arXiv subjects

Marvin Anas Hahn

Publications and source records attributed to Marvin Anas Hahn.

At least 19 recordsLinked to original sources

arXiv2026

Combinatorial geometry of the 2D Toda lattice and Davey Stewartson equation

The KP equation is a prototypical $(2+1)$-dimensional integrable PDE. Its soliton solutions are famously parametrized by the Sato Grassmannian. In seminal work, Kodama and Williams made the surprising discovery that the combinatorics of soliton solutions are intimately related to the combinatorics of the totally positive Grassmannian as pioneered by Postnikov. They introduced novel algorithmic methods inspired by polyhedral structures arising from tropical geometry. Soliton solutions to the 2D Toda lattice and the Davey--Stewartson equation, two closely related integrable systems with soliton solutions, are also classified by the Sato Grassmannian. Kodama suggested that the methods of his work with Williams could generalize to these two integrable equations. In this work, we show that this is indeed the case. We derive algorithms to produce contour plots from elements in the totally nonnegative Grassmannian in both cases. In the asymptotic setting, we recover and refine previous work of Biondini and Wang; as well as Biondini, Kireyev and Maruno.

nlin.SI↗

arXiv2026

Failing to keep the balance: explicit formulae and topological recursion for leaky Hurwitz numbers

Recently a new family of enumerative invariants called leaky Hurwitz numbers was introduced by Cavalieri-Markwig-Ranganathan in the context of logarithmic intersection theory. They admit an interpretation via tropical covers where the balancing condition fails. We employ tropical geometry to prove a generalisation of the piecewise polynomiality of Accadia-Karev-Lewanski for leaky completed cycles Hurwitz numbers, and a different wall crossing that is cubic instead of quadratic. Using tropical combinatorics and generatingfunctionology, we also find closed formulae for one-part and two-part completed cycles leaky Hurwitz numbers in genus $0$. Working more generally with a view towards topological recursion, we use Hamiltonian flows to associate spectral curves to very general cut-and-join operators. Under mild analytic constraints, we find the appropriate spectral curves, and in case the leakiness is fixed, we show that the resulting enumerative invariants satisfy topological recursion. This provides a partial inverse to recent work of Alexandrov-Bychkov-Dunin-Barkowski-Kazarian-Shadrin producing differentials satisfying topological recursion for KP $τ$-functions. In particular these results specialise to completed cycles leaky Hurwitz numbers.

math.AG↗

arXiv2025

Tropicalising hypergeometric $τ$-functions

Weighted Hurwitz numbers arise as coefficients in the power sum expansion of deformed hypergeometric $τ$--functions. They specialise to essentially all known cases of Hurwitz numbers, including classical, monotone, strictly monotone and completed cycles Hurwitz numbers. In this work, we develop a tropical geometry framework for their study, thus enabling a simultaneous investigation of all these cases. We obtain a correspondence theorem expressing weighted Hurwitz numbers in terms of tropical covers. Using this tropical approach, we generalise most known structural results previously obtained for the aforementioned special cases to all weighted Hurwitz numbers. In particular, we study their polynomiality and derive wall--crossing formulae. Moreover, we introduce elliptic weighted Hurwitz numbers and derive tropical mirror symmetry for these new invariants, i.e. we prove that their generating function is quasimodular and that they may be expressed as Feynman integrals.

math.CO↗

arXiv2025

Universal piecewise polynomiality for counting curves in toric surfaces

Inspired by piecewise polynomiality results of double Hurwitz numbers, Ardila and Brugallé introduced an enumerative problem which they call double Gromov--Witten invariants of Hirzebruch surfaces. These invariants serve as a two-dimensional analogue and satisfy a similar piecewise polynomial structure. More precisely, they introduced the enumeration of curves in Hirzebruch surfaces satisfying point conditions and tangency conditions on the two parallel toric boundaries. These conditions are stored in four partitions and the resulting invariants are piecewise polynomial in their entries. Moreover, they found that these expressions also behave polynomially with respect to the parameter determining the underlying Hirzebruch surfaces. Based on work of Ardila and Block, they proposed that such a polynomiality could also hold while changing between more general toric surfaces corresponding to $h$-transverse polygons. In this work, we answer this question affirmatively. Moreover, we express the resulting invariants for $h$-transverse polygons as matrix elements in the two-dimensional bosonic Fock space.

math.AG↗

arXiv2025

A refined twist on Hurwitz numbers

We introduce a two-parameter refinement of the Jucys-Murphy theory, that we call the CJT-refinement, unifying Schur, zonal, and, conjecturally, Jack actions of the ring of symmetric functions on the Fock space. Applications of this formalism include a partial resolution of a recent conjecture of Coulter-Do, as well as cut-and-join recursion for $b$-Hurwitz numbers. The cut-and-join equations enable the derivation of the tropicalization of $b$--Hurwitz numbers. We also provide a first application of this tropical interpretation by answering an open problem of Chapuy-Dołęga on the polynomial structure of $b$-Hurwitz numbers.

