation spaces of Kleinian groups (submitted Feb. 2022)

My Master of Science thesis was completed in 2021-22 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 3-dimensional 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 4-punctured sphere; Robert Riley introduced this object in the 1970s while studying 2-bridge 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 Keen--Series 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 low-dimensional 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:

Approximations of the Riley slice (chapter 8 of the thesis);
The elliptic cousins of the Riley slice (to appear, chapters 4 to 7);
The combinatorics of Farey words and their traces (chapters 6 and 9).

	Slides for talks and other interesting links
Here is a picture of the exterior of the Riley slice:	Riley slice computational package (GitHub). Some animations, and some limit sets. 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 slices. 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. Half-year 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 post-mortem.

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 near-designs

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 near-designs 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 (2019-20). You can view the final report for the scholarship.

Spherical designs and sums of squares

Talk at the UoA algebra and combinatorics seminar, April 2021: Presentation slides.