I this talk we will derive the first black hole solution to the Einstein field equations as was done by Karl Schwarzschild in 1915. We will discuss the strange properties of this spacetime with the help of new coordinate systems, Killing vectors, and Penrose diagrams. We will then present the Birkhoff and uniqueness theorems which tell us there are only four different black hole solutions in 4 dimensions. If time permits we will quickly review the casual structure of these other spacetimes.

The only thing I will be assuming for the talk is that everyone is reasonably familiar with differential geometry.

The theory of framed motives developed by Garkusha and Panin based on ideas by Voevodsky, gives a tool to construct fibrant replacements of spectra in A^1-homotopy category. In the talk we will discuss how this construction gives an identification of the motivic homotopy groups of the base field with its Milnor-Witt K-theory. In fact, this identification can be done in the same manner as the theorem of Suslin-Voevodsky which identifies motivic cohomology of the base field with Milnor K-theory.

The Newton diagram of a polynomial or analytic function is a powerful tool for studying its behaviour near a point. We introduce a "global version" of Newton diagram of a polynomial (or analytic function) f at a subvariety in order to study behaviour of f near generic points of the subvariety. We apply this notion to the "affine Bezout-problem" of counting number of isolated solutions (in C^n) of a system of n polynomials and show that it is possible to arrive at the exact count by a recursive formula which involves at each step mixed volume of the faces of these Newton-type diagrams with respect to various (possibly singular) "flags of subvarieties". This in particular is a natural extension of the Bernstein-Kushnirenko-Khovanskii approach to the affine Bezout-problem.

We will discuss a correspondence between $G$-graded algebras and $S^1$-gerbes on action groupoids.  This was motivated by a question asked during my last visit to Western.

We introduce fiber sequences and the Hopf fibration $S^3 \to S^2$ in HoTT, and we show that the Hopf fibration induces equivalences between the homotopy groups $\pi_k$ of $S^3$ and $S^2$ for $k>2$. We also deduce that $\pi_2(S^2)$ is equivalent to $Z$. We will review the background on truncations and connectedness in this talk too.