math.CO↗

arXiv2025

Order-One Rolling Shutter Cameras

Rolling shutter (RS) cameras dominate consumer and smartphone markets. Several methods for computing the absolute pose of RS cameras have appeared in the last 20 years, but the relative pose problem has not been fully solved yet. We provide a unified theory for the important class of order-one rolling shutter (RS$_1$) cameras. These cameras generalize the perspective projection to RS cameras, projecting a generic space point to exactly one image point via a rational map. We introduce a new back-projection RS camera model, characterize RS$_1$ cameras, construct explicit parameterizations of such cameras, and determine the image of a space line. We classify all minimal problems for solving the relative camera pose problem with linear RS$_1$ cameras and discover new practical cases. Finally, we show how the theory can be used to explain RS models previously used for absolute pose computation.

cs.CV↗

arXiv2024

Tropical twisted Hurwitz numbers for elliptic curves

Hurwitz numbers enumerate branched morphisms between Riemann surfaces. For a fixed elliptic target, Hurwitz numbers are intimately related to mirror symmetry following work of Dijkgraaf. In recent work of Chapuy and Dolega a new variant of Hurwitz numbers with fixed genus $0$ target was introduced that includes maps between between non-orientiable surfaces. These numbers are called $b$-Hurwitz numbers and are polynomials in a parameter $b$ which measures the non-orientability of the involved maps. An interpretation in terms of factorisations of $b$-Hurwitz numbers for $b=1$, so-called twisted Hurwitz numbers, was found in work of Burman and Fesler. In previous work, the authors derived a tropical geometry interpretation of these numbers. In this paper, we introduce a natural generalisation of twisted Hurwitz numbers with elliptic targets within the framework of symmetric groups. We derive a tropical interpretation of these invariants, relate them to Feynman integrals and derive an expression as a matrix element of an operator in the bosonic Fock space.

math.CO↗

arXiv2023

Twisted Hurwitz numbers: Tropical and polynomial structures

Hurwitz numbers count covers of curves satisfying fixed ramification data. Via monodromy representation, this counting problem can be transformed to a problem of counting factorizations in the symmetric group. This and other beautiful connections make Hurwitz numbers a longstanding active research topic. In recent work Chapuy and Dolęga, a new enumerative invariant called b-Hurwitz number was introduced, which enumerates non-orientable branched coverings. For b=1, we obtain twisted Hurwitz numbers which were linked to surgery theory in work of Burman and Fesler and admit a representation as factorisations in the symmetric group. In this paper, we derive a tropical interperetation of twisted Hurwitz numbers in terms of tropical covers and study their polynomial structure.

math.CO↗

arXiv2023

Valued rank-metric codes

In this paper, we study linear spaces of matrices defined over discretely valued fields and discuss their dimension and minimal rank drops over the associated residue fields. To this end, we take first steps into the theory of rank-metric codes over discrete valuation rings by means of skew algebras derived from Galois extensions of rings. Additionally, we model projectivizations of rank-metric codes via Mustafin varieties, which we then employ to give sufficient conditions for a decrease in the dimension.

math.NT↗

arXiv2023

Combinatorics of pruned Hurwitz numbers

Hurwitz numbers enumerate branched morphisms between Riemannn surfaces with fixed numerical data. They represent important objects in enumerative geometry that are accessible by combinatorial techniques. In the past decade, many variants of Hurwitz numbers have appeared in the literature. In this paper, we focus on an exciting such variant that arises naturally from the theory of topological recursion: Pruned Hurwitz numbers. These are defined as an enumeration of a relevant subset of branched morphisms between Riemann surfaces, that yield smaller numbers than their classical counterparts while retaining maximal information. Thus, pruned Hurwitz numbers may be viewed as the core of the Hurwitz problem. In this paper, we develop the combinatorial theory of pruned Hurwitz numbers. In particular, motivated by the successful application of combinatorial techniques to classical Hurwitz numbers, we derive two new combinatorial expressions of pruned Hurwitz numbers. Firstly, we show that they may be expressed in terms of Hurwitz mobiles which are tree-like structure that arise from the theory of random planar maps. Secondly, we prove a tropical correspondence theorem which allows the enumeration of pruned Hurwitz numbers in terms of tropical covers.

math.CO↗

arXiv2023

Subspaces Fixed by a Nilpotent Matrix

The linear spaces that are fixed by a given nilpotent $n \times n$ matrix form a subvariety of the Grassmannian. We classify these varieties for small $n$. Mutiah, Weekes and Yacobi conjectured that their radical ideals are generated by certain linear forms known as shuffle equations. We prove this conjecture for $n \leq 7$, and we disprove it for $n=8$. The question remains open for nilpotent matrices arising from the affine Grassmannian.

math.RA↗

arXiv2022

Intersection numbers on tropical Hassett spaces

We study the intersection of tropical psi-classes on tropical heavy/light Hassett spaces, generalising a result of Kerber--Markwig for tropical moduli spaces of rational stable curves with distinct marked points. Our computation reveals that the weight of a maximal cone in an intersection has a combinatorial intepretation in terms of the underlying tropical curve and it is always nonnegative. In particular, our result specialises to that, in top dimension, the tropical intersection product coincides with its classical counterpart.

