This is a continuation of the talk from October 6. Details will be given of the proof that a Lagrangian surface $X\subset \mathbb C^2$ near an isolated singularity which is a Whitney umbrella is locally polynomially convex. In this talk I will discuss the connection between polynomial convexity of surfaces and their characteristic foliation.

The aim of the talk is to prove the unsolvability by radicals of the quintic (in fact of the general \(n^{\text{th}}\) degree equation for \(n\geq 5\)). This famous theorem was first proved by N. Abel and P. Ruffini around 1821. However, a complete understanding of \(\textit{solvability}\) had to wait for Evariste Galois and his introduction of group theory in a 1831 manuscript that was miraculously found by Liouville in 1843. We will present a proof of the Abel-Ruffini theorem, very close to Galois's own exposition that uses only elementary properties of groups, rings, and fields as they are taught in a first course in abstract algebra.

Geometric operators defined on a compact Riemannian manifold, e.g. Laplacian, Dirac, provide a framework in which we can investigate some geometric properties while we are completely working with algebra of operators on Hilbert spaces and commutators and spectral analysis of operators. In this setting we will have objects called spectral triples introduced by Alain Connes, which will play role of differential calculus on our (noncommutative) spaces.

A spectral triple is a triple (A,H,D) in which A is an involutive algebra (plays role of $C^\infty (M)) and H is Hilbert space on which A acts continuously (it is analogous of the space of the sections of vector bundle which D acts on) and D is an operator (it is our first order elliptic differential operator) which has some properties.

This talk is the first session of a series of talks in which we will investigate different properties and examples and objects related to spectral triples. The talk will start with definition of spectral triples and we shall go through classical examples to show where the ideas come from and at the end a spectral triple defined on NC-torus will be discussed.

A reasonable first approximation of a space X is its homology. Similarly, a first approximation to a functor F may be given by a homology theory, which is a particularly nice linear functor. This is the first stage of a Taylor tower of functors approximating F, developed by Goodwillie. One nice property of this tower is that within the ``radius of convergence'' of F, its value on a space X may be determined by the value of the limit of its Taylor tower on X. Functors of special interest are the identity functor of spaces and Waldhausen's Algebraic K-theory of a space, A(X). We will give an introduction to the calculus of functors, incorporating known results about these two functors.

I describe the theory of the quantum Hall effect in monolayer and bilayer graphene based on the magnetic catalysis effect. The role of the symmetry and its breakdown in this phenomenon is discussed.