The Riemann-Hilbert method: from Toeplitz …

1 The Riemann-Hilbert method: from Toeplitz operators to black holesMaria Cristina C maraCAMGSD-Instituto Superior T cnicoEncontro de Ci ncia 2017 Lisboa, 3-5 JulhoThe Riemann-Hilbert method: from Toeplitz operators to black holesMaria Cristina C maraWhat is a Riemann-Hilbert problem?Problem: To determine such that =0 inD, continuous onT =2fonT(1) = ++ +2, +analytic inD(|z|<1)(2)Let (z) = +(1/ z),|z|>1 ; analytic inC\closD, (t) = +(t)for|t|=1. +(t) + (t) =f(t),t T += f,onTg += f,onTThe Riemann-Hilbert method: from Toeplitz operators to black holesMaria Cristina C maraWhat is a Riemann-Hilbert problem?

2 Problem: To determine such that =0 inD, continuous onT =2fonT(1) = ++ +2, +analytic inD(|z|<1)(2)Let (z) = +(1/ z),|z|>1 ; analytic inC\closD, (t) = +(t)for|t|=1. +(t) + (t) =f(t),t T += f,onTg += f,onTThe Riemann-Hilbert method: from Toeplitz operators to black holesMaria Cristina C maraWhat is a Riemann-Hilbert problem?Problem: To determine such that =0 inD, continuous onT =2fonT(1) = ++ +2, +analytic inD(|z|<1)(2)Let (z) = +(1/ z),|z|>1 ; analytic inC\closD, (t) = +(t)for|t|=1. +(t) + (t) =f(t),t T += f,onTg += f,onTThe Riemann-Hilbert method: from Toeplitz operators to black holesMaria Cristina C maraWhat is a Riemann-Hilbert problem?

3 Problem: To determine such that =0 inD, continuous onT =2fonT(1) = ++ +2, +analytic inD(|z|<1)(2)Let (z) = +(1/ z),|z|>1 ; analytic inC\closD, (t) = +(t)for|t|=1. +(t) + (t) =f(t),t T += f,onTg += f,onTThe Riemann-Hilbert method: from Toeplitz operators to black holesMaria Cristina C marag n n + n 1= n 1+f n 1vectorial RHPg n nM+ n n=M n nmatrix (factorization) RHPB ounded Wiener-Hopf (or Birkhoff) factorization:g=M M 1+M 1+analytic and bounded inDM 1 analytic and bounded inC\closDRH approach:to reduce a problem to the reconstruction of afunction analytic inC\ from jump conditions across.

4 The Riemann-Hilbert method: from Toeplitz operators to black holesMaria Cristina C maraApplications in- Diffraction problems- Elastodynamics- Singular integral equations- Combinatorial probability- Random matrices- Orthogonal polynomials- Integrable systemsDeift, Its, Kapaev, Novokshenov, Fokas, Ablowitz, Bleher, stenssonThe Riemann-Hilbert method: a Swiss Army knifeMarko Bertola (Concordia University), 2012 The Riemann-Hilbert method: from Toeplitz operators to black holesMaria Cristina C maraApplications in- Diffraction problems- Elastodynamics- Singular integral equations- Combinatorial probability- Random matrices- Orthogonal polynomials- Integrable systemsDeift, Its, Kapaev, Novokshenov, Fokas, Ablowitz, Bleher, stenssonThe Riemann-Hilbert method: a Swiss Army knifeMarko Bertola (Concordia University), 2012 The Riemann-Hilbert method.

5 From Toeplitz operators to black holesMaria Cristina C maraThere areno general methods to solve matrix has to develop custom-made methods, case by progress has been made inexplicitfactorisation C. C mara, A. F. dos Santos and P. F. dos Santos:Matrix Riemann-Hilbertproblems and factorization on Riemann surfaces, J. Funct. Anal. (2008).M. C. C mara, C. Diogo and L. Rodman:Fredholmness of Toeplitz operatorsand corona Problems, J. Funct. Anal. (2010).M. C. C mara, C. Diogo, Yu. Karlovich and Spitkovsky:Factorizations,Riemann-Hilber t problems and the corona theorem, J. London Math.

6 Soc. 86(2012).M. C. C mara, C. Diogo and Spitkovsky: Toeplitz operators of finiteinterval type and the table method, J. Math. Anal. Appl. (2014).The Riemann-Hilbert method: from Toeplitz operators to black holesMaria Cristina C maraThere areno general methods to solve matrix has to develop custom-made methods, case by progress has been made inexplicitfactorisation C. C mara, A. F. dos Santos and P. F. dos Santos:Matrix Riemann-Hilbertproblems and factorization on Riemann surfaces, J. Funct. Anal. (2008).M. C. C mara, C. Diogo and L. Rodman:Fredholmness of Toeplitz operatorsand corona Problems, J.

7 Funct. Anal. (2010).M. C. C mara, C. Diogo, Yu. Karlovich and Spitkovsky:Factorizations,Riemann-Hilber t problems and the corona theorem, J. London Math. Soc. 86(2012).M. C. C mara, C. Diogo and Spitkovsky: Toeplitz operators of finiteinterval type and the table method, J. Math. Anal. Appl. (2014).The Riemann-Hilbert method: from Toeplitz operators to black holesMaria Cristina C maraSpectral properties and kernels of Toeplitz operators(TO)In the context ofL2(T)orL2(R):L2=H2+ H 2H 2=FL2(R )P+:L2 H+2 Toeplitz operator:Tg:H+2 H+2g L Tg +=P+g +gis thesymbolof the TO ( it can bematricial ).

8 InL2(R), TO are unitarily equivalent, via the Fourier transform,to convolution operators on the half-lineR+.The Riemann-Hilbert method: from Toeplitz operators to black holesMaria Cristina C maraToeplitz operators are intimately related to RHP:1 Fredholmness, invertibility, the dimension of the kernel andthe cokernel (and therefore their spectral properties) aredetermined by a RH factorisation of their particular:Tgis invertible g=M M 1+Progress in developing methods to explicitly solveRHfactorisation problemsgoes hand in hand with progressin thespectral theory of Toeplitz operators2 Kernels of TO.

9 Many important spaces of functions, such asmodelspaces, can be described as Toeplitz kernels consist of the solutions to a vectorial RHPg += The Riemann-Hilbert method: from Toeplitz operators to black holesMaria Cristina C maraSome recent results taking this RH approach to Toeplitzkernels:- new (and surprising) properties of all Toeplitz kernelsM. C. C mara and J. R. Partington,Near invariance and kernels ofToeplitz operators, J. Anal. Math. (2014).- generalisation of Hitt s and Hayashi s results on nearlyinvariant subspaces to a Banach space C mara and Partington,Finite-dimensional Toeplitz kernelsand nearly-invariant subspaces, J.

10 Operator Theory (2016).- characterisation of the multipliers between Toeplitz C mara and Partington,Multipliers between Toeplitz kernels(2017).The Riemann-Hilbert method: from Toeplitz operators to black holesMaria Cristina C maraConstructing new solutions to Einstein s fieldequations Einstein s field equations arenonlinearPDE s. They are very difficult to solve in general, so one mustconcentrate on special classes of solutions which exhibitsymmetries. Reduced (2 dimensional) field equations:d( A) = denotes the Hodge dual ( d =dv, dv= d )andAis a matrix one-formA=M 1dMwhereM( ,v)determines the solution to Einstein s equations.