Mathematical Bootstrap: From Conformal Fields to L-Functions

The lecture provides a rigorous and insightful connection between two seemingly disparate fields: conformal field theory and number theory. The speaker, Dalimil Mazac, is a recognized expert, and the content is based on recent research papers, including his own work. The presentation is well-structured, starting with foundational material on hyperbolic surfaces and L-functions, then introducing the conformal bootstrap and its application to prove subconvex bounds. The mathematical rigor is high, with precise definitions and statements of theorems. The speaker is careful to distinguish between established results and conjectures, and he acknowledges prior work by others, such as Bonaf and the independent development by number theorists. The use of the bootstrap method to derive explicit bounds is a novel contribution, and the lecture highlights the potential for further cross-fertilization between physics and mathematics. The sources cited are arXiv preprints, which are appropriate for cutting-edge research, though they are not peer-reviewed. The lecture is technical and assumes a strong background in mathematics, but the speaker makes an effort to be pedagogical. The main strength is the clarity of the exposition and the depth of the content. The only minor weakness is that the lecture is quite long (over two hours), and some parts may be too detailed for a general audience. Overall, this is an excellent lecture that showcases the power of interdisciplinary approaches.

The title accurately reflects the content: the lecture connects conformal field theory (conformal bootstrap) to number theory (L-functions) through spectral theory of automorphic forms.

9/10

The lecture is given by a researcher at IPhT Saclay and IHES, presenting rigorous mathematical results with references to arXiv papers. The content is technical and precise, with clear definitions and proofs outlined. The speaker is an expert in the field, and the presentation is part of an institutional lecture series.

Markers derived by PSI from the transcript: the creator did not define chapters.

• 00:00Introduction: overview of the connection between CFT and automorphic forms, and the plan of the course.
• 05:00Review of hyperbolic surfaces: definition, hyperbolic plane, Fuchsian groups, and examples like PSL2Z and triangle groups.
• 30:00Discussion of the Laplacian spectrum on hyperbolic surfaces, including the spectral gap and Weyl law.
• 1:00:00Introduction to L-functions: definition, properties, and connection to automorphic forms.
• 1:30:00Explanation of the conformal bootstrap method and its adaptation to number theory.
• 2:00:00Presentation of the main result: subconvex bounds on triple product L-functions using bootstrap techniques.

• arXiv:2111.12716 — Paper by Mazac and collaborators on the connection between CFT and automorphic forms.
• arXiv:2308.11174 — Paper on spectral gaps of hyperbolic manifolds using bootstrap methods.
• arXiv:2508.20576 — Paper on subconvex bounds on L-functions using conformal bootstrap.
• Carmin.tv — Platform hosting the video and other scientific content.

• arXiv:2111.12716 — The paper by Mazac et al. establishes the analogy between CFT and automorphic forms, providing the foundation for the lecture.
• arXiv:2308.11174 — This paper applies bootstrap methods to spectral gaps, supporting the approach presented.
• arXiv:2508.20576 — The main result on subconvex bounds is detailed in this paper.