@article{66309,
abstract = {{Abstract
Let
$A$
be an abelian variety with totally degenerate reduction over a non-Archimedean field. We describe the moduli space of semi-homogeneous vector bundles on
$A$
from the perspective of non-Archimedean uniformization and show that the essential skeleton may be identified with a tropical analogue of this moduli space. For
$H=0$
our moduli space may be identified with the moduli space
$M_{0,r}(A)$
of semi-stable vector bundles with vanishing Chern classes on
$A$
. In this case we construct a surjective analytic morphism from the character variety of the analytic fundamental group of
$A$
onto
$M_{0,r}(A)$
, which naturally tropicalizes. One may view this construction as a non-Archimedean uniformization of
$M_{0,r}(A)$
.
}},
author = {{Gross, Andreas and Kaur, Inder and Ulirsch, Martin and Werner, Annette}},
issn = {{2949-7647}},
journal = {{Moduli}},
publisher = {{Cambridge University Press (CUP)}},
title = {{{Semi-homogeneous vector bundles on abelian varieties: moduli spaces and their tropicalization}}},
doi = {{10.1017/S2949764726100228}},
volume = {{3}},
year = {{2026}},
}
@unpublished{66304,
abstract = {{Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic zero carrying the trivial valuation. In this article we discuss two candidates for what could be the tropicalization of $G$.
Our first suggestion is the extended affine building associated to $G$. This perspective makes makes use of Berkovich's embedding of the extended affine building into the Berkovich analytic space $G^{\textrm{an}}$ and expands on work of Mumford by associating a toroidal bordification of $G$ to the choice of stacky fan in the building. We show that the natural retraction onto the building is compatible with the tropicalization map associated to a toroidal bordification.
Our second suggestion is a Weyl chamber of $G$, a special instance of spherical tropicalization, where we think of $G$ as a spherical $G\times G$-variety with respect to left-right-multiplication. We show that the spherical tropicalization map may be identified with the toroidal tropicalization map associated to a wonderful compactification of $G$. This map also has a moduli-theoretic interpretation expanding on the compactifications of $G$ as moduli spaces of framed $\mathbb{G}_m$-equivariant principal bundles on chains of projective lines introduced by Martens and Thaddeus.}},
author = {{Coles, Desmond and Ulirsch, Martin}},
booktitle = {{arXiv:2503.21654}},
title = {{{Towards the tropicalization of reductive groups}}},
year = {{2025}},
}
@unpublished{66303,
abstract = {{We propose an elementary tropical analogue of a reductive group that combines the datum of a Weyl group and the tropicalization of a fixed maximal torus. For the classical groups, as well as $G_2$, these tropical reductive groups admit descriptions as tropical matrix groups that resemble their classical counterparts. Employing this perspective, we introduce tropical principal bundles on metric graphs and study their explicit presentations as pushforwards of line bundles along covers with symmetries and extra data. Our main result identifies the essential skeleton of the moduli space of semistable principal bundles on a Tate curve with its tropical analogue.}},
author = {{Gross, Andreas and Kuhrs, Arne and Ulirsch, Martin and Zakharov, Dmitry}},
booktitle = {{arXiv:2511.05422}},
title = {{{Tropical reductive groups and principal bundles on metric graphs}}},
year = {{2025}},
}
@article{66308,
abstract = {{Abstract
A bimatroid is a matroid-like generalization of the collection of regular minors of a matrix. In this article, we use the theory of Lorentzian polynomials to study the logarithmic concavity of natural sequences associated with bimatroids. Bimatroids can be used to characterize morphisms of matroids and this observation (originally due to Kung) allows us to prove a weak version of logarithmic concavity of the number of bases of a morphism of matroids. This is weaker than the original result by Eur and Huh; it nevertheless provides us with a new perspective on Mason’s log-concavity conjecture for independent sets of matroids. We finally show that for realizable bimatroids, the regular minor polynomial is a volume polynomial. Applied to morphisms of matroids, this shows that the weak basis generating polynomial of a morphism is a volume polynomial; this confirms a conjecture of Eur–Huh for morphisms of nullity
$$\le 1$$
≤
1
and gives an algebro-geometric explanation for Mason’s log-concavity conjecture in the realizable case.
