arXiv subjects

Anna Wienhard

Publications and source records attributed to Anna Wienhard.

At least 19 recordsLinked to original sources

arXiv2026

Illustrating Hyperbolic Surfaces with Mesh Embeddings

Hyperbolic geometry exhibits geometric phenomena, such as fast area growth, that are difficult to visualize faithfully in Euclidean space, and which standard models like the Poincar\'e disk can obscure. To bring hyperbolic geometry to life, we embed hyperbolic surfaces in Euclidean space by discretizing the surfaces into meshes, and minimizing a distortion energy so that the edge lengths in the embeddings match those in the hyperbolic plane. The resulting surfaces buckle and ruffle to accommodate the extra area, making visible what flat models hide. We present exemplary illustrations, such as embedded disks, equidistant strips, diverging geodesics, and also artistic organic-like renders. We discuss our use of these models, as renders and 3D prints, in research talks, public engagement, outreach, and education.

math.HO↗

arXiv2026

Noncommutative Cluster Varieties and Moduli Spaces of Local Systems

In this article, we construct noncommutative cluster varieties, $\mathcal{A}_{R,S}$, for each reduced root system $R$ and marked surface $S$ simultaneously generalizing the cluster varieties of Fock-Goncharov, Li, Goncharov-Shen, Berenstein-Retakh, Goncharov-Kontsevich, and our previously introduced polygonal cluster algebras. Additionally, we define a large class of algebraic groups, we call Jordan split groups. Given a reduced root system $R$ and a family of Jordan algebras, the Lie algebra for $G$ is constructed by unifying the Tits-Kantor-Koecher construction for a single Jordan algebra with the construction of a split Lie algebra. The notion of Jordan split groups is closely related to a grading of its Lie algebra by the root system $R$. We show that these gradings are usually induced by a choice of standard parabolic subalgebra $\mathfrak{p}_\Theta$ and we classify $R$-graded pairs $(G,\Theta)$ via a condition depending only on the subset $\Theta\subset \Delta$ of the set of simple roots. Next, we define Jordan algebra points of $\mathcal{A}_{R,S}$ which parameterize $G$-local systems on $S$ with boundary decoration related to cosets $G/U_\Theta$ when $G$ is Jordan split of type $R$. When $S$ is a disk, points of $\mathcal{A}_{R,S}$ parameterize configurations of decorated flags. We use this to give noncommutative cluster structures on the double $R$-Bruhat cells of $G$, generalizing the cluster algebras of Berenstein-Fomin-Zelevinsky. When each Jordan algebra is formally real, we say that $G$ has a positive structure with respect to $\Theta$. This defines a positive semigroup in $G$. For real algebraic groups, the pairs $(G,\Theta)$ which have positive structures are exactly those which admit a positive structure as defined by Guichard-Wienhard and we give algebraic proofs of many of the properties of positive configurations of flags and of positive representations.

math.RT↗

arXiv2026

Generalizing Lusztig's total positivity II : geometric properties

Positive structures in Lie groups with respect to a subset $\Theta$ of the set of positive roots provide a generalization of Lusztig's total positivity in split real Lie groups to the setting of general real semisimple Lie groups. In [GW25] Lie groups G admitting a positive structure were classified and many key properties of the unipotent positive semigroups were established. In this article we focus on the positive semigroup in G. We establish key geometric properties of elements in the positive semigroup. Further we introduce corresponding positive and non-negative parts of flag varieties and determine the topology of the non-negative parts of flag varieties in many cases. For symplectic flag varieties we provide explicit descriptions of the positive and non-negative flag varieties.

math.DG↗

arXiv2026

Certifying Arithmeticity for Two Degree-Six Symplectic Hypergeometric Monodromy Groups

We prove arithmeticity for two degree-six symplectic hypergeometric monodromy groups, called C-47, and C-55 in the paper \cite{BajpaiDonaNitsche2025Thin} by Bajpai-Dona-Nitsche. This settles two of the three remaining cases, whose classification was left open by \cite{BajpaiDonaNitsche2025Thin}. The arithmeticity certificates were found with AlphaEvolve and then independently verified with exact matrix arithmetic over $\QQ$ using a computer. We include illustrations of limit sets of several degree-six symplectic hypergeometric monodromy groups. Based on these illustration we conjecture C-32 to be thin.

