Michelle Chu - Salem numbers in arithmetic hyperbolic manifolds - IPAM at UCLA

Michelle Chu - Salem numbers in arithmetic hyperbolic manifolds - IPAM at UCLA

by Michelle Chu · View the channel on PSI

Formal & Physical Sciences Mathematics PBMathematicsPBHNumber theory
🎙 Michelle Chu 👥 44K 📅 October 12, 2026 ⏱ 47 min 👁 4 📄 original study 🧭 2026-10-12
Available in: English (current) Français

Keywords

Salem numberarithmetic manifoldgeodesic lengthlength spectrumcommensurability

Summary

In this talk, Michelle Chu presents recent results on Salem numbers and arithmetic hyperbolic manifolds. She begins by defining arithmetic hyperbolic manifolds using quadratic forms over number fields, explaining how they give rise to finite volume manifolds. She then introduces Salem numbers, real algebraic integers greater than 1 whose other conjugates lie on the unit circle, and connects them to lengths of closed geodesics via eigenvalues of hyperbolic elements. She discusses Lehmer’s conjecture and its link to the existence of a minimal Salem number, along with the equivalence to a uniform lower bound on systoles for arithmetic hyperbolic manifolds. The talk then focuses on the length spectrum and the rational length spectrum as commensurability invariants, citing known results for various dimensions. The main new theorems give explicit conditions, based on the arithmetic data of the quadratic form, to determine whether a given Salem number appears in the rational length spectrum of a commensurability class, and show that any admissible Salem number occurs in infinitely many pairwise incommensurable classes. The talk concludes with a brief discussion of open questions and a Q&A session.

182 words

Critical Evaluation

Value of the Information & Strength of the Argument

The information presented is valuable for researchers in hyperbolic geometry and arithmetic groups. The argumentation is rigorous, with precise definitions and clear logical progression. The talk builds on established results and presents new theorems with concrete conditions, supported by arguments based on the arithmetic structure of quadratic forms. The connection between Salem numbers and geodesic lengths is well-motivated through the eigenvalue of hyperbolic elements, and the discussion of Lehmer’s conjecture provides context for the significance of these numbers.

Scientific Rigor, Source Quality, Title Accuracy

The talk demonstrates scientific rigor by carefully defining terms and citing prior work (Emory–Radcliffe–Shance, Prasad–Rapinchuk, etc.) without giving full bibliographic details but acknowledging contributions. The sources mentioned are credible within the field. The title accurately matches the content. No comments were provided to analyze public reception.

139 words

Title / Content Match

The title accurately reflects the content: the talk focuses on Salem numbers and their appearance in arithmetic hyperbolic manifolds.

Quality & Reliability

8/10

The talk is rigorous, well-structured, and presents original research results at an advanced level. The speaker is a researcher at a reputable institution, and the content aligns with known literature. However, the presentation is dense and assumes prior knowledge, which may limit accessibility.

Key Moments

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

Cited Sources

Contribution & Novelties

The talk provides explicit arithmetic conditions that characterize when a Salem number appears in the rational length spectrum of a commensurability class of arithmetic hyperbolic manifolds, extending prior results. It also shows that any such Salem number can be realized in infinitely many pairwise incommensurable classes, a strong statement about the density of these numbers. The novel aspect is the explicit use of the quadratic form’s arithmetic data (splittings of the minimal polynomial, Hasse invariants) to determine membership.

Pour aller plus loin :

  • Salem number — Definition and properties of Salem numbers, central to the talk.
  • Lehmer’s conjecture — The conjecture on the gap in Mahler measure, related to the existence of a smallest Salem number.
  • Hyperbolic manifold — Background on hyperbolic manifolds, where closed geodesics correspond to hyperbolic elements. Moreover, the concept of length spectrum is relevant: see Length spectrum for a general overview.

145 words

Radar Profile

The profile shows high technical level and reliability, with slightly lower quantity due to the specialized focus, but still substantial content.

Reliability 9/10