Cambridge

The strong cosmic censorship conjecture is a fundamental open problem in classical general relativity, first put forth by Roger Penrose in the early 70s. This is essentially the question of whether general relativity is a deterministic theory. Perhaps the most exciting arena where the validity of the conjecture is challenged is the interior of rotating black holes, and there has been a lot of work in the past 50 years in identifying mechanisms ensuring that at least some formulation of the conjecture be true. It turns out that when a nonzero cosmological constant Λ is added to the Einstein equations, these underlying mechanisms change in an unexpected way, and the validity of the conjecture depends on a detailed understanding of subtle aspects of black hole scattering theory, surprisingly involving, in the case of negative Λ, some number theory. Does strong cosmic censorship survive the challenge of non-zero Λ? This talk will try to address this Question!

In this talk I will present a new class of algorithms for separable nonlinear inverse problems based on inexact Krylov methods. In particular, I will focus in semi-blind deblurring applications. In this setting, inexactness stems from the uncertainty in the parameters defining the blur, which are computed throughout the iterations. After giving a brief overview of the theoretical properties of these methods, as well as strategies to monitor the amount of inexactness that can be tolerated, the performance of the algorithms will be shown through numerical examples. This is joint work with Silvia Gazzola (University of Bath).

I will describe joint work with Tyler Kelly and Ran Tessler. FJRW (Fan-Jarvis-Ruan-Witten) theory is an enumerative theory of quasi-homogeneous singularities, or alternatively, of Landau-Ginzburg models. It associates to a potential W:C^n -> C given by a quasi-homogeneous polynomial moduli spaces of (orbi-)curves of some genus and marked points along with some extra structure, and these moduli spaces carry virtual fundamental classes as constructed by Fan-Jarvis-Ruan. Here we specialize to the case W=x^r+y^s and construct an analogous enumerative theory for disks. We show that these open invariants provide perturbations of the potential W in such a way that mirror symmetry becomes manifest. Further, these invariants are dependent on certain choices of boundary conditions, but satisfy a beautiful wall-crossing formalism.

A loop of Hamiltonian diffeomorphisms of a symplectic manifold $X$ defines, by clutching, a symplectic fibration over the two-sphere with fibre $X$.  We prove that the integral cohomology of the total space splits additively, answering a question of McDuff, and extending the rational cohomology analogue proved by Lalonde-McDuff-Polterovich in the late 1990’s. The proof uses a virtual fundamental class of moduli spaces of sections of the fibration in Morava K-theory. This talk reports on joint work with Mohammed Abouzaid and Mark McLean.

An $r$-uniform tight cycle of length $k>r$ is a hypergraph with vertices $v_1,\ldots,v_k$ and edges $\{v_i,v_{i+1},…,v_{i+r-1}\}$ (for all $i$), with the indices taken modulo $k$. Sós, and independently Verstraëte, asked the following question: how many edges can there be in an $n$-vertex $r$-uniform hypergraph if it contains no tight cycles of any length? In this talk I will review some known results, and present recent progress on this problem.

Given a fixed poset $\mathcal P$, we say that a family $\mathcal F$ of subsets of $[n]$ is $\mathcal P$-free if it does not contain an (induced) copy of $\mathcal P$. And we say that $F$ is $\mathcal P$-saturated if it is maximal $\mathcal P$-free. How small can a $\mathcal P$-saturated family be? The smallest such size is the induced saturation number of $\mathcal P$, $\text{sat}^*(n, \mathcal P)$. Even for very small posets, the question of the growth speed of $\text{sat}^*(n,\mathcal P)$ seems to be hard. We present background on this problem and some recent results.

Ardakov and Wadsley developed a theory of D-modules on rigid analytic spaces and established a Beilinson-Bernstein style localisation theorem for coadmissible modules over the locally analytic distribution algebra. Using this theory, they obtained admissible locally analytic representations of SL_2 by taking global sections of Drinfeld line bundles. In this talk, we will extend their techniques to SL_3 by studying the Drinfeld upper half-space \Omega^{(3)} of dimension 2.

The injection of CO2 into porous subsurface reservoirs is a technological means for removing anthropogenic emissions, which relies on a series of complex porous flow properties. During injection of CO2 small-scale heterogeneities, often in the form of sedimentary layering, can play a significant role in focusing the flow of less viscous CO2 into high permeability pathways, with large-scale implications for the overall motion of the CO2 plume. In these settings, capillary forces between the CO2 and water preferentially rearrange CO2 into the most permeable layers (with larger pore space), and may accelerate plume migration by as much as 200%. Numerous factors affect overall plume acceleration, including the structure of the layering, the permeability contrast between layers, and the playoff between the capillary, gravitational and viscous forces that act upon the flow. However, despite the sensitivity of the flow to these heterogeneities, it is difficult to acquire detailed field measurements of the heterogeneities owing to the vast range of scales involved, presenting an outstanding challenge. As a first step towards tackling this uncertainty, we use a simple modelling approach, based on an upscaled thin-film equation, to create ensemble forecasts for many different types and arrangements of sedimentary layers. In this way, a suite of predictions can be made to elucidate the most likely scenarios for injection and the uncertainty associated with such predictions. 

Penrose's proposal of smooth conformal compactification is not only of geometric elegance, it also makes concrete predictions on physically measurable objects such as the "late-time tails" of gravitational waves.  At the same time, the physical motivation for a smooth null infinity remains itself unclear. In this talk, building on arguments due to Christodoulou, Damour and others, I will show that, in generic gravitational collapse, the "peeling property" of gravitational radiation is violated (so one cannot attach a smooth null infinity). Moreover, I will explain how this violation of peeling is in principle measurable in the form of leading-order deviations from the usual late-time tails of gravitational radiation.

This talk is based on https://arxiv.org/abs/2105.08079, https://arxiv.org/abs/2105.08084 and … .

It will be a hybrid seminar on both zoom and in-person in L5. 

A central theme of complex geometry is the relationship between differential-geometric PDEs and algebro-geometric notions of stability. Examples include Hermitian Yang-Mills connections and Kähler-Einstein metrics on the PDE side, and slope stability and K-stability on the algebro-geometric side. I will describe a general framework associating geometric PDEs on complex manifolds to notions of stability, and will sketch a proof showing that existence of solutions is equivalent to stability in a model case. The framework can be seen as an analogue in the setting of varieties of Bridgeland's stability conditions on triangulated categories.