Tuesday, September 17, 2019
09/17/2019 - 4:15pm
Abstract: A fundamental principle in number theory is that the cohomology of locally symmetric spaces carries interesting Galois representations. However, locally symmetric spaces are also associated with other kinds of algebraic invariants, such as algebraic K-theory, and one could also try to study the Galois action on them. In joint work with Soren Galatius and Akshay Venkatesh, we define a symplectic version of algebraic K-theory for the integers, and compute the Galois action on it. The proofs use the fundamental theorem of complex multiplication and the fundamental theorem of hermitian K-theory.
09/17/2019 - 4:15pm
The marked length spectrum of a metric on a compact Riemannian manifold records the length of the shortest closed curve in each free homotopy class. It is known that a negatively curved metric on a compact Riemannian manifold is uniquely determined by its marked length spectrum up to isometry. My preliminary results show that under certain conditions on the excluded homotopy classes, a partial marked length spectrum also uniquely determines the metric.
09/17/2019 - 6:00pm
Abstract: Take a polynomial of degree n having all its roots on the real axis. Suppose n is really large and that those roots are distributed roughly like, say, a Gaussian. If you differentiate this polynomial n/2 times, the result is a polynomial of degree n/2 having n/2 roots on the real line – what can be said about the distribution of these n/2 roots? Do they still look like a Gaussian? An old conjecture, dating back at least to Polya, is that they become more regular. Even though this is a question about polynomials, extremely little is known. I am describing the state of the art and a recently discovered dynamical system that seems to be able to predict the evolution. An old idea of Gauss plays a role; in the end, there are surprising connections to fluid dynamics. Do roots of polynomials flow like a fluid? The entire talk is elementary (we know, nothing, really – what we can prove requires only high school algebra and calculus) and there are many open research problems.