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.

Gravitational instantons are non-compact Calabi-Yau metrics with L^2 bounded curvature and are categorized into six types. We will discuss one such type called ALH* metrics which has a non-compact end modelled by the Calabi ansatz with inhomogeneous collapsing near infinity. Such metrics  appeared recently in the works on SYZ conjecture, as well as the scaling bubble limits for codimension-3 collapsing of K3 surfaces, where the study of its Laplacian played a central role. In this talk I will talk about the Fredholm mapping property and L^2 cohomology of such metrics. This is ongoing work joint with Rafe Mazzeo.

Zoom: https://harvard.zoom.us/j/7855806609

Password: cmsa

In characteristic zero, Ueno’s conjecture states that if X is a smooth projective variety with Kodaira dimension zero, then its Albanese morphism in an algebraic fiber space and the Kodaira dimension of the general fiber is again zero. This was proven by Cao and Păun in 2016.

Building on the generic vanishing techniques of Hacon and Patakfalvi, we prove a positive characteristic version of this result. We use it to deduce new cases of Iitaka’s subadditivity conjecture in positive characteristics. The goal of this talk is to explain how these techniques work, and how we can use them to prove such results.

For more information, please see https://researchseminars.org/seminar/harvard-mit-ag-seminar