Research Projects - William Chuang

Research Projects

Critical Scaling in Hyperbolic Attention Mechanisms

This project presents a comprehensive, mathematically rigorous framework for hyperbolic attention mechanisms in transformer architectures, linking them to statistical mechanics, spectral theory, and fractal geometry. It offers an explicit derivation of the critical inverse temperature $ \beta_c(\delta, \kappa, \mathcal{T}) $ in terms of fractal dimension $ \delta $, curvature $ \kappa $, and topological connectivity $ \mathcal{T} $.

The manuscript unifies concepts from hyperbolic geometry, partition functions, Laplace–Beltrami operators, and transformer design. Key contributions include:

An exact formula for $ \beta_c \sim \exp(C(\kappa)\,\delta\,r_{\mathrm{eff}})/\lambda_{\max}(\mathcal{T}) $
Spectral density derivations based on fractal boundaries
Dynamic attention scaling protocols minimizing energy dissipation
Extended discussions on quantum security, Langlands correspondence, and Lorentz adaptations

Download the full paper: Critical Scaling in Hyperbolic Attention Mechanisms (PDF)

Supplementary Notes on Thermodynamic Formalism and Hyperbolic Dynamics

In follow-up to the explicit dimension formula $ \dim \mathcal{H}(\Lambda_\Gamma) = \frac{\ln(2m - 1)}{r_{\mathrm{eff}}} $, I include supplementary materials that frame the result within the broader context of symbolic dynamics, thermodynamic formalism, and Lie-theoretic flows. These connections provide a more unified and rigorous perspective on the structure of limit sets, their self-similarity, and the role of PSL(2,$\mathbb{R}$) isometries.

Key Topics Covered in These Notes

The sum $ \sum_{|g| = n} |g'(z)|^\delta \sim 1 $ as a bridge between symbolic dynamics and fractal geometry.
A derivation of critical exponents via pressure-zero arguments, connecting partition functions to Hausdorff dimension.
A proof that all Schottky group orbit branches approach the boundary circle at a uniform exponential rate, ensuring well-formed fractal limit sets.
Differential equations and flow models in the upper half-plane and Poincaré disk that interpolate discrete isometries.
Rigorous constructions of Patterson–Sullivan measures and their decay properties under the group action.

These results are particularly powerful when analyzing the dynamics of Schottky subgroups of PSL(2,$\mathbb{R}$) through the lens of the Lie algebra $ \mathfrak{sl}(2,\mathbb{R}) $. The uniform convergence to the boundary and equivalence of hyperbolic displacement among conjugates ensures that side-branch instabilities do not distort the limit set’s dimension.

Additional Lecture Notes:

Together, these documents provide a rich and self-contained exposition suitable for advanced study in geometric group theory, dynamical systems, spectral theory, and their applications to mathematical physics and quantum information.

Supplement: First-Level Symmetry and Exact Hausdorff Dimension

This supplementary note highlights a key insight: if the initial generators of a Schottky group exhibit complete first-level symmetry—that is, the magnitudes of their derivatives at a common base point $ z_0 $ satisfy $ |T_i'(z_0)| = \text{const} $—then the entire Hausdorff dimension of the limit set can be determined using only this first-level data.

Specifically, under these conditions, the zero-pressure equation \[ \sum_{|T_i| = 1} |T_i'(z_0)|^{-\delta} = 1 \] yields an exact solution for the Hausdorff dimension $\dim_H(\Lambda_\Gamma) = \delta$, without requiring data from deeper iterates.

Even when perfect symmetry breaks at higher levels, as long as bounded distortion holds, the contribution of higher iterates remains controlled. The result is robust: full symmetry at the first level ensures the validity of the explicit formula throughout the group’s dynamical hierarchy.

This observation strengthens the theoretical justification for using well-distributed Schottky generators to derive explicit, closed-form dimension formulas.

This work provides a novel and explicit closed‐form formula for computing the Hausdorff dimension of limit sets associated with Schottky groups that are well‐distributed—that is, those with uniformly arranged generators. In this framework, the Hausdorff dimension is given by $$\dim \mathcal{H}(\Lambda_\Gamma) = \frac{\ln(2m - 1)}{r_{\mathrm{eff}}},$$ where m is the number of free generators and r_eff is the effective translation length determined via a rigorous two‐step displacement method.

