Mathematics

Lecture 1: On counting solutions of polynomial systems

•   Bézout's theorem
•   Smale's 17-th problem
•   Shortcomings of Bézout's theorem
•   Sparse polynomial systems, and the mixed volume

Lecture 2: Differential forms

•   Multilinear algebra over R
•   Complex differential forms
•   Kähler geometry
•   The coarea formula, using bundles.
•   Projective space

Lecture 3: Reproducing kernel spaces

•   Spaces of complex fewnomials
•   Metric structure on root space
•   Unmixed systems
•   Random analytic functions

Lecture 4: Mixed systems of equations

•   Root density
•   Affine and multi-homogeneous setting
•   The semigroup of fewnomial spaces

Lecture 5: Exponential sums

•   Legendre transform
•   The momentum map
•   Proof of Bernstein's theorem