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.

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.

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.

Einstein's equation E=mc^2 is probably the most famous equation in history. But what it really means and why is it true? In this lecture we shall explore the roots of this equation in geometry of spacetime and how experiment informs this geometry. Basic notions of mathematics like linearity, invariance and symmetry, and the notion of a group, play a big role here.

This talk is aimed at graduate students and is more or less an advertisement for the power of algebraic geometry. Two diverse theorems in completely different subjects will be unified in this talk. First we have the Galois correspondence in field theory. Second we have a topological theorem, the correspondence between covering spaces and subgroups of the fundamental group. Riemann's existence theorem, due to Grothendieck, is a kind of meta-theorem unifying these results. It leads to the notion of the etale fundamental group.

This talk will introduce the concept of orbifold as a generalization of that of manifold and some basic notions about groupoids. This alternate perspective will be helpful in answering some questions about orbifolds, as for example, what is a necessary condition for an orbifold to be a global quotient.

This talk is a report on ongoing joint work with M. Hassanzadeh and I. Shapiro, where we extend the definition of Hopf cyclic cohomology with coefficients in a Yetter-Drinfeld type module to braided monoidal categories and fusion categories. There are many algebraic structures, e.g. Drinfeld's quasi-Hopf algebras, that share some of the axioms of Hopf algebras, but not all of them. In many cases these structures are objects of a monoidal category. I shall introduce a class of monoidal categories where one can in fact define a cyclic module for its objects.