Abstracts and Schedule

Schedule:

Day 1 - 04/07/2022 - Monday

13:00 - 14:00 Claire Gilson

14:00 - 15:00 George Papamikos

15:00 -15:30 Coffee/Tea Break.

15:30 - 16:30 Rod Halburd

16:30 - 17:30 Benoit Vicedo

19:00 Reception and Dinner at The Green Room.

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Day 2 - 05/07/2022 - Tuesday

9:30 - 10:30 Maria Carmen Reguera

10:30 - 11:30 Murat Akman

11:30 - 12:00 Coffee/Tea Break

12:00 - 13:00 Andrew Morris

13:00 - 14:00 Linhan Li

All the talks and coffee/tea breaks will be at Lecture Theatre Building: LTB 9. This webpage would be useful to find your way on the campus https://findyourway.essex.ac.uk/.

The talks will also be broadcasted on zoom. Please contact the organizers for the zoom link.

Abstracts:

Murat Akman (University of Essex):

Title: Perturbation of elliptic operators on domains with rough boundaries.

Abstract: In this talk, we study perturbations of elliptic operators on domains with rough boundaries. In particular, we focus on the following problem: suppose that we have ``good estimates'' for the Dirichlet problem for a uniformly elliptic operator $L_0$ (with corresponding elliptic measure $\omega_{L_0}$), under what optimal conditions, are those good estimates transferred to the Dirichlet problem for uniformly elliptic operator $L$ (with corresponding elliptic measure $\omega_{L}$) which is a ``perturbation'' of $L_0$?

When the domain is 1-sided NTA satisfying the capacity density condition, we show that if the discrepancy of the corresponding matrices satisfies a natural Carleson measure condition with respect to $\omega_{L_0}$ then $\omega_L\in A_\infty(\omega_{L_0})$. Moreover, we obtain that $\omega_L\in RH_q(\omega_{L_0})$ for any given $1<q<\infty$ if the Carleson measure condition is assumed to hold with a sufficiently small constant.

This is a joint work with Steve Hofmann, Jose Maria Martell, and Tatiana Toro.

Claire Gilson (University of Glasgow):

Title: Non-Commutative Integrable Systems, Determinants and Pfaffians.

Abstract: There are a number of non-commutative integrable systems that have solutions in terms of quasi- determinants that are of so called Wronskian and Grammian type. One such system is the non-commutative KP equation.

In this talk I will look at the possibility of finding non-commutative integrable systems not of Wronskian and Grammian type but of Pfaffian type. For this I will review the properties of Pfaffians and quasi-determinants and show how you can define a non-commuting Pfaffian like object which we shall called a quasi-pfaffian. Like the normal quasi-determinants the quasi-pfaffians satisfy various identities, these identities could potentially be useful for finding equations with quasi-pfaffian solutions, however it appears that we might not need to use these identities. I will finish by considering a known integrable system with Pfaffian solutions first written down by Hirota and Ohta and, as a first step towards getting a non-commutative counterpart, recast this system in terms of quasi-pfaffians.

Rod Halburd (University College London):

Title: Finding meromorphic solutions of functional equations.

Abstract: We use global results about functions that are meromorphic in regions of the plane to find individual solutions of differential, difference and delay-differential equations whose only movable singularities are poles. We also allow for simple global branching. In this way we can find or describe subsets of solutions of equations that are in general non-integrable.

Linhan Li (University of Minnesota/University of Edinburgh):

Title: Quantitative properties of the Green function for elliptic operators with non-smooth coefficients.

Abstract: In the upper half-space, the distance function to the boundary is a positive solution to Laplace's equation that vanishes on the boundary, which can be interpreted as the Green function with pole at infinity for the Laplacian. We are interested in generalizing this result to a bigger class of operators and understanding the exact relations between the behavior of the Green function and the structure of the underlying operator. In joint work with G. David and S. Mayboroda, we obtain a precise and quantitative control of the proximity of the Green function and the distance function on the upper half-space by the oscillation of the coefficients of the operator. We shall see that the class of the operators that we consider is of the nature of the best possible for the Green function to behave like a distance function. We shall also discuss how this result can be generalized to other settings. If time permits, we shall discuss how this kind of result can be applied to derive optimal estimates for elliptic kernels.

Andrew Morris (University of Birmingham):

Title: A first-order approach to solvability for singular Schrödinger equations.

Abstract: We will first give a brief overview of the first-order approach to boundary value problems, which factorises second-order divergence-form equations into Cauchy-Riemann systems. The advantage is that the holomorphic functional calculus for such systems can provide semigroup solution operators in tremendous generality, extending classical harmonic measure and layer potential representations. We will then show how recent developments now allow for the incorporation of singular perturbations in the associated quadratic estimates. This allows us to solve Dirichlet and Neumann problems for Schrödinger equations with potentials in reverse Hölder spaces.

This is joint work with Andrew Turner.

George Papamikos (University of Essex):

Title: Multicomponent generalisations of known integrable equations and their Darboux transformations.

Abstract: I will present multicomponent generalisations of the modified KdV and sine-Gordon equations based on Lax operators over high rank Lie algebras with a prescribed reduction group. The corresponding Darboux transformation is constructed and the corresponding Backlund-dressing transformations are given. The inverse problem related to a Darboux transformation is also discussed and an integrable Volterra flow over the sphere is derived.

Maria Carmen Reguera (University of Birmingham):

Title: Quadratic sparse domination beyond the integral realm.

Abstract: We discuss quadratic sparse domination. A form of sparse domination that is suited for square functions. In this talk we will focus on building the theory for non-integral square functions. That includes square functions associated to non-elliptic PDEs.

This talk is based on joint work with Gianmarco Brocchi and Julian Bailey.

Benoit Vicedo (University of York):

Title: The Gaudin model from 3d mixed BF theory.

Abstract: Integrable systems are very special dynamical systems characterised by having a maximal set of integrals of motion. The Gaudin model provides an important and general class of finite-dimensional integrable systems. After reviewing the basic theory of finite-dimensional classical Hamiltonian integrable systems, focusing on the particular example of the classical Gaudin model, I will show how the latter can be obtained from the gauge fixing of a certain 3 dimensional gauge theory known as 3d mixed BF theory.

The talk will be based on the joint work arXiv:2201.07300 with J. Winstone.