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)
Hausdorff Dimension of Well-Distributed Schottky Groups: A Summary
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 reff 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.
What I Learned Today
A daily update of the ideas and results I encountered today.
What I Learned Today?
My Notes on Semisimple Rings and Radicals in Coding Theory and Cryptography
Semisimple Rings and Radicals in Coding Theory and Cryptography
My Notes on Lifting Brauer Characters and Generalized Class Functions
Lifting Brauer Characters and Generalized Class Functions
My Notes on Decomposition Matrices and Projective Indecomposable Characters
Decomposition Matrices and Projective Indecomposable Characters
My Notes on Block Theory, Defect Groups, and Decomposition Matrix Structure
Block Theory, Defect Groups, and Decomposition Matrix Structure
My Notes on Representations and Characters of Finite Groups
Representations and Characters of Finite Groups
My Notes on Modular Representations and Brauer Characters
Modular Representations and Brauer Characters
My Notes on Irreducibility over Finite Fields and Frobenius Automorphisms
Irreducibility over Finite Fields and Frobenius Automorphisms
My Notes on Brauer Character Tables and Decomposition Matrices
Brauer Character Tables and Decomposition Matrices
My Notes on Wedderburn Decomposition and Group Algebras in Modular Settings
Wedderburn Decomposition and Group Algebras in Modular Settings
My Notes on Block Theory, Primitive Idempotents, and Defect Groups
Block Theory, Primitive Idempotents, and Defect Groups
My Notes on The Structure and Properties of Irr(G)
The Structure and Properties of Irr(G)
My Notes on Projective Modules and Projective Covers
Projective Modules and Projective Covers
My Notes on Using GAP for Modular Character Theory
Using GAP for Modular Character Theory
My Notes on Green Correspondence and Subgroup Relations
Green Correspondence and Subgroup Relations
My Notes on Synthesis, Advanced Examples, and Research Directions in Modular Character Theory
Synthesis, Advanced Examples, and Research Directions in Modular Character Theory
My Notes on The Set IBr(G) — Irreducible Brauer Characters
The Set IBr(G) — Irreducible Brauer Characters
My Notes on Modular Representations and Brauer Characters of A5
Modular Representations and Brauer Characters of A5
My Notes on Decomposition Matrices, GAP Computations, and Frobenius--Schur Indicators
Decomposition Matrices, GAP Computations, and Frobenius--Schur Indicators
My Notes on Irreducibility Criteria, Frobenius--Schur Analysis, and Modular Representation Implications
Irreducibility Criteria, Frobenius--Schur Analysis, and Modular Representation Implications
My Notes on Semisimple Modules, Jacobson Radicals, and Related Results
A brief note I wrote based on Prof. Lux's lecture.
Semisimple Modules, Jacobson Radicals, and Related Results
My Notes on Semisimple Algebras, Jacobson Radical, and Group Algebras
A brief note I wrote based on Prof. Lux's lecture.
Notes on Semisimple Algebras, Jacobson Radical, and Group Algebras
My Notes on Nakayama's Lemma, Jacobson Radicals, and Related Topics
A brief note I wrote based on Prof. Lux's lecture.
Notes on Nakayama's Lemma, Jacobson Radicals, and Related Topics
My Notes on Representation Theory, Character Theory, and Applications to Random Walks
A brief note I wrote based on Prof. Lux's lecture.
Introduction to Representation Theory, Character Theory, and Applications to Random Walks
My Notes on Random Walks and Representation Theory
A brief note I wrote based on Prof. Lux's lecture.
Lecture Notes on Random Walks and Representation Theory
My Notes on Probability and Representation Theory of Finite Groups
A brief note I wrote based on Prof. Lux's lecture.
Notes on Probability and Representation Theory of Finite Groups
My Notes on Finite Group Representations and Total Variation Distance
A brief note I wrote based on Prof. Lux's lecture.
Basic Notes on Finite Group Representations and Total Variation Distance
My Notes on Jacobson Radical in Artinian \(\mathbb{Z}\)-Algebras
Jacobson Radical in Artinian Z-Algebras-Nilpotency and Centrality
My Notes on Character Tables
A brief note I wrote based on Prof. Lux's lecture.
Lecture Notes on the Character Table of A5
My Notes on Nakayama's Lemma
A brief note I took based on Prof. Lux's lecture.
An Introduction to Nakayama's Lemma
My Notes on Double Coset Random Walks
A brief note I took based on a talk by Prof. Persi Diaconis.
My Notes on Double Coset Random Walks
What I Learned from March 2023 to March 2024
A summary of the topics and papers I studied from March 2023 to March 2024.
What I Learned from March 2023 to March 2024?
RTG Meeting Notes with Prof. Ning Hao
Notes and references from my presentations in RTG meetings.
2023 RTG Meetings
What I Learned in 2023 Summer
A summary of the topics and results I studied during the summer of 2023.
What I Learned in 2023 Summer?
Old Notes and Projects
A collection of previous notes and projects.
My Old Notes
Additional Resources