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In this talk, we present low-regularity numerical schemes for nonlinear dispersive equations, with a particular focus on the Zakharov system (ZS) and the “good” Boussinesq (GB) equation. These models exhibit strong nonlinear interactions and are known to pose significant analytical and numerical challenges when the solution has limited regularity.

We concentrate on our recent results for the Zakharov system, where we construct and analyze an explicit filtered Lie splitting scheme applied directly to its original coupled form. This method successfully overcomes the essential difficulty of derivative loss in the nonlinear terms, which not only obstructs low-regularity analysis, but has long prevented rigorous error estimates for explicit Lie splitting schemes based directly on the original Zakharov system. By developing multilinear estimates in discrete Bourgain spaces, we rigorously prove the first explicit low-regularity error estimate for the original Zakharov system, and also the first such result for a coupled system within the Bourgain framework. The analytical strategy developed here can also be extended to other dispersive equations with derivative loss, offering a way to overcome both low-regularity difficulties and the fundamental obstacle posed by derivative-loss nonlinearities. Numerical experiments confirm the theoretical predictions.

• Speaker: Hang Li (Sorbonne Université)
• Thursday 01 May 2025, 15:00-16:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Georg Maierhofer.

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In this talk, we present low-regularity numerical schemes for nonlinear dispersive equations, with a particular focus on the Zakharov system (ZS) and the “good” Boussinesq (GB) equation. These models exhibit strong nonlinear interactions and are known to pose significant analytical and numerical challenges when the solution has limited regularity.

We concentrate on our recent results for the Zakharov system, where we construct and analyze an explicit filtered Lie splitting scheme applied directly to its original coupled form. This method successfully overcomes the essential difficulty of derivative loss in the nonlinear terms, which not only obstructs low-regularity analysis, but has long prevented rigorous error estimates for explicit Lie splitting schemes based directly on the original Zakharov system. By developing multilinear estimates in discrete Bourgain spaces, we rigorously prove the first explicit low-regularity error estimate for the original Zakharov system, and also the first such result for a coupled system within the Bourgain framework. The analytical strategy developed here can also be extended to other dispersive equations with derivative loss, offering a way to overcome both low-regularity difficulties and the fundamental obstacle posed by derivative-loss nonlinearities. Numerical experiments confirm the theoretical predictions.

• Speaker: Hang Li (Sorbonne Université)
• Thursday 01 May 2025, 15:00-16:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Georg Maierhofer.

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Abstract not available

• Speaker: Jianbo Cui (The Hong Kong Polytechnic University)
• Thursday 26 June 2025, 16:00-17:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Georg Maierhofer.

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Abstract not available

• Speaker: Jianbo Cui (The Hong Kong Polytechnic University)
• Thursday 26 June 2025, 16:00-17:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Georg Maierhofer.

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Abstract not available

• Speaker: Liying Sun (Chinese Academy of Sciences)
• Thursday 26 June 2025, 15:00-16:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Georg Maierhofer.

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Abstract not available

• Speaker: Liying Sun (Chinese Academy of Sciences)
• Thursday 26 June 2025, 15:00-16:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Georg Maierhofer.

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In this talk, we focus on the scattering of time-harmonic acoustic, elastic, and polarized electromagnetic waves by multiple finite-length open arcs in an unbounded two-dimensional domain. We begin by reformulating the corresponding boundary value problems with Dirichlet or Neumann conditions as weakly and hypersingular boundary integral equations (BIEs), respectively. We then introduce a family of fast spectral Galerkin methods for solving these BIEs. The discretization bases are built from weighted Chebyshev polynomials that accurately capture the solutions’ edge behavior. Under the assumption of analyticity of the sources and arc geometries, we show that these bases yield exponential convergence with respect to the polynomial degree.

Numerical examples will illustrate the accuracy and robustness of the proposed methods, with respect to both the number of arcs and the wavenumber. Additionally, we demonstrate that, for general weakly and hypersingular BIEs, the solutions depend holomorphically on perturbations of the arc parametrizations. These results are crucial for establishing the shape holomorphy of domain-to-solution maps arising in boundary integral equations, with applications in uncertainty quantification, inverse problems, and deep learning, among others. They also raise new questions—some of which you may have the answer to!

• Speaker: Carlos Jerez Hanckes (University of Bath)
• Thursday 15 May 2025, 15:00-16:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Georg Maierhofer.

