# Seminars & Events for 2011-2012

##### An Introduction to Knot Homologies

Knot homology theories associate to a knot or link a complex of graded modules whose graded Euler characteristic is a classical knot polynomial. This type of knot invariant has been increasingly influential in low-dimensional topology in the ten years or so since the first one was developed.

##### Perturbations of geodesic flows producing unbounded growth of energy

We consider a geodesic flow on a manifold endowed with some generic Riemannian metric. We couple the geodesic flow with a time-dependent potential driven by an external dynamical system, which is assumed to satisfy some recurrence condition. We prove that there exist orbits whose energy grows unboundedly at a linear rate with respect to time; this growth rate is optimal.

##### On some fully non-linear PDEs and their applications in Kaehler geometry

In this talk, we will discuss the so-called inverse $\sigma_k$ equation on Kahler manifolds. We will concentrate on examples of singular solutions and their application in Kahler geometry. Parts of the work are joint with Mijia Lai, Jian Song and Ben Weinkove.

##### Deforming $G_2$ conifolds.

A theorem of Dominic Joyce says that the moduli space of compact $G_2$ manifolds is smooth of dimension equal to the third Betti number of the manifold. We study the moduli space question for noncompact $G_2$ manifolds with one end, asymptotic to a metric cone of $G_2$ holonomy. This includes the explicit Bryant--Salamon manifolds as examples.

##### Moment estimates for square-free integers on short intervals

Square-free integers are known to have asymptotic density 6/(pi^2). Fix some x and let n be distributed uniformly on the integers between 1 and x. Consider the corresponding variance of the number of square-free integers on a short interval [n+1, n+N] and let x tend to infinity.