# Registration

Registration is required for attendance (for catering and other purposes)!!!

Please register before 23:59 on 06/03/2023 by emailing lashi.bandara@brunel.ac.uk

# What, Why, Who...

What?

This is a one day conference around global analysis, with a particular emphasis on methods from modern harmonic analysis, geometry, mathematical physics and their interaction.

Why?

Global analysis sits on the intersection of geometry, operator theory and analysis. Many of the questions are encoded using differential operators on vector bundles, which then are the objects of study typically using methods from analysis. These problems themselves often arise from mathematical physics.

In the recent past, methods from real-variable harmonic analysis have found a footing in global analysis. Coupled with the bounded holomorphic functional calculus, they have allowed for the resolution of problems where there is non-compactness and non-smoothness.

Who?

The aim of this meeting is to bring together participants who have interest in global analysis and related areas.

This event is open to everyone and in particular, PhD students and postdocs are welcome and invited to attend the event. Please remember to register for this event by emailing lashi.bandara@brunel.ac.uk before 06/03/2023.

There will be four talks given in total, with the host (Lashi Bandara) being one of the speakers, along with invited talks by:

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# Event info!

Event location

The talks will be held in the Eastern Gateway Building, Room 003a.

Timetable

 10:00-10:30 Coffee (Eastern Gateway Building) 10:30-11:30 Manifolds, Operators, Estimates Lashi Bandara 11:30-13:00 Lunch 13:00-14:00 New progress on solvability of Regularity problem for elliptic operators with coefficients satisfying large Carleson condition Martin Dindoš 14:00-15:00 Coffee (Eastern Gateway Building) 15:00-16:00 Global stability of spacetimes with supersymmetric compactifications Zoe Wyatt 16:00-17:00 Algebraic and analytic aspects of renormalisation Kasia Rejzner 17:00-18:30 Social event (Tower A, Room 203) 19:00- Dinner (TBA)

Talks

• Manifolds, Operators, Estimates

Lashi Bandara

Broadly speaking, global analysis is the term used to describe the way geometric problems can be formulated, understood, and resolved through the use of differential operators and mathematical analysis. More often that not, these geometric problems arise from physical situations, such as the study of the electron or other fundamental particle in the presence of gravity.

I will discuss a particular aspect: when we consider a differential operator acting between sections of vector bundles over a manifold with boundary. In this case, the operator has two distinguished extensions, the maximal and the minimal, with the latter contained in the former. The study of all possible extensions of the minimal operator contained in the maximal one is equivalent to the study of boundary conditions. Obtaining trace theorems provide a tangible two way road between these extensions and boundary conditions. My ambition will be to relate this example to the areas of the three speakers in the afternoon.

• New progress on solvability of Regularity problem for elliptic operators with coefficients satisfying large Carleson condition

Martin Dindoš

Recall that when $Lu=\mathrm{div}(A\nabla u)$ and coefficients A satisfy Carleson condition (that is $|\nabla A|^2\delta$ is a Carleson measure) then the corresponding elliptic measure is $A_\infty$ and hence the $L^p$ Dirichlet problem is solvable for some large $p < \infty$. There is also an analogous smallness result, namely that on $C^1$ domains $L^p$ Dirichlet problem is solvable for all $1<p<\infty$, provided $|\nabla A|^2\delta$ is a vanishing Carleson measure.

In the case of Regularity problem, the analogous smallness result was established in Dindos-Pipher-Rule. We have recently (jointly with S. Hofmann and J. Pipher) established an n-dimensional reduction that relates solvability of Regularity problem to the solvability of Regularity problem for a block-form operator.

This reduction has allowed us to fully resolve the question of solvability of Regularity problem with coefficients satisfying large Carleson condition in all dimensions and also the Neumann problem in dimension 2 in the interval for $1<p<1+\epsilon$.

Remarkably, Mourgoglou, Poggi and Tolsa have considered the same question from a different perspective using new ideas that allow extrapolation of the small Carleson result of Dindos-Pipher-Rule using improved version of Mourgoglou Tolsa decomposition of domains to Lipschitz subdomains (developed for originally for the Regularity problem for Laplacian). This method allows to consider the Regularity problem for more general domains (beyond Lipschitz). I compare the two methods and outline possible further open questions that might be addressed thanks to these exciting developments.

• Algebraic and analytic aspects of renormalisation

Kasia Rejzner

In this talk I will discuss recent progress in understanding mathematical foundations of renormalization. The approach I'm going to focus on is Epstein-Glaser renormalisation combined with ideas of local covariance, to allow for construction of quantum field theory models on curved spacetimes. The resulting framework is called perturbative algebraic quantum field theory (pAQFT).

• Global stability of spacetimes with supersymmetric compactifications

Zoe Wyatt

Spacetimes with compact directions which have special holonomy, such as Calabi-Yau spaces, play an important role in supergravity and string theory. In this talk I will discuss a work with Andersson, Blue and Yau, where we show the global, nonlinear stability a spacetime which is a cartesian product of a high dimensional Minkowski space with a compact Ricci flat internal space with special holonomy. This stability result is related to a conjecture of Penrose concerning the validity of string theory.

# Funding and acknowledgements

This event is funded by the London Mathematical Society through their Celebrating New Appointments Grant Scheme as well as the Department of Mathematics, Brunel University London