Course Number:

Credit Hours:

4

Course Title:

Atiyah-Singer Index Theory

Course Description:

Introduces the Atiyah-Singer index theorem, one of the most impressive achievements of mathematics of the twentieth century. Connects analysis, geometry, and topology, and has numerous applications in mathematical physics, such as a calculation of the dimensions of moduli spaces of instantons. Topics include elliptic operators in sections of vector bundles, their index, and heat-kernel invariants; the Atiyah-Bott formula and local expression for the index; Chern-Weil construction of characteristic classes; and invariants of representations of orthogonal and unitary groups with applications to the heat kernel invariants of Laplacians. Also covers index formulas for classical elliptic operators and elliptic complexes (Gauss-Bonnet theorem, Hirzebruch signature theorem, the Riemann-Roch-Hirzebruch theorem, and Lefschetz-type theorems). Studies elements of K-theory and index theorem for general elliptic operators.