Tuesday, October 28 at 4:00 p.m.

After a general introduction to sums-of-squares problems, this lecture focuses on polynomials in two variables. In this setting, sums-of-squares questions are governed by Hodge theory applied equivariantly with respect to complex conjugation. Applications include the optimality of Pfister’s bound in dimension two, the Cassels–Ellison–Pfister theorem, and a density theorem for sums of three squares.

Wednesday, October 29 at 4:00 p.m.

This lecture explores the optimality of Pfister’s bound in dimension three and higher. In dimension three, the problem is closely related to existence questions for algebraic cycles in the spirit of the integral Hodge conjecture and the Griffiths–Harris conjecture. Using this cycle-theoretic perspective together with algebraic topology, the lecture explains why Pfister’s bound can always be improved for low-degree polynomials in any number of variables. Based on joint work with Olivier Wittenberg.

Thursday, October 30 at 2:15 p.m.

Is every nonnegative real-analytic function on ℝⁿ a sum of squares of real-analytic meromorphic functions? This analytic analogue of Hilbert’s 17th problem remains open. The lecture presents counterparts of the Artin and Pfister theorems under an additional compactness hypothesis, drawing on the algebraic geometry of Stein spaces.

Tuesday, October 8

The derived category of coherent sheaves began as a technical tool for sheaf cohomology but has become a powerful invariant of algebraic varieties. This introductory lecture surveys known and conjectural connections between derived categories and birational geometry, especially for Calabi–Yau and Fano varieties, and motivates a noncommutative enlargement of classical algebraic geometry. No previous knowledge of derived categories or birational geometry is assumed.

Wednesday, October 9

Hyperkähler varieties are one of the three building blocks for projective varieties with trivial canonical bundle. Their still-open classification appears closely related to the classification of noncommutative Calabi–Yau surfaces, whose Bridgeland-stable objects can produce hyperkähler moduli spaces. The lecture discusses this circle of ideas, including work with Arend Bayer, Laura Pertusi, and Xiaolei Zhao on varieties of Kummer type.

Thursday, October 10

The period–index conjecture proposes a precise bound for the dimension of a division algebra over a function field in terms of its order in the Brauer group. This lecture explains recent progress, including a result for unramified division algebras over the function field of an abelian threefold, based on joint work with James Hotchkiss. The approach recasts the problem through a noncommutative integral Hodge conjecture and uses enumerative geometry for noncommutative Calabi–Yau threefolds.

Tuesday, November 7 · Colloquium

Periods connect diophantine geometry, differential algebra, and algebraic geometry. The lecture introduces André’s generalization of the Grothendieck period conjecture, which also encompasses phenomena related to Schanuel’s conjecture, then turns to its geometric function-field analogue and its close relationship with Ax–Schanuel functional transcendence.

Wednesday, November 8

O-minimal geometry is a theory of tame functions with major applications in number theory and a natural role in Hodge theory. Although period domains are not algebraic, they carry definable o-minimal structures that often allow algebraicity arguments to be imitated. Applications include Griffiths’ algebraicity conjecture and relationships between periods in families and derivatives of their Hodge coordinates.

Thursday, November 9

This lecture explains the role of transcendence in unlikely-intersection problems, with an emphasis on geometric examples. One guiding question asks for the smallest genus of a curve contained in a very general abelian variety of dimension g. The lecture compares what is known over the complex numbers with the additional functional-transcendence techniques required over the algebraic closure of the rational numbers.