Given a (real or complex) analytic map $f=(f_1,\dots,f_n):X\to\mathbb{K}^n$, one can consider its approximations by maps $T_d(f)$ whose coordinates are Taylor polynomials of the $f_i$ of degree $d$. We will show that the continuity of the family of fibres of $f$ is finitely determined. That is, it is already determined by the polynomial maps $T_d(f)$ for $d$ sufficiently large. As a consequence, we also obtain finite determinacy of analytic complete intersections. This is joint work with H. Seyedinejad.

In this talk, we will discuss Noncommutative Poisson Structure which was introduced by Crawley-Boevey and how it fits nicely with Kontsevich-Rosenberg principle. We will also give some examples. If time allows, we will also discuss its relation to Van den Bergh's Double Poisson Algebras.

The spectral action principle was introduced by Connes and Chamseddine to formulate the notion of action for spectral triples. This talk will be an introduction to this principle and to the so called spectral action functional. After a quick review of the theory of Dirac operators and spectral triples, we will show how one can produce classical actions, such as Einstein-Hilbert or Yang-Mills action, by considering the spectral action functional for a suitable Dirac operator.

I will sketch Adams' original proof of the Hopf invariant one problem.

We investigate the rationality of four-dimensional algebraic tori and the associated equivariant birational linearisation problem. We connect these problems to the question of determining when an algebraic group is (stably) Cayley - that is (stably) equivariantly birationally isomorphic to its Lie algebra and earlier joint work on the (stably) Cayley problem with Popov, Reichstein, Borovoi and Kunyavskii.

A formula that expresses a Laplace-type operator, i.e an operator with the metric tensor as their leading symbol, as sum of the Laplacian of a connection and an endomorphism, is called Bochner-type formula. The goal of this talk is to review the general form of Bochner type formulas and focus on some specific examples from the spin geometry, such as Lichnerowicz formula for Dirac and twisted Dirac operators.