Mathematics / research
Spectral geometry and geometric analysis
Relate eigenvalues and heat flow to geometry through variational bounds, asymptotics, curvature, and inverse questions.
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Module 01 research
Spectral geometry and geometric analysis
Relate eigenvalues and heat flow to geometry through variational bounds, asymptotics, curvature, and inverse questions.
- 01
Laplace–Beltrami operators and self-adjointness
Fix the positive Laplacian, its energy form, and the domain that makes its spectrum meaningful.
29 min - 02
Rayleigh quotients and min–max principles
Translate trial functions and trial subspaces into bounds with the correct form domain and eigenvalue index.
30 min - 03
Dirichlet–Neumann bracketing
Use artificial interfaces to bracket spectra and reverse the comparison correctly for counting functions.
30 min - 04
Weyl law and eigenvalue counting
Recover dimension and volume from the high-energy count without treating an asymptotic as an exact finite formula.
31 min - 05
Heat-kernel asymptotics and geometric coefficients
Read volume and curvature from the short-time trace while keeping boundary and zero-mode corrections explicit.
33 min - 06
Cheeger constants and spectral gaps
Relate isoperimetric bottlenecks to the first positive eigenvalue without reversing a one-sided inequality.
32 min - 07
Eigenfunction nodal domains and Courant theorem
Count connected sign regions of real eigenfunctions with explicit multiplicity and zero-mode indexing.
32 min - 08
Trace formulas and closed geodesics
Interpret the wave trace as a distribution and qualify how closed-geodesic lengths enter its singularities.
34 min - 09
Isospectral manifolds and inverse spectral limits
Separate audible invariants from complete geometric recovery and state a qualified isospectral construction.
34 min - 10
Spectral determinants and zeta regularization
Remove zero modes, continue a spectral zeta function, and compute determinants without multiplying divergent infinite products.
35 min