}},
author = {{Röhrle, Felix and Ulirsch, Martin}},
journal = {{Annals of Combinatorics}},
number = {{2}},
pages = {{501--523}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{Logarithmic concavity of bimatroids}}},
doi = {{10.1007/s00026-025-00780-z}},
volume = {{30}},
year = {{2025}},
}
@article{66315,
abstract = {{Abstract
Motivated by the recent surge of interest in the geometry of hybrid spaces, we prove an Abel–Jacobi theorem for a metrized complex of Riemann surfaces, generalizing both the classical Abel–Jacobi theorem and its tropical analogue.}},
author = {{Hofmann, Maximilian C. E. and Ulirsch, Martin}},
issn = {{1615-7168}},
journal = {{Advances in Geometry}},
number = {{2}},
pages = {{263--278}},
publisher = {{Walter de Gruyter GmbH}},
title = {{{An Abel-Jacobi theorem for metrized complexes of Riemann surfaces}}},
doi = {{10.1515/advgeom-2025-0010}},
volume = {{25}},
year = {{2025}},
}
@inbook{66318,
author = {{Küronya, Alex and Souza, Pedro and Ulirsch, Martin}},
booktitle = {{EMS Series of Congress Reports}},
isbn = {{9783985470969}},
issn = {{2523-515X}},
publisher = {{EMS Press}},
title = {{{Tropicalization of toric prevarieties}}},
doi = {{10.4171/ECR/22/16}},
year = {{2025}},
}
@unpublished{66306,
abstract = {{Recently, several proofs of the Mason--Welsh conjecture for matroids have been found, which asserts the log-concavity of the sequence that counts independent sets of a given size. In this article we use the theory of Lorentzian polynomials, developed by Brändén and Huh, to prove a generalization of the Mason-Welsh conjecture to the context of valuated matroids. In fact, we provide a log-concavity result in the more general setting of valuated discrete polymatroids, or equivalently, M-convex functions. Our approach is via the construction of a generic extension of a valuated matroid or M-convex function, so that the bases of the extension are related to the independent sets of the original matroid. We also provide a similar log-concavity result for valuated bimatroids, which, we believe, might be of independent interest.}},
author = {{Giansiracusa, Jeffrey and Rincón, Felipe and Schleis, Victoria and Ulirsch, Martin}},
booktitle = {{arXiv:2407.05808}},
title = {{{Log-concavity for independent sets of valuated matroids}}},
year = {{2024}},
}
@unpublished{66312,
abstract = {{Let $X$ be a complex abelian variety. We prove an analogue of both the (cohomological) $P=W$ conjecture and the geometric $P=W$ conjecture connecting the finer topological structure of the Dolbeault moduli space of topologically trivial semistable Higgs bundles on $X$ and the Betti moduli space of characters of the fundamental group of $X$. The geometric heart of our approach is the spectral data morphism for Dolbeault moduli spaces on abelian varieties that naturally factors the Hitchin morphism and whose target is not an affine space of pluricanonical sections, but a suitable symmetric product.}},
author = {{Bolognese, Barbara and Küronya, Alex and Ulirsch, Martin}},
booktitle = {{arXiv:2303.03734}},
title = {{{$P=W$ phenomena on abelian varieties}}},
year = {{2023}},
}
@article{66325,
abstract = {{AbstractLet$\mathfrak{A}$be a finite abelian group. In this paper, we classify harmonic$\mathfrak{A}$-covers of a tropical curve$\Gamma$(which allow dilation along edges and at vertices) in terms of the cohomology group of a suitably defined sheaf on$\Gamma$. We give a realisability criterion for harmonic$\mathfrak{A}$-covers by patching local monodromy data in an extended homology group on$\Gamma$. As an explicit example, we work out the case$\mathfrak{A}=\mathbb{Z}/p\mathbb{Z}$and explain how realisability for such covers is related to the nowhere-zero flow problem from graph theory.}},
author = {{Len, Yoav and Ulirsch, Martin and Zakharov, Dmitry}},
issn = {{0305-0041}},
journal = {{Mathematical Proceedings of the Cambridge Philosophical Society}},
number = {{2}},
pages = {{395--416}},
publisher = {{Cambridge University Press (CUP)}},
title = {{{Abelian tropical covers}}},
doi = {{10.1017/S0305004123000518}},
volume = {{176}},
year = {{2023}},
}
@article{66358,