The study begins with an in‐depth review of classical hyperbolic geometry and builds upon foundational results by Patterson, Sullivan, and Bowen. By using the Bowen–Series expansion alongside symbolic dynamics and ergodic theory, the work shows that the symmetry in generator placement yields a uniform contraction ratio. This uniformity allows for an exact calculation of the fractal dimension of the limit set, overcoming the need for purely numerical methods.

A key insight of the research is that every finitely generated convex-cocompact Fuchsian group can be approximated arbitrarily closely by a well-distributed Schottky group. This approximation not only validates the theoretical approach but also provides a practical method for computing the Hausdorff dimension of more general hyperbolic groups. The paper further extends these ideas to higher-dimensional hyperbolic spaces, opening up new avenues for studying Kleinian groups and their fractal limit sets.

Beyond its theoretical contributions, the explicit dimension formula has significant interdisciplinary implications. In mathematical physics, it connects the fractal geometry of limit sets with the spectral properties of hyperbolic manifolds. In cryptography, the computability of these fractal dimensions can be leveraged to design robust, quantum-resistant protocols. Moreover, the work’s insights into the Fourier decay properties of Patterson–Sullivan measures contribute to a deeper understanding of chaotic scattering and resonances in dynamical systems.

This comprehensive study not only deepens the theoretical understanding of fractal dimensions in hyperbolic geometry but also bridges abstract mathematical theory with practical computational techniques. The explicit formula for the Hausdorff dimension serves as a powerful tool for researchers in geometric group theory, dynamical systems, and related fields.

For a complete and rigorous exposition—including all derivations and proofs—please refer to the full document: Hausdorff Dimension of Well-Distributed Schottky Groups.

Simple Geodesics on Hyperbolic Surfaces: Theory and Applications

My recent note on simple geodesics explores various techniques for understanding geodesics on hyperbolic surfaces. For further details, see the full document Simple Geodesics on Hyperbolic Surfaces: Theory and Applications.

In this survey, we explore the fascinating interplay between number theory, geometry, and dynamical systems. To set the stage, we begin by recalling the classical Prime Number Theorem which describes the asymptotic distribution of prime numbers. This fundamental result motivates analogous asymptotic counting problems in geometry, such as the enumeration of closed geodesics on hyperbolic surfaces.

Several key works form the backbone of our approach. Mirzakhani's groundbreaking study established deep connections between the asymptotic growth of simple closed geodesics on hyperbolic surfaces and the geometry of moduli spaces, while Arana-Herrera provides a modern ergodic-theoretic perspective on counting problems ranging from primitive integer points to simple closed curves. Foundational background on surface topology and mapping class groups is supplied by Farb and Margalit’s A Primer on Mapping Class Groups as well as Martelli’s An Introduction to Geometric Topology. Comprehensive treatments of hyperbolic geometry and its spectral theory are available in Ratcliffe’s Foundations of Hyperbolic Manifolds, Borthwick’s Spectral Theory of Infinite-Area Hyperbolic Surfaces, and Dal’Bo’s work on geodesic and horocyclic trajectories. For additional background in measure theory and the geometry of numbers, see Cassels and Einsiedler--Ward.

References

Dal'Bo, F. (2011). Geodesic and horocyclic trajectories. Springer-Verlag London, Ltd. DOI: 10.1007/978-0-85729-073-1.
Ratcliffe, John. G. (2019). Foundations of Hyperbolic Manifolds. Springer. DOI: 10.1007/978-3-030-31597-9.
Borthwick, D. (2016). Spectral Theory of Infinite-Area Hyperbolic Surfaces. Birkhäuser/Springer. DOI: 10.1007/978-3-319-33877-4.
Martelli, B. (2016). An Introduction to Geometric Topology. arXiv:1610.02592.
Farb, B., & Margalit, D. (2012). A Primer on Mapping Class Groups. Princeton University Press.
Mirzakhani, M. (2004). Simple geodesics on hyperbolic surfaces and the volume of the moduli space of curves. Harvard University.
Arana-Herrera, F. (2022). Counting problems from the viewpoint of ergodic theory: from primitive integer points to simple closed curves. arXiv:2202.04156.