Extremal and typical results in Real Algebraic Geometry

In the first part of the dissertation we show that $2((d-1)^n-1)/(d-2)$ is the maximum possible number of critical points that a generic $(n-1)$-dimensional spherical harmonic of degree $d$ can have. Our result in particular shows that there exist generic real symmetric tensors whose all eigenvectors are real. The results of this part are contained in Chapter \ref{ch:harmonics}. In the second part of the thesis we are interested in expected outcomes in three different problems of probabilistic real algebraic and differential geometry. First, in Chapter \ref{ch:discriminant} we compute the volume of the projective variety $\Delta\subset \txt{P}\txt{Sym}(n,\K{R})$ of real symmetric matrices with repeated eigenvalues. Our computation implies that the expected number of real symmetric matrices with repeated eigenvalues in a uniformly distributed projective $2$-plane $L\subset \txt{P}\txt{Sym}(n,\K{R})$ equals $\mean\#(\Delta\cap L) = {n\choose 2}$. The sharp upper bound on the number of matrices in the intersection $\Delta\cap L$ of $\Delta$ with a generic projective $2$-plane $L$ is ${n+1 \choose 3}$. Second, in Chapter \ref{ch:pevp} we provide explicit formulas for the expected condition number for the polynomial eigenvalue problem defined by matrices drawn from various Gaussian matrix ensembles. Finally, in Chapter \ref{ch:tangents} we are interested in the expected number of lines that are simultaneously tangent to the boundaries of several convex sets randomly positioned in the sphere. We express this number in terms of the integral mean curvatures of the boundaries of the convex sets.