math.CO↗

arXiv2021

Orders and Polytropes: Matrix Algebras from Valuations

We apply tropical geometry to study matrix algebras over a field with valuation. Using the shapes of min-max convexity, known as polytropes, we revisit the graduated orders introduced by Plesken and Zassenhaus. These are classified by the polytrope region. We advance the ideal theory of graduated orders by introducing their ideal class polytropes. This article emphasizes examples and computations. It offers first steps in the geometric combinatorics of endomorphism rings of configurations in affine buildings.

math.CO↗

arXiv2020

One-parameter families of multiview varieties via quotient lattices

We develop a novel theory of one-parameter families of multi-view varieties. These families are induced by quotient lattices over discrete valuation rings and generalise the notion of \textit{Mustafin varieties}. We study the geometry and the combinatorics of the limit of these families.

math.AG↗

arXiv2020

Mustafin models of projective varieties and vector bundles

Mustafin varieties are well-studied degenerations of projective spaces induced by a choice of integral points in a Bruhat--Tits building. In recent work, Annette Werner and the author initiated the study of degenerations of plane curves obtained by Mustafin varieties by means of arithmetic geometry. Moreover, we applied these techniques to construct models of vector bundles on plane curves with strongly semistable reduction. In this work, we take a Groebner basis approach to the more general problem of studying degenerations of projective varieties. Our methods include determining the behaviour of Groebner bases under substitution over unique factorisation rings. Finally, we outline applications to the $p-$adic Simpson correspondence, when the respective projective variety is a curve.

math.AG↗

arXiv2019

A short note on Cayley-Salmon equations

A Cayley-Salmon equation for a smooth cubic surface $S$ in $\mathbb P^3$ is an expression of the form $l_1l_2l_3 - m_1m_2m_3 = 0$ such that the zero set is $S$ and $l_i$, $m_j$ are homogeneous linear forms. This expression was first used by Cayley and Salmon to study the incidence relations of the 27 lines on $S$. There are 120 essentially distinct Cayley-Salmon equations for $S$. In this note we give an exposition of a classical proof of this fact. We illustrate the explicit calculation to obtain these equations and we apply it to Clebsch surface and to the octanomial model. Finally we show that these $120$ Cayley-Salmon equations can be directly computed using recent work by Cueto and Deopurkar.

math.AG↗

arXiv2019

Strongly semistable reduction of syzygy bundles on plane curves

We investigate degenerations of syzygy bundles on plane curves over $p$-adic fields. We use Mustafin varieties which are degenerations of projective spaces to find a large family of models of plane curves over the ring of integers such that the special fiber consists of multiple projective lines meeting in one point. On such models we investigate vector bundles whose generic fiber is a syzygy bundle and which become trivial when restricted to each projective line in the special fiber. Hence these syzygy bundles have strongly semistable reduction. This investigation is motivated by the fundamental open problem in $p$-adic Simpson theory to determine the category of Higgs bundles corresponding to continuous representations of the étale fundamental group of a curve. Faltings' $p$-adic Simpson correspondence and work of Deninger and the second author shows that bundles with Higgs field zero and potentially strongly semistable reduction fall into this category. Hence the results in the present paper determine a class of syzygy bundles on plane curves giving rise to a $p$-adic local system. We apply our methods to a concrete example on the Fermat curve suggested by Brenner and prove that this bundle has potentially strongly semistable reduction.

math.AG↗

arXiv2019

Triply mixed coverings of arbitrary base curves: Quasimodularity, quantum curves and a mysterious topological recursions

Simple Hurwitz numbers enumerate branched morphisms between Riemann surfaces with fixed ramification data. In recent years, several variants of this notion for genus $0$ base curves have appeared in the literature. Among them are so-called monotone Hurwitz numbers, which are related to the HCIZ integral in random matrix theory and strictly monotone Hurwitz numbers which count certain Grothendieck dessins d'enfants. We generalise the notion of Hurwitz numbers to interpolations between simple, monotone and strictly monotone Hurwitz numbers to any genus and any number of arbitrary but fixed ramification profiles. This yields generalisations of several results known for Hurwitz numbers. When the target surface is of genus one, we show that the generating series of these interpolated Hurwitz numbers are quasimodular forms. In the case that all ramification is simple, we refine this result by writing this series as a sum of quasimodular forms corresonding to tropical covers weighted by Gromov-Witten invariants. Moreover, we derive a quantum curve for monotone and Grothendieck dessins d'enfants Hurwitz numbers for arbitrary genera and one arbitrary but fixed ramification profile. Thus, we obtain spectral curves via the semiclassical limit as input data for the CEO topological recursion. Astonishingly, we find that the CEO topological recursion for the genus $1$ spectral curve of the strictly monotone Hurwitz numbers compute the monotone Hurwitz numbers in genus $0$. Thus, we give a new proof that monotone Hurwitz numbers satisfy CEO topological recursion. This points to an unknown relation between those enumerants. Finally, specializing to target surface $\mathbb{P}^1$, we find recursions for monotone and Grothendieck dessins d'enfants double Hurwitz numbers, which enables the computation of the respective Hurwitz numbers for any genera with one arbitrary but fixed ramification profile.

math.AG↗