Abstract: This week in NCG seminar: ---Miraculous cancellations, Getzler's supersymmetric proof of the Atiyah-Singer index theorem, special cases: Gauss-Bonnet-Chern, Hirzebruch signature theorem, and Riemann-Roch, --- Approach via path integrals and quantum mechanics.

Abstract: The inverse Galois problem is a major question in mathematics. For a given base field and a given finite group G, one would like to list all Galois extensions L=F such that the Galois group of L=F is G. In this work we shall solve this problem for all fields F, and for group G of unipotent 4 x 4 matrices over F2. We also list all 16 U4(F2)- extensions of Q2. The importance of these results is that they answer the inverse Galois problem in some specific cases. This is joint work with Jan Minac and Nguyen Duy Tan.

Abstract: Toeplitz operators are linear operators that act on spaces of holomorphic sections of powers of a hermitian holomorphic line bundle on a Kaehler manifold. When the Kaehler manifold (say, $M$) is a compact smooth quotient of an irreducible bounded symmetric domain $D$, holomorphic sections of powers of the canonical bundle on $M$ are in correspondence with holomorphic automorphic forms on $D$. This will be mostly review. I will also mention some recent results, including several results from joint work in progress with N. Alluhaibi.