Abstracts of the lecture series:
Michel Brion, On automorphism groups of projective varieties
It is known that the automorphism group
of a projective algebraic variety is an extension of
a discrete group (the group of components) by a smooth
connected algebraic group (the connected automorphism
group). Very little is known on the group of components;
by contrast, the connected automorphism group can be
any prescribed smooth connected algebraic group. The lectures
will present old and new results on these automorphism groups,
in the setting of normal projective varieties over an
algebraically closed field.
Ana-Maria Castravet, Mori Dream Spaces and Blow-ups
I will start these lectures with an introduction to Mori Dream Spaces and explain the close connections with Hilbert's 14th problem.
Then I will present a sketch of the proof that the Grothendieck-Knudsen moduli space of stable rational curves with n markings is not a Mori Dream
Space when n is large, by reducing the question to a study of blow-ups of weighted projective planes. I will then discuss the latter problem and its
fascinating consequences in detail.
Zach Teitler, Waring Rank
The Waring rank of a homogeneous form is the number of terms needed to write the form as a sum of powers of linear forms. I will give an introduction and overview of some of the questions around Waring rank. First, motivated by conjectures in complexity theory, it is of interest to determine and compare the Waring ranks of the determinant and permanent. At this time very little is known; in fact, there are very few forms whose Waring rank is known. I will discuss some of the known lower bounds for Waring rank and approaches to determining Waring ranks. Second, the maximum value of Waring rank is unknown in all but a handful of cases; it is even unknown in most cases whether forms with higher than generic rank exist. Third, the symmetric analogue of Strassen's conjecture asserts that Waring rank is additive for forms in independent variables. This remains open, despite Shitov's recent counterexample for Strassen's conjecture in the tensor (non-symmetric) case. I will discuss some sufficient conditions for forms to satisfy the assertion of the conjecture.