math.GR↗

arXiv2026

Sheaf Neural Networks on SPD Manifolds: Second-Order Geometric Representation Learning

Graph neural networks face two fundamental challenges rooted in the linear structure of Euclidean vector spaces: (1) Current architectures represent geometry through vectors (directions, gradients), yet many tasks require matrix-valued representations that capture relationships between directions-such as how atomic orientations covary in a molecule. These second-order representations are naturally captured by points on the symmetric positive definite matrices (SPD) manifold; (2) Standard message passing applies shared transformations across edges. Sheaf neural networks address this via edge-specific transformations, but existing formulations remain confined to vector spaces and therefore cannot propagate matrix-valued features. We address both challenges by developing the first sheaf neural network operates natively on the SPD manifold. Our key insight is that the SPD manifold admits a Lie group structure, enabling well-posed analogs of sheaf operators without projecting to Euclidean space. Theoretically, we prove that SPD-valued sheaves are strictly more expressive than Euclidean sheaves: they admit consistent configurations (global sections) that vector-valued sheaves cannot represent, directly translating to richer learned representations. Empirically, our sheaf convolution transforms effectively rank-1 directional inputs into full-rank matrices encoding local geometric structure. Our dual-stream architecture achieves SOTA on 6/7 MoleculeNet benchmarks, with the sheaf framework providing consistent depth robustness.

cs.LG↗

arXiv2025

Positivity and Non-commutativity

In this article we revisit a new notion of positivity in real semisimple Lie groups that at the same time generalizes total positivity in split real Lie groups as well as positive Lie semigroups in Hermitian Lie groups of tube type. We shortly discuss the relationship with higher rank Teichm\"uller spaces, and then focus on describing different aspects of positivity as well as open questions. In the second part we describe a non-commutative perspective on Hermitian Lie groups of tube type that is suggested by positivity and leads to interesting applications, such as non-commutative generalizations of Markov numbers.

math.GR↗

arXiv2025

Symmetric spaces for groups over involutive algebras and applications to Higgs bundles

We study symplectic groups and indefinite orthogonal groups over involutive, possibly noncommutative, algebras $(A, \sigma)$. In the case when the algebra $(A, \sigma)$ is Hermitian, or the complexification $(A_{\mathbb C}, \sigma_{\mathbb C})$ of a Hermitian involutive algebra, one can identify maximal compact subgroups of such groups, and consider their associated Riemannian symmetric spaces. This new perspective allows for the realization of various geometric models for the symmetric space. We describe explicitly the complexified tangent space for each of the models, as well as the diffeomorphisms between them and their differentials. In the second part of the article, we give a number of applications of this theory. The geometric realizations of the Riemannian symmetric spaces described in the first part provide new geometric interpretations of Higgs bundle data that can be used for the study of fundamental group representations into symplectic or into indefinite orthogonal groups over Hermitian involutive algebras. We give an exact component count for the moduli spaces of $\rm{Sp}_2(A_{\mathbb C}, \sigma_{\mathbb C})$-Higgs bundles and of $\rm O(A_{\mathbb C}, \sigma_{\mathbb C})$-Higgs bundles, using the topology of the corresponding maximal compact subgroups rather than Morse-Bott theory techniques. Furthermore, we use the noncommutative symmetric-space models to construct a factorization of the Hitchin morphism for $\rm{Sp}_2(A_{\mathbb C},\sigma_{\mathbb C})$-Higgs bundles, together with analogous factorizations for the real groups $\rm{Sp}_2(A,\sigma)$ and $\rm O_{(1,1)}(A,\sigma)$. These factorizations are induced by quadratic norm maps from the corresponding tangent models to Jordan-algebraic targets and pass through intermediate affine GIT quotients. As a consequence, they reduce the algebraic complexity required in order to characterize the Hitchin base explicitly.