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In this talk, we focus on the scattering of time-harmonic acoustic, elastic, and polarized electromagnetic waves by multiple finite-length open arcs in an unbounded two-dimensional domain. We begin by reformulating the corresponding boundary value problems with Dirichlet or Neumann conditions as weakly and hypersingular boundary integral equations (BIEs), respectively. We then introduce a family of fast spectral Galerkin methods for solving these BIEs. The discretization bases are built from weighted Chebyshev polynomials that accurately capture the solutions’ edge behavior. Under the assumption of analyticity of the sources and arc geometries, we show that these bases yield exponential convergence with respect to the polynomial degree.

Numerical examples will illustrate the accuracy and robustness of the proposed methods, with respect to both the number of arcs and the wavenumber. Additionally, we demonstrate that, for general weakly and hypersingular BIEs, the solutions depend holomorphically on perturbations of the arc parametrizations. These results are crucial for establishing the shape holomorphy of domain-to-solution maps arising in boundary integral equations, with applications in uncertainty quantification, inverse problems, and deep learning, among others. They also raise new questions—some of which you may have the answer to!

• Speaker: Carlos Jerez Hanckes (University of Bath)
• Thursday 15 May 2025, 15:00-16:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Georg Maierhofer.

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As is known, bounds of the resolvent of a matrix in the right complex half-plane yield bounds of solutions of homogeneous and inhomogeneous linear differential equations with this matrix. We ask two basic questions:

- Up to which size of structured perturbations are the resolvent norms of the perturbed matrices within a given bound in the right complex half-plane?

- For a given size of structured perturbations, what is the smallest common bound for the resolvent norms of the perturbed matrices in the right complex half-plane?

This is considered for general linear structures such as complex or real matrices with a given sparsity pattern or with restricted range and corange, or special classes such as Toeplitz or Hankel matrices. Conceptually, we combine unstructured and structured pseudospectra in a joint pseudospectrum, allowing for the use of resolvent bounds as with unstructured pseudospectra and for structured perturbations as with structured pseudospectra. The above questions are addressed by an algorithm which solves eigenvalue optimization problems via suitably discretized rank-1 matrix differential equations. The talk is based on joint work with Nicola Guglielmi.

• Speaker: Christian Lubich
• Thursday 20 March 2025, 15:00-16:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Matthew Colbrook.

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As is known, bounds of the resolvent of a matrix in the right complex half-plane yield bounds of solutions of homogeneous and inhomogeneous linear differential equations with this matrix. We ask two basic questions:

- Up to which size of structured perturbations are the resolvent norms of the perturbed matrices within a given bound in the right complex half-plane?

- For a given size of structured perturbations, what is the smallest common bound for the resolvent norms of the perturbed matrices in the right complex half-plane?

This is considered for general linear structures such as complex or real matrices with a given sparsity pattern or with restricted range and corange, or special classes such as Toeplitz or Hankel matrices. Conceptually, we combine unstructured and structured pseudospectra in a joint pseudospectrum, allowing for the use of resolvent bounds as with unstructured pseudospectra and for structured perturbations as with structured pseudospectra. The above questions are addressed by an algorithm which solves eigenvalue optimization problems via suitably discretized rank-1 matrix differential equations. The talk is based on joint work with Nicola Guglielmi.

• Speaker: Christian Lubich
• Thursday 20 March 2025, 15:00-16:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Matthew Colbrook.

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In her fundamental 1918 paper, written whilst at Göttingen at the invitation of Klein and Hilbert to help them resolve an apparent paradox concerning the conservation of energy in general relativity, Emmy Noether proved two fundamental theorems relating symmetries and conservation laws of variational problems. Her First Theorem, as originally formulated, relates strictly invariant variational problems and conservation laws of their Euler—Lagrange equations. The Noether correspondence was extended by her student Bessel-Hagen to divergence invariant variational problems. A key issue is when is a divergence invariant variational problem equivalent to a strictly invariant one. Here, I illustrate these issues using a very basic example from her original paper, and then highlight the role of Lie algebra cohomology in resolving this question in general. This part includes some provocative remarks on the role of invariant variational problems in the modern formulation of fundamental physics.

Noether’s Second Theorem concerns variational problems admitting an infinite-dimensional symmetry group depending on an arbitrary function. I first recall the two well-known classes of partial differential equations that admit infinite hierarchies of higher order generalized symmetries: 1) linear and linearizable systems that admit a nontrivial point symmetry group; 2) integrable nonlinear equations such as Korteweg—de Vries, nonlinear Schrödinger, and Burgers’. I will then introduce a new general class: 3) underdetermined systems of partial differential equations that admit an infinite-dimensional symmetry algebra depending on one or more arbitrary functions of the independent variables. An important subclass of the latter are the underdetermined Euler—Lagrange equations arising from a variational principle that admits an infinite-dimensional variational symmetry algebra depending on one or more arbitrary functions of the independent variables. According to Noether’s Second Theorem, the associated Euler—Lagrange equations satisfy Noether dependencies and are hence underdetermined and the conservation laws corresponding to such symmetries are trivial; examples include general relativity, electromagnetism, and parameter-independent variational principles.

