Oxford

One knows, for example by proving well-posedness for an initial value problem with data at the singularity, that there exist many cosmological solutions of the Einstein equations with an initial curvature singularity but for which the conformal metric can be extended through the singularity. Here we consider a converse, a local extension problem for the conformal structure: given an incomplete causal curve terminating at a curvature singularity, when can one extend the conformal structure to a set containing a neighbourhood of a final segment of the curve?

We obtain necessary and sufficient conditions based on boundedness of tractor curvature components. (Based on work with Christian Luebbe: arXiv:0710.5552, arXiv:0710.5723.)

The complement of a knot or link is a 3-manifold which admits a geometric

structure. However, given a diagram of a knot or link, it seems to be a

difficult problem to determine geometric information about the link

complement. The volume is one piece of geometric information. For large

classes of knots and links with complement admitting a hyperbolic

structure, we show the volume of the link complement is bounded by the

number of twist regions of a diagram. We prove this result for a large

collection of knots and links using a theorem that estimates the change in

volume under Dehn filling. This is joint work with Effie Kalfagianni and

David Futer

This is the second of two talks, and probably will not be comprehensible unless you came to last week's talk.

A Kuranishi space is a topological space equipped with a Kuranishi structure, defined by Fukaya and Ono. Kuranishi structures occur naturally on many moduli spaces in differential geometry, and in particular, in moduli spaces of stable $J$-holomorphic curves in symplectic geometry.

Let $Y$ be an orbifold, and $R$ a commutative ring. We define four topological invariants of $Y$: two kinds of Kuranishi bordism ring $KB_*(Y;R)$, and two kinds of Kuranishi homology ring $KH_*(Y;R)$. Roughly speaking, they are spanned over $R$ by isomorphism classes $[X,f]$ with various choices of relations, where $X$ is a compact oriented Kuranishi space, which is without boundary for bordism and with boundary and corners for homology, and $f:X\rightarrow Y$ is a strong submersion. These theories are powerful tools in symplectic geometry.

Today we discuss the definition of Kuranishi homology, and the proof that weak Kuranishi homology is isomorphic to the singular homology.

A Kuranishi space is a topological space equipped with a Kuranishi structure, defined by Fukaya and Ono. Kuranishi structures occur naturally on many moduli spaces in differential geometry, and in particular, in moduli spaces of stable $J$-holomorphic curves in symplectic geometry.

Let $Y$ be an orbifold, and $R$ a commutative ring. We shall define four topological invariants of $Y$: two kinds of Kuranishi bordism ring $KB_*(Y;R)$, and two kinds of Kuranishi homology ring $KH_*(Y;R)$. Roughly speaking, they are spanned over $R$ by isomorphism classes $[X,f]$ with various choices of relations, where $X$ is a compact oriented Kuranishi space, which is without boundary for bordism and with boundary and corners for homology, and $f:X\rightarrow Y$ is a strong submersion. Our main result is that weak Kuranishi homology is isomorphic to the singular homology of $Y$.

These theories are powerful tools in symplectic geometry for several reasons. Firstly, using them eliminates the issues of virtual cycles and perturbation of moduli spaces, yielding technical simplifications. Secondly, as $KB_*,KH_*(Y;R)$ are very large, invariants defined in these groups contain more information than invariants in conventional homology. Thirdly, we can define Gromov-Witten type invariants in Kuranishi bordism or homology groups over $\mathbb Z$, not just $\mathbb Q$, so they can be used to study the integrality properties of Gromov-Witten invariants.

This is the first of two talks. Today we deal with motivation from symplectic geometry, and Kuranishi bordism. Next week's talk discusses Kuranishi homology.

The convergence of Markov processes to stationary distributions is a basic topic of introductory courses in stochastic processes, and the theory has been thoroughly developed. What happens when we add killing to the process? The process as such will not converge in distribution, but the survivors may; that is, the distribution of the process, conditioned on survival up to time t, converges to a "quasistationary distribution" as t goes to infinity.

This talk presents recent work with Steve Evans, proving an analogue of the transience-recurrence dichotomy for killed one-dimensional diffusions. Under fairly general conditions, a killed one-dimensional diffusion conditioned to have survived up to time t either escapes to infinity almost surely (meaning that the probability of finding it in any bounded set goes to 0) or it converges to the quasistationary distribution, whose density is given by the top eigenfunction of the adjoint generator.

These theorems arose in solving part of a longstanding problem in biological theories of ageing, and then turned out to play a key role in a very different problem in population biology, the effect of unequal damage inheritance on population growth rates.