math.DG↗

arXiv2024

Noncommutative Polygonal Cluster Algebras

We define a new family of noncommutative generalizations of cluster algebras called polygonal cluster algebras. These algebras generalize the noncommutative surfaces of Berenstein-Retakh, and are inspired by the emerging theory of $\Theta$-positivity for the groups $\mathrm{Spin}(p,q)$. They are generated by mutations of quivers which we call ST-compatible, and which encode the order of the products that appear in the exchange relations. We show that these ST-compatible quivers can be represented by tilings of surfaces by polygons, a generalization of the description of surface type cluster algebras. As examples, we construct tilings which produce ST-compatible versions of the Del Pezzo quivers and the quivers first described by Le for Fock-Goncharov coordinates for Lie groups of type $B$. We show that polygonal cluster algebras have natural evaluations in Clifford algebras, which we use to produce noncommutative generalizations of the Somos sequences and to parameterize the $\Theta$-positive semigroup of $\mathrm{Spin}(2,n)$. We indicate how this will be done for the semigroup in $\mathrm{Spin}(p,q)$ and how one will give coordinates for general $\Theta$-positive representations into $\mathrm{Spin}(p,q)$.

math.RT↗

arXiv2024

Invariant multi-functions and Hamiltonian flows for surface group representations

Goldman defined a symplectic form on the smooth locus of the $G$-character variety of a closed, oriented surface $S$ for a Lie group $G$ satisfying very general hypotheses. He then studied the Hamiltonian flows associated to $G$-invariant functions $G \to \mathbb R$ obtained by evaluation on a simple closed curve and proved that they are generalized twist flows. In this article, we investigate the Hamiltonian flows on (subsets of the) $G$-character variety induced by evaluating a $G$-invariant multi-function $G^k \to \mathbb R$ on a tuple $ \underline{\alpha} \in \pi_1(S)^k$. We introduce the notion of a subsurface deformation along a supporting subsurface $S_0$ for $\underline{\alpha}$ and prove that the Hamiltonian flow of an induced invariant multi-function is of this type. We also give a formula for the Poisson bracket between two functions induced by invariant multi-functions and prove that they Poisson commute if their supporting subsurfaces are disjoint. We give many examples of functions on character varieties that arise in this way and discuss applications, for example, to the flow associated to the trace function for non-simple closed curves on $S$.

math.GT↗

arXiv2024

Positivity, cross-ratios and the Collar Lemma

We prove that $Θ$-positive representations of fundamental groups of surfaces (possibly cusped or of infinite type) satisfy a collar lemma, and their associated cross-ratios are positive. As a consequence we deduce that $Θ$-positive representations form closed subsets of the representation variety.

math.DG↗

arXiv2024

Thurston's asymmetric metrics for Anosov representations

We provide a good dynamical framework allowing to generalize Thurston's asymmetric metric and the associated Finsler norm from Teichmüller space to large classes of Anosov representations. In many cases, including the space of Hitchin representations, this gives a (possibly asymmetric) Finsler distance. In some cases we explicitly compute the associated Finsler norm.

math.DG↗

arXiv2024

Generalizing Lusztig's total positivity

We introduce the notion of $Θ$-positivity in real simple Lie groups. This notion at the same time generalizes Lusztig's total positivity in split real Lie groups and invariant orders in Lie groups of Hermitian type. We show that there are four families of Lie groups which admit $Θ$-positive structures, and investigate basic properties of $Θ$-positivity.

math.DG↗

arXiv2024

$\mathrm{SL}_2$-like Properties of Matrices Over Noncommutative Rings and Generalizations of Markov Numbers

We study $2\times 2$ matrices over noncommutative rings with anti-involution, with a special focus on the symplectic group $\mathrm{Sp}_2(\mathcal{A},σ)$. We define traces and determinants of such matrices and use them to prove a Cayley Hamilton identity and trace relations which generalize well known relations for elements of $\mathrm{SL}_2(R)$ over a commutative ring. We compare the structure of elements of $\mathrm{Sp}_2(\mathcal{A},σ)$ with Manin matrices over general noncommutative rings; this naturally leads to a quantization $\mathrm{Sp}_2(\mathcal{A},σ)_q$. In contrast to the usual definition of the quantum group as a deformation of the ring of matrix functions on $\mathrm{SL}_2(R)$, this quantization produces a group of matrices over a new noncommutative ring with involution. We finish the comparison by constructing a generalization of a Hopf algebra structure on the noncommutative ring of matrix functions of our quantum group. Finally, we use the noncommutative surface-type cluster algebras of Berenstein and Retakh to give a geometric interpretation of our Hopf algebra structure and to produce noncommutative generalizations of Markov numbers over many rings with involution including the complex numbers, dual numbers, matrix rings, and group rings.