• Speaker: Peter Olver
• Monday 17 March 2025, 16:00-17:00
• Venue: Centre for Mathematical Sciences, MR13.
• Series: Applied and Computational Analysis; organiser: Matthew Colbrook.

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In her fundamental 1918 paper, written whilst at Göttingen at the invitation of Klein and Hilbert to help them resolve an apparent paradox concerning the conservation of energy in general relativity, Emmy Noether proved two fundamental theorems relating symmetries and conservation laws of variational problems. Her First Theorem, as originally formulated, relates strictly invariant variational problems and conservation laws of their Euler—Lagrange equations. The Noether correspondence was extended by her student Bessel-Hagen to divergence invariant variational problems. A key issue is when is a divergence invariant variational problem equivalent to a strictly invariant one. Here, I illustrate these issues using a very basic example from her original paper, and then highlight the role of Lie algebra cohomology in resolving this question in general. This part includes some provocative remarks on the role of invariant variational problems in the modern formulation of fundamental physics.

Noether’s Second Theorem concerns variational problems admitting an infinite-dimensional symmetry group depending on an arbitrary function. I first recall the two well-known classes of partial differential equations that admit infinite hierarchies of higher order generalized symmetries: 1) linear and linearizable systems that admit a nontrivial point symmetry group; 2) integrable nonlinear equations such as Korteweg—de Vries, nonlinear Schrödinger, and Burgers’. I will then introduce a new general class: 3) underdetermined systems of partial differential equations that admit an infinite-dimensional symmetry algebra depending on one or more arbitrary functions of the independent variables. An important subclass of the latter are the underdetermined Euler—Lagrange equations arising from a variational principle that admits an infinite-dimensional variational symmetry algebra depending on one or more arbitrary functions of the independent variables. According to Noether’s Second Theorem, the associated Euler—Lagrange equations satisfy Noether dependencies and are hence underdetermined and the conservation laws corresponding to such symmetries are trivial; examples include general relativity, electromagnetism, and parameter-independent variational principles.

• Speaker: Peter Olver
• Monday 17 March 2025, 16:00-17:00
• Venue: Centre for Mathematical Sciences, MR13.
• Series: Applied and Computational Analysis; organiser: Matthew Colbrook.

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Quantum graphs are (usually self-adjoint) second-order differential operators over collections of intervals that are glued at their endpoints (“metric graphs”). We survey recent and past advances in quantum graph eigenvalue estimates, focusing on how geometry, topology, and diffusion-transport phenomena interact. Specifically, we examine surgery techniques—localized graph modifications—to manipulate eigenvalues. We also explore transport and transport-diffusion equations on directed graphs, highlighting how directional flow and non-symmetric operators impact spectral properties. We discuss adapting surgery techniques to control these effects, emphasizing the link between edge transmission/boundary conditions, diffusion-transport dynamics, and resulting eigenvalues.

• Speaker: Delio Mugnolo
• Thursday 13 March 2025, 15:00-16:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Matthew Colbrook.

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Quantum graphs are (usually self-adjoint) second-order differential operators over collections of intervals that are glued at their endpoints (“metric graphs”). We survey recent and past advances in quantum graph eigenvalue estimates, focusing on how geometry, topology, and diffusion-transport phenomena interact. Specifically, we examine surgery techniques—localized graph modifications—to manipulate eigenvalues. We also explore transport and transport-diffusion equations on directed graphs, highlighting how directional flow and non-symmetric operators impact spectral properties. We discuss adapting surgery techniques to control these effects, emphasizing the link between edge transmission/boundary conditions, diffusion-transport dynamics, and resulting eigenvalues.

• Speaker: Delio Mugnolo
• Thursday 13 March 2025, 15:00-16:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Matthew Colbrook.

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Abstract not available

• Speaker: Carlos Jerez Hanckes (University of Bath)
• Thursday 15 May 2025, 15:00-16:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Georg Maierhofer.

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Abstract not available

• Speaker: Carlos Jerez Hanckes (University of Bath)
• Thursday 15 May 2025, 15:00-16:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Georg Maierhofer.

