Nathan Geer - Non-Semisimple Skien TQFTs - IPAM at UCLA

Value of the Information & Strength of the Argument

The talk provides an accessible yet precise introduction to a cutting-edge topic. Geer clearly explains why non-semisimple categories require new tools, such as the modified trace, and demonstrates their power through the recovery of the Alexander polynomial. The argumentation is logical and builds step-by-step, connecting abstract categorical notions to concrete invariants. The inclusion of motivation (quantum computing, line operators, faithful mapping class group representations) strengthens the value of the presentation. The theoretical framework is well-supported by references to joint papers and ongoing collaborations.

Scientific Rigor, Source Quality, Title Accuracy

The presentation is rigorous, with careful definitions and proofs of key statements (e.g., vanishing of quantum trace, modified-trace properties). Geer explicitly cites his joint work and points to arXiv preprints. He also acknowledges uncertainty in some conventions, which shows scientific honesty. The title is accurate, though ‘Skien’ is a minor typo for ‘Skein’. No comments are provided, so public trends cannot be assessed; the talk appears designed for an expert audience at a workshop tutorial.

The title accurately reflects the content, which focuses on constructing TQFTs from non-semisimple categories. The typo 'Skien' likely should be 'Skein', but this does not affect the overall adequacy.

8/10

High-quality seminar by an expert, based on peer-reviewed joint work with explicit references. The presentation is rigorous but assumes advanced background, and some algebraic conventions (e.g., root-of-unity orders) are stated with caution. Overall reliable and well-structured.

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

• 01:00Introduction and thanks; motivation from quantum computing and a connection to line operators in earlier talks.
• 05:00Overview of non-semisimple invariants: recovery of Alexander polynomial vs Jones polynomial; potential relation to Floer homology.
• 10:00Setup of pivotal categories and the Reshetikhin–Turaev functor; definition of quantum trace and the vanishing property for projective objects.
• 20:00Introduction of the modified trace: properties of partial closure, independence of cut; example of modified dimension.
• 30:00Examples with quantum sl(2) at a root of unity: ADO invariant, calculation of zero quantum dimension, and link invariant properties.
• 40:00Transition to TQFT construction: ingredients, admissible skein modules, and the role of spherical tensor categories.
• 1:00:00Summary, acknowledgments, and discussion of open problems; joint work and future directions.

• IPAM Workshop Schedule — Provides details of the tutorial program and the talk's context.
• arXiv:2302.04509 — Joint work mentioned in the description; likely the main paper on non-semisimple TQFTs.
• arXiv:2306.03225 — Second joint paper cited for the construction of admissible skein modules.

• arXiv:2302.04509 — Independently states the construction details and results presented in the talk.