abstract = {{AbstractAffine Bruhat–Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of parameterizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite‐dimensional vector space, up to homothety. It has first been studied by Goldman and Iwahori as a piecewise‐linear analogue of symmetric spaces. The space of seminorms compactifies the space of norms and admits a natural surjective restriction map from the Berkovich analytification of projective space that factors the natural tropicalization map. Inspired by Payne's result that the analytification is the limit of all tropicalizations, we show that the space of seminorms is the limit of all tropicalized linear embeddings and prove a faithful tropicalization result for compactified linear spaces. The space of seminorms is in fact the tropical linear space associated to the universal realizable valuated matroid.}},
author = {{Battistella, Luca and Kühn, Kevin and Kuhrs, Arne and Ulirsch, Martin and Vargas, Alejandro}},
issn = {{0024-6107}},
journal = {{Journal of the London Mathematical Society}},
number = {{1}},
publisher = {{Wiley}},
title = {{{Buildings, valuated matroids, and tropical linear spaces}}},
doi = {{10.1112/jlms.12850}},
volume = {{109}},
year = {{2023}},
}
@article{66316,
author = {{Gross, Andreas and Ulirsch, Martin and Zakharov, Dmitry}},
issn = {{0001-8708}},
journal = {{Advances in Mathematics}},
publisher = {{Elsevier BV}},
title = {{{Principal bundles on metric graphs: the $\mathrm{GL}_n$ case}}},
doi = {{10.1016/j.aim.2022.108775}},
volume = {{411}},
year = {{2022}},
}
@article{66329,
abstract = {{
We show that the non-Archimedean skeleton of the
d
d
-th symmetric power of a smooth projective algebraic curve
X
X
is naturally isomorphic to the
d
d
-th symmetric power of the tropical curve that arises as the non-Archimedean skeleton of
X
X
. The retraction to the skeleton is precisely the specialization map for divisors. Moreover, we show that the process of tropicalization naturally commutes with the diagonal morphisms and the Abel-Jacobi map and we exhibit a faithful tropicalization for symmetric powers of curves. Finally, we prove a version of the Bieri-Groves Theorem that allows us, under certain tropical genericity assumptions, to deduce a new tropical Riemann-Roch-Theorem for the tropicalization of linear systems.
}},
author = {{Brandt, Madeline and Ulirsch, Martin}},
issn = {{2330-0000}},
journal = {{Transactions of the American Mathematical Society, Series B}},
number = {{20}},
pages = {{586--618}},
publisher = {{American Mathematical Society (AMS)}},
title = {{{Symmetric powers of algebraic and tropical curves: a non-Archimedean perspective}}},
doi = {{10.1090/btran/113}},
volume = {{9}},
year = {{2022}},
}
@article{66317,
abstract = {{In this article we provide a stack-theoretic framework to study the universal tropical Jacobian over the moduli space of tropical curves. We develop two approaches to the process of tropicalization of the universal compactified Jacobian over the moduli space of curves -- one from a logarithmic and the other from a non-Archimedean analytic point of view. The central result from both points of view is that the tropicalization of the universal compactified Jacobian is the universal tropical Jacobian and that the tropicalization maps in each of the two contexts are compatible with the tautological morphisms. In a sequel we will use the techniques developed here to provide explicit polyhedral models for the logarithmic Picard variety.Comment: 51 pages, 2 figures, v3: published version}},
author = {{Melo, Margarida and Molcho, Samouil and Ulirsch, Martin and Viviani, Filippo}},
issn = {{2491-6765}},
journal = {{Épijournal de Géométrie Algébrique}},
publisher = {{Centre pour la Communication Scientifique Directe (CCSD)}},
title = {{{Tropicalization of the universal Jacobian}}},
doi = {{doi.org/10.46298/epiga.2022.8352}},
volume = {{Volume 6}},
year = {{2022}},
}
@article{66326,
author = {{Len, Yoav and Ulirsch, Martin}},
issn = {{1944-7833}},
journal = {{Algebra & Number Theory}},
number = {{3}},
pages = {{785--820}},
publisher = {{Mathematical Sciences Publishers}},
title = {{{Skeletons of Prym varieties and Brill--Noether theory}}},
doi = {{10.2140/ant.2021.15.785}},
volume = {{15}},
year = {{2021}},
}
@article{66361,
author = {{Len, Yoav and Ulirsch, Martin}},
issn = {{1944-7833}},
