It is planned to present certain necessary and sufficient conditions for approximability of functions on an arbitrary plane compact set by functions, bianalytic on neighborhoods of the set, in the norms of Whitney spaces $C^m$ for all $m \geq 0$. The formulated criteria are given in the so-called individual form and for each $m$ they are formulated in terms of geometric measure theory.

See you next week.

This is a quick introduction to gauge theory as a classical field theory of elementary particles. Ideas of symmetry breaking and Higgs mechanism will be emphasized.

I shall define the Fredholm index (resp. Euler characteristic) of an elliptic PDE (res. elliptic complex) acting on smooth sections of bundles over a closed manifold. This is the analytic index map. I shall also sketch a proof of the fact, due to Atiyah and Jaenich, that the space of Fredholm operators is a classifying space for K-theory. Time permitting, I shall indicate the classifying space of odd K-theory (space of self adjoint Fredholm operators and spectral flow). Basic tools of functional analysis like compact operators and pseudodifferential calculus will be recalled.

It is planned to present certain necessary and sufficient conditions for approximability of functions on an arbitrary plane compact set by functions, bianalytic on neighborhoods of the set, in the norms of Whitney spaces $C^m$ for all $m \geq 0$. The formulated criteria are given in the so-called individual form and for each $m$ they are formulated in terms of geometric measure theory.

This is joint work with J$\mathrm{\acute{a}}$n Min$\mathrm{\acute{a}\check{c}}$. Let $F$ be a field. We show that any triple Massey product in Galois cohomology of $F$ with coefficient $\mathbb{F}_2$ contains 0 whenever it is defined. This extends a recent theorem of M. J. Hopkins and K. G. Wickelgren, from a global field to an arbitrary field.