arXiv subjects

Thomas Baier

Publications and source records attributed to Thomas Baier.

11 recordsLinked to original sources

arXiv2025

Abelianization of the $\operatorname{SL}_2$ Hitchin connection at level four

We prove that the Hitchin connection for $\operatorname{SL}_2$ at level four can be understood in terms of the Mumford-Welters connections on bundles of abelian theta functions for Prym torsors of all unramified double covers, and use this to show that its monodromy is finite. This builds on earlier works, for individual curves, of the last named author with Oxbury and Ramanan. The key ingredients in making this work on the level of connections are equivariant conformal embeddings, and anti-invariant level-rank duality.

math.AG↗

arXiv2024

The Prym-Hitchin Connection and Anti-Invariant Level-Rank Duality

We construct a "Hitchin-type" connection on bundles of non-abelian theta functions on higher-rank Prym varieties, for unramified double covers of curves. We formulate a version of level-rank duality in this Prym setting (building on work of Zelaci), show it holds for level one, and establish that the duality respects the flat connections at all levels.

math.AG↗

arXiv2024

Fibering polarizations and Mabuchi rays on symmetric spaces of compact type

In this paper, we describe holomorphic quantizations of the cotangent bundle of a symmetric space of compact type $T^*(U/K)\cong U_\mathbb{C}/K_\mathbb{C}$, along Mabuchi rays of $U$-invariant K\"ahler structures. At infinite geodesic time, the K\"ahler polarizations converge to a mixed polarization $\mathcal{P}_\infty$. We show how a generalized coherent state transform relates the quantizations along the Mabuchi geodesics such that holomorphic sections converge, as geodesic time goes to infinity, to distributional $\mathcal{P}_\infty$-polarized sections. Unlike in the case of $T^*U$, the gCST mapping from the Hilbert space of vertically polarized sections are not asymptotically unitary due to the appearance of representation dependent factors associated to the isotypical decomposition for the $U$-action. In agreement with the general program outlined in [Bai+23], we also describe how the quantization in the limit polarization $\mathcal{P}_\infty$ is given by the direct sum of the quantizations for all the symplectic reductions relative to the invariant torus action associated to the Hamiltonian action of $U$.

math.SG↗

arXiv2023

Quantization in fibering polarizations, Mabuchi rays and geometric Peter--Weyl theorem

In this paper we use techniques of geometric quantization to give a geometric interpretation of the Peter--Weyl theorem. We present a novel approach to half-form corrected geometric quantization in a specific type of non-Kähler polarizations and study one important class of examples, namely cotangent bundles of compact semi-simple groups $K$. Our main results state that this canonically defined polarization occurs in the geodesic boundary of the space of $K\times K$-invariant Kähler polarizations equipped with Mabuchi's metric, and that its half-form corrected quantization is isomorphic to the Kähler case. An important role is played by invariance of the limit polarization under a torus action. Unitary parallel transport on the bundle of quantum states along a specific Mabuchi geodesic, given by the coherent state transform of Hall, relates the non-commutative Fourier transform for $K$ with the Borel--Weil description of irreducible representations of $K$.

math.SG↗

arXiv2022

Evaluating Agent Interactions Through Episodic Knowledge Graphs

We present a new method based on episodic Knowledge Graphs (eKGs) for evaluating (multimodal) conversational agents in open domains. This graph is generated by interpreting raw signals during conversation and is able to capture the accumulation of knowledge over time. We apply structural and semantic analysis of the resulting graphs and translate the properties into qualitative measures. We compare these measures with existing automatic and manual evaluation metrics commonly used for conversational agents. Our results show that our Knowledge-Graph-based evaluation provides more qualitative insights into interaction and the agent's behavior.

cs.AI↗

arXiv2022

A modular architecture for creating multimodal agents

The paper describes a flexible and modular platform to create multimodal interactive agents. The platform operates through an event-bus on which signals and interpretations are posted in a sequence in time. Different sensors and interpretation components can be integrated by defining their input and output as topics, which results in a logical workflow for further interpretations. We explain a broad range of components that have been developed so far and integrated into a range of interactive agents. We also explain how the actual interaction is recorded as multimodal data as well as in a so-called episodic Knowledge Graph. By analysing the recorded interaction, we can analyse and compare different agents and agent components.

