Hello 👋 I'm Samuël Borza. I am a Marie Skłodowska-Curie (MSCA) Postdoctoral
Fellow in Mathematics at the
University of Vienna. My project
SUBLOR is devoted to sub-Lorentzian geometry, the
non-holonomic
analogue of Lorentzian geometry and the Lorentzian counterpart of sub-Riemannian geometry.
From 2021 to 2024, I was a Postdoctoral Researcher in Mathematics at the Scuola
Internazionale Superiore di Studi Avanzati (SISSA) in
Trieste, Italy. There, my research was supported by the ERC
Starting Grant (grant agreement
No. 945655) Project GEOSUB -
Geometric
analysis of sub-Riemannian spaces through interpolation inequalities. I was part of the
research team
led by
Luca Rizzi.
My research lies at the intersection of differential geometry, geometric analysis, and optimal transport, with
some applications to mathematical physics. More specifically, I study Carnot–Carathéodory geometry, by which I
mean not only sub-Riemannian but also sub-Finsler and sub-Lorentzian spaces. These geometric structures model
systems with non-holonomic constraints and constitute a broad generalisation of Riemannian, Finsler, and
Lorentzian geometry. While sub-Riemannian geometry has become a well-established and active area of research,
sub-Lorentzian geometry remains comparatively underexplored, and the MSCA project SUBLOR aims to help address that gap. More recently, I have also started
working on causal set theory, a mathematical approach to quantum gravity in which spacetime is modelled as a
discrete
causal graph.
List of Publications
Samuël Borza, Chiara Rigoni and Omar Zoghlami, Hausdorff dimension and failure of synthetic curvature
bounds in the sub-Lorentzian Heisenberg group, 2025. [arXiv]
Michael Albert, Samuël Borza and Maria Gordina, Geodesics on Grushin spaces, 2025.
[arXiv]
Samuël Borza and Luca Rizzi, Failure of the measure contraction property via quotients in higher-step
sub-Riemannian structures, 2025. [arXiv, CVGMT]
Samuël Borza, Wilhelm Klingenberg and Patrick Wood, Optimal transport on the sub-Lorentzian Heisenberg
group, 2025. [arXiv]
Samuël Borza and Kenshiro Tashiro, Curvature-dimension condition of sub-Riemannian \(\alpha\)-Grushin
half-spaces, Transactions of the London Mathematical Society, Volume 12, Issue 1, 2025. [DOI, arXiv, CVGMT]
Samuël Borza, Mattia Magnabosco, Tommaso Rossi and Kenshiro Tashiro, Curvature exponent of sub-Finsler
Heisenberg groups, SIAM Journal on Mathematical Analysis, Volume 57, Issue 4, pp. 3561–3586,
2025. [DOI, arXiv, CVGMT]
Samuël Borza, Mattia Magnabosco, Tommaso Rossi and Kenshiro Tashiro, Measure contraction property and
curvature-dimension condition on sub-Finsler Heisenberg groups, 2024.
[arXiv, CVGMT]
Samuël Borza and Kenshiro Tashiro, Measure contraction property, curvature exponent and geodesic
dimension of sub-Finsler \(\ell^p\)-Heisenberg groups, Annales de l’Institut Fourier
(accepted), 2023. [arXiv, CVGMT]
Samuël Borza, Normal forms for the sub-Riemannian exponential map of \(\mathbb{G}_\alpha\),
\(\mathrm{SU}(2)\), and \(\mathrm{SL}(2)\), New Trends in Sub-Riemannian Geometry, Contemporary
Mathematics, Volume 809, American Mathematical Society, pp. 23–48, 2025. [DOI, arXiv, CVGMT]
Samuël Borza and Wilhelm Klingenberg, Local non-injectivity of the exponential map at critical points
in
sub-Riemannian geometry, Nonlinear Analysis, Volume 239, pp. 113421, 2024. [DOI, arXiv, CVGMT]
Samuël Borza and Wilhelm Klingenberg, Regularity and Continuity Properties of the Sub-Riemannian
Exponential
Map, Journal of Dynamical and Control Systems, Volume 29, pp. 1385–1407, 2023. [DOI, arXiv, CVGMT]
Samuël Borza, Distortion coefficients of the \(\alpha\)-Grushin plane, Journal of Geometric
Analysis, Volume 32, Paper No. 78, 2022. [DOI, arXiv, CVGMT]
Samuël Borza, Distortion coefficients and exponential map in sub-Riemannian geometry,
Ph.D. thesis, Durham University, 2021. [Durham
e-Theses]
Lean and mathlib
I have recently become interested in formalised mathematics and more precisely in the development of the
mathematical library mathlib, an open-source library of
formal mathematics
written in Lean. I have formalised a direct,
self-contained, and
elegant proof of the transcendence of \(\pi\) that does not rely on the Lindemann-Weierstrass theorem.
The repository is available on GitHub. I plan to contribute
more to the mathlib initiative in the future, especially in geometry.
In the summer semester 2026, I am a lecturer for the course Principal
Fiber Bundles, together with Michael Kunzinger, at the University of Vienna. The course provides an introduction to principal fibre bundles, a
fundamental mathematical object in
differential geometry and global analysis, with important applications to gauge theory and mathematical physics.
The course covers transformation groups and the theory of fibre, principal, associated, and vector bundles.
Connections,
parallel transport, and curvature on principal bundles are also treated, together with holonomy theory and the
Yang-Mills equation, including its relation to Maxwell's equations. The lecture notes for the course are
available here.
I was a teaching assistant of the course Mathematical
Analysis
II for second-year engineering students at the University
of Trieste. One distinctive feature of the course is that, among other things, it focuses on introducing
students to the Henstock-Kurzweil
integral, a broader theory of integration that is easier to understand compared to what is typically
taught
at
the undergraduate level. Its construction is simple and geometrically comprehensible, akin to Riemann sums, yet
more
general than Lebesgue's theory of integration. The course investigator is Alessandro Fonda.
I was one of the teaching assistants helping in the delivery of the course Elementary Mathematics (Fiche
ECTS)
to first-year students majoring in Mathematics, Physics, and Computer Science at the Faculty of Sciences of the
University of Mons. The course, offered by the Department of Mathematics, aims to facilitate the
transition from high school to university. Further information on the teaching methodology can be found here (in
French).
I was a teaching assistant for the courses Mathematics - Supplementary Course and Introduction to
Differential Manifolds during the preparation for the summer exam session at the Department of
Mathematics of the University of Mons. I helped third-year students in Mathematics and Physics in passing their
exams by thoroughly reviewing the course
material and conducting practice sessions using past papers. The curriculum of the first course included topics
on convergence of
functions, Fourier transform, Hilbert space theory, and an introduction to the theory of distributions. The
second course focused on the differential geometry of curves and surfaces: regular curves and surfaces in
Euclidean spaces, Gauss map, First and Second fundamental forms, Gauss's Theorema Egregium.
Contact
Office address
Office 123, 3rd floor
Faculty of Mathematics, University of Vienna
Oskar-Morgenstern-Platz 1
1090 Vienna
AUSTRIA
Office address
Office 123, 3rd floor
Email and social media
