Mathematics / research
Elliptic operators and index theory
Connect analytic kernels to global topology through elliptic complexes, Fredholm indices, characteristic classes, and heat methods.
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Module 01 research
Elliptic operators and index theory
Connect analytic kernels to global topology through elliptic complexes, Fredholm indices, characteristic classes, and heat methods.
- 01
Elliptic symbols and Fredholm operators
Choose Sobolev realizations before relating an invertible principal symbol to a finite analytic index.
32 min - 02
Index stability and homotopy invariance
Track a Fredholm index under norm-continuous deformation while allowing kernel and cokernel dimensions to jump together.
31 min - 03
The de Rham complex and Euler characteristic
Roll an elliptic complex into a graded operator and compute its index from harmonic forms and product topology.
33 min - 04
Dirac operators and Clifford modules
Fix the Riemannian Clifford sign, principal symbol and chirality before forming a chiral index.
34 min - 05
Spin structures and spinor bundles
Separate orientation from the spin lifting obstruction and track how different lifts change spinor zero modes.
34 min - 06
Topological K-theory and symbol classes
Place an elliptic symbol in complex K-theory with compact support and normalize the Bott generator before computing winding examples.
35 min - 07
The Atiyah–Singer index formula
Identify analytic and topological indices and compute the appropriate characteristic pairing for spin and Dolbeault operators.
36 min - 08
Heat supertraces and local index density
Pair positive chiral spectra and distinguish a constant integrated supertrace from a generally varying pointwise heat kernel.
36 min - 09
Families index and parameter dependence
Replace jumping kernels by a stabilized finite-rank index bundle and retain topology invisible to the numerical index.
37 min - 10
Boundary operators and the Atiyah–Patodi–Singer index
Fix a product collar and spectral domain, then account for boundary spectral asymmetry and zero modes using reduced eta.
38 min