I will present a sufficient condition, expressed in terms of the condition numbers of underlying matrices, for a product of positive definite quadratic forms to be convex. The condition is weaker than previously known sufficient conditions, and is also necessary in the case of a product of two forms.

I introduce an equivalence of norms on Banach spaces that is finer than the usual one. In addition to generating the same topology, angularly equivalent norms share certain geometric properties. Equivalent Hilbert space norms are always angularly equivalent, with the constants of equivalence related to the condition number of the matrix relating their inner products

I'll discuss the notion of a model of dependent type theory. After outlining the general algebraic semantics, I'll show how such models can be constructed from categories that we encounter in the so-called "mathematical practice".

Ever since their first description in the 1965 PhD thesis of Bruno Buchberger, Gr$\mathrm{\ddot{o}}$bner bases have been an important tool for computational algebra. We can view Gr$\mathrm{\ddot{o}}$bner bases as a nonlinear version of Gaussian Elimination or a multivariate version of Euclid's Algorithm. They allow to answer problems in ideal theory, polynomial system solving, algebraic geometry, homological algebra, graph theory, diophantine equations and many other areas.

In this talk we will discuss the mathematical definition of Gr$\mathrm{\ddot{o}}$bner bases of polynomial ideals, computation of Gr$\mathrm{\ddot{o}}$bner bases with Buchberger's algorithm, conversion of Gr$\mathrm{\ddot{o}}$bner bases using the FGLM algorithm and the Gr$\mathrm{\ddot{o}}$bner walk, generalisations of Gr$\mathrm{\ddot{o}}$bner bases beyond commutative polynomials, and a selected number of applications including ideal comparison as well as symbolic summation.