cs.AI↗

arXiv2022

The Hitchin connection in arbitrary characteristic

We give an algebro-geometric construction of the Hitchin connection, valid also in positive characteristic (with a few exceptions). A key ingredient is a substitute for the Narasimhan-Atiyah-Bott Kähler form that realizes the Chern class of the determinant-of-cohomology line bundle on the moduli space of bundles on a curve. As replacement we use an explicit realisation of the Atiyah class of this line bundle, based on the theory of the trace complex due to Beilinson-Schechtman and Bloch-Esnault.

math.AG↗

arXiv2021

EMISSOR: A platform for capturing multimodal interactions as Episodic Memories and Interpretations with Situated Scenario-based Ontological References

We present EMISSOR: a platform to capture multimodal interactions as recordings of episodic experiences with explicit referential interpretations that also yield an episodic Knowledge Graph (eKG). The platform stores streams of multiple modalities as parallel signals. Each signal is segmented and annotated independently with interpretation. Annotations are eventually mapped to explicit identities and relations in the eKG. As we ground signal segments from different modalities to the same instance representations, we also ground different modalities across each other. Unique to our eKG is that it accepts different interpretations across modalities, sources and experiences and supports reasoning over conflicting information and uncertainties that may result from multimodal experiences. EMISSOR can record and annotate experiments in virtual and real-world, combine data, evaluate system behavior and their performance for preset goals but also model the accumulation of knowledge and interpretations in the Knowledge Graph as a result of these episodic experiences.

cs.HC↗

arXiv2018

Picard group and quantization of toric orbifolds

In the classical theory of toric manifolds polytopes appear in two guises -- as Newton polytopes of line bundles on the complex, and as moment polytopes on the symplectic side, the link between the two being established by the prequantizability condition on the cohomology class of the symplectic form. Here we give a combinatorial description of the orbifold Picard group for complete toric orbifolds, with the aim of detailing the relation between complex and symplectic aspects in the orbifold setting. In particular this permits to illustrate the breakdown of identification of (orbifold) line bundles by their Chern class (or moment polytope up to translations in $\mathfrak{t}^\ast$), and non-constancy of $h^0$ on representatives of the same Chern class. As an application, we discuss symplectic reduction with respect to restrictions of the action to sub-tori, and the associated Bohr--Sommerfeld conditions in mixed polarizations.

math.DG↗

arXiv2011

Toric Kähler metrics seen from infinity, quantization and compact tropical amoebas

We consider the metric space of all toric Kähler metrics on a compact toric manifold; when "looking at it from infinity" (following Gromov), we obtain the tangent cone at infinity, which is parametrized by equivalence classes of complete geodesics. In the present paper, we study the associated limit for the family of metrics on the toric variety, its quantization, and degeneration of generic divisors. The limits of the corresponding Kähler polarizations become degenerate along the Lagrangian fibration defined by the moment map. This allows us to interpolate continuously between geometric quantizations in the holomorphic and real polarizations and show that the monomial holomorphic sections of the prequantum bundle converge to Dirac delta distributions supported on Bohr-Sommerfeld fibers. In the second part, we use these families of toric metric degenerations to study the limit of compact hypersurface amoebas and show that in Legendre transformed variables they are described by tropical amoebas. We believe that our approach gives a different, complementary, perspective on the relation between complex algebraic geometry and tropical geometry.

math.DG↗

arXiv2010

Quantization of Abelian Varieties: distributional sections and the transition from Kähler to real polarizations

We study the dependence of geometric quantization of the standard symplectic torus on the choice of invariant polarization. Real and mixed polarizations are interpreted as degenerate complex structures. Using a weak version of the equations of covariant constancy, and the Weil-Brezin expansion to describe distributional sections, we give a unified analytical description of the quantization spaces for all nonnegative polarizations. The Blattner-Kostant-Sternberg (BKS) pairing maps between half-form corrected quantization spaces for different polarizations are shown to be transitive and related to an action of $Sp(2g,\R)$. Moreover, these maps are shown to be unitary.

math.SG↗