journal = {{Algebra & Number Theory}},
number = {{3}},
pages = {{785--820}},
publisher = {{Mathematical Sciences Publishers}},
title = {{{Skeletons of Prym varieties and Brill–Noethertheory}}},
doi = {{10.2140/ant.2021.15.785}},
volume = {{15}},
year = {{2021}},
}
@article{66321,
abstract = {{AbstractIn this article we use techniques from tropical and logarithmic geometry to construct a non-Archimedean analogue ofTeichmüller space$$\overline{{{\mathcal {T}}}}_g$$T¯gwhose points are pairs consisting of a stable projective curve over a non-Archimedean field and a Teichmüller marking of the topological fundamental group of its Berkovich analytification. This construction is closely related to and inspired by the classical construction of a non-Archimedean Schottky space for Mumford curves by Gerritzen and Herrlich. We argue that the skeleton of non-Archimedean Teichmüller space is precisely the tropical Teichmüller space introduced by Chan–Melo–Viviani as a simplicial completion of Culler–Vogtmann Outer space. As a consequence, Outer space turns out to be a strong deformation retract of the locus of smooth Mumford curves in$$\overline{{\mathcal {T}}}_g$$T¯g.}},
author = {{Ulirsch, Martin}},
issn = {{1022-1824}},
journal = {{Selecta Mathematica}},
number = {{3}},
pages = {{39}},
publisher = {{Springer Science and Business Media LLC}},
title = {{{A non-Archimedean analogue of Teichmüller space and its tropicalization}}},
doi = {{10.1007/s00029-021-00651-4}},
volume = {{27}},
year = {{2021}},
}
@article{66333,
abstract = {{
We use recent results by Bainbridge–Chen–Gendron–Grushevsky–Möller on compactifications of strata of abelian differentials to give a comprehensive solution to the realizability problem for effective tropical canonical divisors in equicharacteristic zero. Given a pair
(\Gamma, D)
consisting of a stable tropical curve
\Gamma
and a divisor
D
in the canonical linear system on
\Gamma
, we give a purely combinatorial condition to decide whether there is a smooth curve
X
over a non-Archimedean field whose stable reduction has
\Gamma
as its dual tropical curve together with an effective canonical divisor
K_X
that specializes to
D
.
}},
author = {{Moeller, Martin and Ulirsch, Martin and Werner, Annette}},
issn = {{1435-9855}},
journal = {{Journal of the European Mathematical Society}},
number = {{1}},
pages = {{185--217}},
publisher = {{European Mathematical Society - EMS - Publishing House GmbH}},
title = {{{Realizability of tropical canonical divisors}}},
doi = {{10.4171/JEMS/1009}},
volume = {{23}},
year = {{2020}},
}
@article{66324,
author = {{Brandt, Madeline and Ulirsch, Martin}},
issn = {{0026-2285}},
journal = {{Michigan Mathematical Journal}},
number = {{2}},
publisher = {{Michigan Mathematical Journal}},
title = {{{Divisorial motivic zeta functions for marked stable curves}}},
doi = {{10.1307/mmj/20195792 }},
volume = {{71}},
year = {{2020}},
}
@article{66336,
abstract = {{We contribute to the foundations of tropical geometry with a view toward formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show that it is representable by a geometric stack over the category of rational polyhedral cones. In this framework, the natural forgetful morphisms between moduli spaces of curves with marked points function as universal curves.
Our approach to tropical geometry permits tropical moduli problems—moduli of curves or otherwise—to be extended to logarithmic schemes. We use this to construct a smooth tropicalization morphism from the moduli space of algebraic curves to the moduli space of tropical curves, and we show that this morphism commutes with all of the tautological morphisms.}},
author = {{Cavalieri, Renzo and Chan, Melody and Ulirsch, Martin and Wise, Jonathan}},
issn = {{2050-5094}},
journal = {{Forum of Mathematics, Sigma}},
publisher = {{Cambridge University Press (CUP)}},
title = {{{A MODULI STACK OF TROPICAL CURVES}}},
doi = {{10.1017/fms.2020.16}},
volume = {{8}},
year = {{2020}},
}
@article{66339,
author = {{Ulirsch, Martin}},
issn = {{0001-8708}},
journal = {{Advances in Mathematics}},
pages = {{346--381}},
publisher = {{Elsevier BV}},
title = {{{Non-Archimedean geometry of Artin fans}}},
doi = {{10.1016/j.aim.2019.01.008}},
volume = {{345}},
year = {{2019}},
}