Abstract: Hodge structures are linear-algebraic invariants of non-linear objects (algebraic varieties over the complex numbers). The Hodge conjecture predicts that we can detect when two algebraic varieties are related, purely from this linear-algebraic data. An amazing instance of such a phenomenon is the Torelli theorem, which states that a K3 surface is uniquely determined, up to isomorphism, by the Hodge structure on its second cohomology group. Thus, moduli spaces of K3 surfaces can be understood as moduli spaces of Hodge structures, also known as Shimura varieties. But these spaces are not compact. I will discuss joint work with Alexeev, relating Hodge-theory and minimal model program approaches to their compactification.

Abstract: The Mandelbrot set M has been studied for many years but continues to baffle mathematicians. By definition, M is the set of all complex numbers c for which the orbit of 0 is bounded under iteration of f(z) = z^2 + c. Within M, there is a distinguished subset of "special" parameters, where the orbit of 0 is finite. These parameters are special from both a dynamical and - somewhat surprisingly - a number-theoretic point of view. In this talk, we'll explore how the geometry of these special parameters is controlled by the number theory, and how this relates to the big remaining open questions about M. This is joint work with Myrto Mavraki.

• Alexander Bertoloni Meli: abertolo@bu.edu
• Ryan Goh: rgoh@bu.edu
• Yu-Shen Lin: yslin@bu.edu
• Debhargya Mukherjee: mdeb@bu.edu
• Padmavathi Srinivasan: padmask@bu.edu