We will construct Buser's example of two Schreier graphs that are isospectral but not isomprphic. The proof of this has a wonderful connection to representation theory, and some useful pre-trace formulae will be reviewed. We will then start the construction of isospectral, non-isometric planar domains that will be concluded in part 5 of this series of talks.

Originally aimed at understanding the relationship between classical and quantum mechanics, geometric quantization is a construction whose ideas originated in representation theory, in the work of Kirillov, Kostant and Souriau during the late 1960's. From the symplectic geometry perspective, geometric quantization produces representations of compact Lie groups starting from `geometric' (i.e. Hamiltonian) group actions of such Lie groups. Some ideas surrounding this construction will be discussed, including their adaptation to the theory of 'group-valued' moment maps, which is a finite-dimensional model of the theory of Hamiltonian loop group actions. Applications in this context (re)produce so-called Verlinde formulas of conformal field theory for simply connected compact Lie groups. For non-simply connected Lie groups, Verlinde formulas have been conjectured, and this approach verifies the conjectured formulas for $G=SO(3)$.

This is joint work with Jiri Lebl and Liz Vivas. We prove that the rank of a Hermitian form on the space of holomorphic polynomials can be bounded by a constant depending only on the maximum rank of the form restricted to affine manifolds. As an application we prove a result along the lines of the Baouendi-Huang and Baouendi-Ebenfelt-Huang rigidity theorems for CR mappings between hyperquadrics. If we have a real-analytic CR mapping of a hyperquadric not equivalent to a sphere to another hyperquadric Q(A,B), then either the image of the mapping is contained in a complex affine subspace or A is bounded by a constant depending only on B.

The irreducible (real) Clifford modules play a very important role in the theory of Dirac operators. The obstruction for the existence of such a module in real and complex case is different. For the real one, vector bundle should have vanishing second Stiefel-Whitney class, however, in the complex case the second Stiefel-Whitney class only needs to be mod 2 reduction of an integral class, equivalently, the vector bundle admits a $spin^c$ structure. In this talk we will examine the obstruction for the existence of $spin^c$ structure, then the construction of complex spinor bundle and the $spin^c$ connection on it and finally Dirac operator on the spinor fields will be discussed.

The Grothendieck ring of varieties over a field is a bookkeeping device for invariants of varieties which preserve the relation [X] = [Z]+[X-Z] whenever Z is a closed subvariety of X. Examples of such invariants include counting points if the field in question is finite, or the topological Euler characteristic if the field is the complex numbers. After introducing the Grothendieck ring and some invariants, I will discuss a certain invariant which involves the A^1-homotopy type of Morel and Voevodsky.

What do the 'picture hanging problem' and 'Borromean rings' have in common? Their solution can be described by Milnor invariants in link theory, or equivalently by higher cohomological Massey products.

As noticed by B. Mazur, M. Morishita et al, there is a remarkable analogy between the theory of links and pro-$p$-extensions of number fields with ramification restricted to a finite set of primes. We discuss this analogy and give an arithmetic interpretation of Massey products in low degrees. It turns out that certain symmetry relations in the topological world carry over to number theory in special cases only.

We report on the work on applications of higher Massey products in order to construct so-called mild pro-$p$-groups and investigate recent progress in the theory of tamely ramified pro-$p$-extensions by J. Labute and A. Schmidt.

In this talk, we will show how to obtain mild pro-$p$-groups in the arithmetic context.

This talk is dedicated to the proof of Bott periodicity. First we generalize the clutching function to vector bundles over $X \times S^2$, then we can simplify the clutching function, first to a Laurent polynomial, then to a polynomial, finally to a linear function. And by the discussion on the linear clutching function, we can finally decompose it into a vector bundle with trivial clutching function and another one with clutching function $z$. Finally, we can construct a inverse by the simplified clutching function for the external product to prove that it is an isomorphism.