Calendar

In 1973, Lemmens and Seidel posed the problem of determining the maximum number of equiangular lines in R^r with angle arccos(alpha) and gave a partial answer in the regime r <= 1/alpha^2 – 2. At the other extreme where r is at least exponential in 1/alpha, recent breakthroughs have led to an almost complete resolution of this problem. In this talk, we introduce a new method for obtaining upper bounds which unifies and improves upon all previous approaches, thereby yielding bounds which bridge the gap between the aforementioned regimes and are best possible either exactly or up to a small multiplicative constant. A crucial new ingredient of our approach is orthogonal projection of matrices with respect to the Frobenius inner product and it also yields the first extension of the Alon-Boppana theorem to dense graphs, with equality for strongly regular graphs corresponding to r(r+1)/2 equiangular lines in R^r. Applications of our method in the complex setting will be discussed as well.

The fractional Dehn twist coefficient (FDTC) is an invariant of a self-map of a surface which is some measure of how the map twists near a boundary component of the surface. It has been studied for compact (or finite-type) surfaces; in this setting the invariant is always a fraction. I will discuss work to extend this invariant to infinite-type surfaces and show that it has surprising properties in this setting. In particular, the invariant no longer needs to be a fraction – any real number amount of twisting can be achieved! I will also discuss a new set of examples of (tame) big mapping classes called wagon wheel maps which exhibit irrational twisting behavior. This is joint work with Diana Hubbard and Peter Feller.