math.RA↗

arXiv2023

The self-similar evolution of stationary point processes via persistent homology

Persistent homology provides a robust methodology to infer topological structures from point cloud data. Here we explore the persistent homology of point clouds embedded into a probabilistic setting, exploiting the theory of point processes. We introduce measures on the space of persistence diagrams and the self-similar scaling of a one-parameter family of these. As the main result we prove a packing relation between the occurring scaling exponents.

math.PR↗

arXiv2023

Fiber bundles associated with Anosov representations

Anosov representations $ρ$ of a hyperbolic group $Γ$ into a semisimple Lie group $G$ are known to admit cocompact domains of discontinuity in flag varieties $G/Q$, endowing the compact quotient manifolds $M_ρ$ with a $(G,G/Q)$-structure. In general the topology of $M_ρ$ can be quite complicated. In this article, we consider the case when $Γ$ is the fundamental group of a closed (real or complex) hyperbolic manifold $N$ and $ρ$ is a deformation of a (twisted) lattice embedding $Γ\to \mathrm{Isom}(\mathbb H_\mathbb K) \to G$ through Anosov representations. We prove that, in this situation, $M_ρ$ is alway a smooth fiber bundle over $N$. Determining the topology of the fiber seems hard in general. The second part of the paper focuses on the special case when $N$ is a surface, $ρ$ a quasi-Hitchin representation into $\mathrm{Sp}(4,\mathbb C)$, and $M_ρ$ is modelled on the space of complex Lagrangians in $\mathbb C^4$. We show that, in this case, the fiber is homeomorphic to $\mathbb{CP}^2 \sharp \overline{\mathbb{CP}^2}$.

math.GT↗

arXiv2022

Parametrizing spaces of positive representations

Using Lusztig's total positivity in split real Lie groups V. Fock and A. Goncharov have introduced spaces of positive (framed) representations. For general semisimple Lie groups a generalization of Lusztig's total positivity was recently introduced by O. Guichard and A. Wienhard. They also introduced the associated space of positive representations. Here we consider the corresponding spaces of positive framed representations of the fundamental group of a punctured surface. We give several parametrizations of the spaces of framed positive representations. Using these parametrizations, we describe their topology and their homotopy type. We show that the number of connected components of the space of framed positive representations agrees with the number of connected components of the space of positive representations, and determine this number for simple Lie groups. Along the way, we also parametrize, for an arbitrary semisimple Lie group, the space of representations of the fundamental group of a punctured surface which are transverse with respect to a fixed ideal triangulation of the surface.

math.DG↗

arXiv2022

Noncommutative coordinates for symplectic representations

We introduce coordinates on the spaces of framed and decorated representations of the fundamental group of a surface with nonempty boundary into the symplectic group $Sp(2n,\mathbf R)$. These coordinates provide a noncommutative generalization of the parametrizations of the spaces of representations into $SL(2,\mathbf R)$ or $PSL(2,\mathbf R)$ given by Thurston, Penner, Kashaev, and Fock-Goncharov. On the space of decorated symplectic representations the coordinates give a geometric realization of the noncommutative cluster-like structures introduced by Berenstein-Retakh. The locus of positive coordinates maps to the space of framed maximal representations. We use this to determine an explicit homeomorphism between the space of framed maximal representations and a quotient by the group $O(n)$. This allows us to describe the homotopy type and, when $n=2$, to give an exact description of the singularities. Along the way, we establish a complete classification of pairs of nondegenerate quadratic forms.

math.DG↗

arXiv2021

Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices

We propose the use of the vector-valued distance to compute distances and extract geometric information from the manifold of symmetric positive definite matrices (SPD), and develop gyrovector calculus, constructing analogs of vector space operations in this curved space. We implement these operations and showcase their versatility in the tasks of knowledge graph completion, item recommendation, and question answering. In experiments, the SPD models outperform their equivalents in Euclidean and hyperbolic space. The vector-valued distance allows us to visualize embeddings, showing that the models learn to disentangle representations of positive samples from negative ones.

cs.LG↗