2026-09-21T13:49:36Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/129410 2025-10-20 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

application/pdf en text Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-10-19 without embargo terms The student, Karthik Vasu, accepted the attached license on 2025-04-18 at 11:32. The student, Karthik Vasu, submitted this Dissertation for approval on 2025-04-18 at 11:36. This Dissertation was approved for publication on 2025-04-18 at 15:56. DSpace SAF Submission Ingestion Package generated from Vireo submission #21832 on 2025-10-19 at 18:18:26 A geometric description of the heat kernel of the Witten Laplacian and the Cheeger-Müller theorem Vasu, Karthik 2025-04-18 Albin, Pierre Dunfield, Nathan M Laugesen, Richard La Nave, Gabriele Cheeger Muller theorem Blow up heat kernels eng In 1935, Reidemeister introduced the Reidemeister torsion, an invariant on cochain complexes with fixed basis. These invariants were used to distinguish homotopy equivalent manifolds that are not homeomorphic. Ray and Singer in 1971, introduced an analytic analogue of the torsion on smooth manifolds, the Ray-Singer analytic torsion. They noticed these two torsions exhibit similar properties and conjectured that they must be equal. The conjecture was proved independently by Cheeger and Müller in 1979. Later in 1994, Bismut and Zhang introduced a new proof of the Cheeger-Müller theorem using the heat kernel of the Witten Laplacian. The proof was based on the study of the asymptotic expansion of the heat kernel. In this thesis, we provide a geometric description of the heat kernel of the Witten Laplacian by using Melrose’s blow up techniques. We construct a blown up heat space and show that the heat kernel is polyhomogeneous on it. We then use Getzler’s rescaling argument to get a explicit expression for the anomaly term when the representation is not unimodular. 2025-05 Thesis https://hdl.handle.net/2142/129410 Copyright 2025 Karthik Vasu Mathematics Mathematics University of Illinois Urbana-Champaign Ph.D. Dissertation