Talk

It follows from the work of Artin (1927, 1958) and Hooley (1967) that, under assumption of the generalized Riemann hypothesis, every non-square rational number different from -1 is a primitive root modulo infinitely many primes. Moreover, the set of these primes has a natural density that can be written as the product of a `naive density' and a somewhat complicated correction factor reflecting the entanglement of the number fields that underly the density statement.

The problem of separation of prime numbers presentable as values of quadratic polynomials, for example, of primes having the form $n^2+1$, naturally leads to the  question on Euler product factorization of zeta-functions corresponding to theta-series with harmonic coefficients.

Let $G$ be the group of unipotent uppertriangular $n \times n$ matrices. For a matrix of this group, being totally positive means to have all its non-trivial minors positive (those which are non-vanishing in $\mathbb{C}[G]$). From an algorithmic point of view, it is interesting to find a subset of the set of the minors, seen as elements of $\mathbb{C}[G]$, which fully characterizes the total positivity. Such subsets, with only $\left( \begin{matrix} n \\ 2 \end{matrix} \right)$ minors, exist.

(Joint work with Shrawan Kumar and Nicolas Ressayre) It is an interesting open problem to connect the structure constants in the cohomology ring of homogeneous spaces (in the Schubert basis), and those in the invariant theory of groups (generalizing the two appearances of Littlewood Richardson coefficients: in the cohomology of Grassmannians and the invariant theory of $GL_n$). I will talk about a generalization of a conjecture of Fulton which is a step in this direction.