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In recent years, progress has been made in constructing methods for approximating curves in various domains, such as manifolds. This talk will explore our recent advances in curve approximation within Wasserstein spaces, covering key concepts, ongoing developments in analyzing our approximation operators, and illustrative examples.

• Speaker: Nir Sharon
• Thursday 20 February 2025, 15:00-16:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Matthew Colbrook.

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In recent years, progress has been made in constructing methods for approximating curves in various domains, such as manifolds. This talk will explore our recent advances in curve approximation within Wasserstein spaces, covering key concepts, ongoing developments in analyzing our approximation operators, and illustrative examples.

• Speaker: Nir Sharon
• Thursday 20 February 2025, 15:00-16:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Matthew Colbrook.

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In her fundamental 1918 paper, written whilst at Göttingen at the invitation of Klein and Hilbert to help them resolve an apparent paradox concerning the conservation of energy in general relativity, Emmy Noether proved two fundamental theorems relating symmetries and conservation laws of variational problems. Her First Theorem, as originally formulated, relates strictly invariant variational problems and conservation laws of their Euler—Lagrange equations. The Noether correspondence was extended by her student Bessel-Hagen to divergence invariant variational problems. A key issue is when is a divergence invariant variational problem equivalent to a strictly invariant one. Here, I illustrate these issues using a very basic example from her original paper, and then highlight the role of Lie algebra cohomology in resolving this question in general. This part includes some provocative remarks on the role of invariant variational problems in the modern formulation of fundamental physics.

Noether’s Second Theorem concerns variational problems admitting an infinite-dimensional symmetry group depending on an arbitrary function. I first recall the two well-known classes of partial differential equations that admit infinite hierarchies of higher order generalized symmetries: 1) linear and linearizable systems that admit a nontrivial point symmetry group; 2) integrable nonlinear equations such as Korteweg—de Vries, nonlinear Schrödinger, and Burgers’. I will then introduce a new general class: 3) underdetermined systems of partial differential equations that admit an infinite-dimensional symmetry algebra depending on one or more arbitrary functions of the independent variables. An important subclass of the latter are the underdetermined Euler—Lagrange equations arising from a variational principle that admits an infinite-dimensional variational symmetry algebra depending on one or more arbitrary functions of the independent variables. According to Noether’s Second Theorem, the associated Euler—Lagrange equations satisfy Noether dependencies and are hence underdetermined and the conservation laws corresponding to such symmetries are trivial; examples include general relativity, electromagnetism, and parameter-independent variational principles.

• Speaker: Peter Olver
• Monday 17 March 2025, 15:00-16:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Matthew Colbrook.

Add to your calendar or Include in your list

In her fundamental 1918 paper, written whilst at Göttingen at the invitation of Klein and Hilbert to help them resolve an apparent paradox concerning the conservation of energy in general relativity, Emmy Noether proved two fundamental theorems relating symmetries and conservation laws of variational problems. Her First Theorem, as originally formulated, relates strictly invariant variational problems and conservation laws of their Euler—Lagrange equations. The Noether correspondence was extended by her student Bessel-Hagen to divergence invariant variational problems. A key issue is when is a divergence invariant variational problem equivalent to a strictly invariant one. Here, I illustrate these issues using a very basic example from her original paper, and then highlight the role of Lie algebra cohomology in resolving this question in general. This part includes some provocative remarks on the role of invariant variational problems in the modern formulation of fundamental physics.

Noether’s Second Theorem concerns variational problems admitting an infinite-dimensional symmetry group depending on an arbitrary function. I first recall the two well-known classes of partial differential equations that admit infinite hierarchies of higher order generalized symmetries: 1) linear and linearizable systems that admit a nontrivial point symmetry group; 2) integrable nonlinear equations such as Korteweg—de Vries, nonlinear Schrödinger, and Burgers’. I will then introduce a new general class: 3) underdetermined systems of partial differential equations that admit an infinite-dimensional symmetry algebra depending on one or more arbitrary functions of the independent variables. An important subclass of the latter are the underdetermined Euler—Lagrange equations arising from a variational principle that admits an infinite-dimensional variational symmetry algebra depending on one or more arbitrary functions of the independent variables. According to Noether’s Second Theorem, the associated Euler—Lagrange equations satisfy Noether dependencies and are hence underdetermined and the conservation laws corresponding to such symmetries are trivial; examples include general relativity, electromagnetism, and parameter-independent variational principles.

• Speaker: Peter Olver
• Monday 17 March 2025, 15:00-16:00
• Venue: Centre for Mathematical Sciences, MR14.
• Series: Applied and Computational Analysis; organiser: Matthew Colbrook.

Add to your calendar or Include in your list