
Mon 17 Mar 15:00: Two New Developments concerning Noether's Two Theorems
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.
Thu 22 May 15:00: Title to be confirmed
Abstract not available
- Speaker: Sheehan Olver (Imperial College London)
- Thursday 22 May 2025, 15:00-16:00
- Venue: Centre for Mathematical Sciences, MR14.
- Series: Applied and Computational Analysis; organiser: Georg Maierhofer.
Thu 06 Feb 15:00: Numerical analysis of high frequency wave scattering via semiclassical analysis: a case study with non-uniform meshes
In recent years, semiclassical analysis has significantly advanced our understanding of numerical algorithms for high-frequency wave scattering. This talk will begin with an overview of how semiclassical methods have influenced the theory of numerical methods for frequency-domain wave problems. As a case study, we will then focus on the finite element method (FEM), a classical approach for approximating solutions to high-frequency scattering problems. In FEM , the solution is typically approximated using piecewise polynomials of degree p on a mesh of width h. A fundamental question is then: how should h be chosen (as a function of the frequency, k) so that the error in the numerical solution is small? It has been known since the seminal work of Babuska and Ihlenberg that the natural conjecture hk<
- Speaker: Jeffrey Galkowski (UCL)
- Thursday 06 February 2025, 15:00-16:00
- Venue: Centre for Mathematical Sciences, MR14.
- Series: Applied and Computational Analysis; organiser: Matthew Colbrook.
Thu 05 Jun 15:00: Title to be confirmed
Abstract not available
- Speaker: Sergio Blanes (Universidad Politécnica de Valencia)
- Thursday 05 June 2025, 15:00-16:00
- Venue: Centre for Mathematical Sciences, MR14.
- Series: Applied and Computational Analysis; organiser: Georg Maierhofer.
Thu 12 Jun 15:00: Title to be confirmed
Abstract not available
- Speaker: Leonardo Tolomeo (University of Edinburgh)
- Thursday 12 June 2025, 15:00-16:00
- Venue: Centre for Mathematical Sciences, MR14.
- Series: Applied and Computational Analysis; organiser: Georg Maierhofer.
Thu 23 Jan 15:00: Detecting and Attributing Change in Climate and Complex Systems: Foundations, Green's Functions, and Nonlinear Fingerprints
Detection and attribution (D&A) studies are cornerstones of climate science, providing crucial evidence for policy decisions. Their goal is to link observed climate change patterns to anthropogenic and natural drivers via the optimal fingerprinting method (OFM). We show that response theory for nonequilibrium systems offers the physical and dynamical basis for OFM , including the concept of causality used for attribution. Our framework clarifies the method’s assumptions, advantages, and potential weaknesses. We use our theory to perform D&A for prototypical climate change experiments performed on an energy balance model and on a low-resolution coupled climate model. We also explain the underpinnings of degenerate fingerprinting, which offers early warning indicators for tipping points. Finally, we extend the OFM to the nonlinear response regime. Our analysis shows that OFM has broad applicability across diverse stochastic systems influenced by time-dependent forcings, with potential relevance to ecosystems, quantitative social sciences, and finance, among others.
Key References V. Lucarini and M. D. Chekroun, Detecting and Attributing Change in Climate and Complex Systems: Foundations, Green’s Functions, and Nonlinear Fingerprints, Phys. Rev. Lett. 133, 244201 (2024) https://doi.org/10.1103/PhysRevLett.133.244201 V. Lucarini and M. D. Chekroun, Theoretical tools for understanding the climate crisis from Hasselmann’s programme and beyond, Nat. Rev. Phys. 5, 744 (2023) https://doi.org/10.1038/s42254-023-00650-8
- Speaker: Valerio Lucarini (University of Leicester)
- Thursday 23 January 2025, 15:00-16:00
- Venue: Centre for Mathematical Sciences, MR14.
- Series: Applied and Computational Analysis; organiser: Matthew Colbrook.
Thu 29 May 15:00: Title to be confirmed
Abstract not available
- Speaker: Alberto Paganini (University of Leicester)
- Thursday 29 May 2025, 15:00-16:00
- Venue: Centre for Mathematical Sciences, MR14.
- Series: Applied and Computational Analysis; organiser: Georg Maierhofer.
Thu 01 May 15:00: Title to be confirmed
Abstract not available
- 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.
Thu 05 Dec 15:00: Can Humans Supervise Increasingly Ultracrepidarian AI?
Large language models have evolved to solve increasingly complex problems but still fail at many simple ones—from a human point of view. This discordance with human difficulty expectations strongly affects the reliability of these models, as users cannot identify a safe operating condition where the model is expected to be correct. With the extensive use of scaling up and shaping up (such as RLHF ) in newer generations of LLMs, we question whether this is the case. In a recent Nature paper, we examined several LLM families and showed that instances that are easy for humans are usually easy for the models. However, scaled-up, shaped-up models do not secure areas of low difficulty in which either the model does not err or human supervision can spot the errors. We also found that early models often avoid user questions, whereas scaled-up, shaped-up models tend to give apparently sensible yet wrong answers much more often, including errors on difficult questions that human supervisors frequently overlook. Finally, we disentangled whether this behaviour arises from scaling up or shaping up, and discovered new scaling laws showing that larger models become more incorrect and especially more ultracrepidarian, operating beyond their competence. These findings highlight the need for a fundamental shift in the design and development of general-purpose artificial intelligence, particularly in high-stakes areas where a predictable distribution of errors is paramount.
The talk will be based on the recent paper: L Zhou, W Schellaert, FM Plumed, YM Daval, C Ferri, JH Orallo (2024) “Larger and more instructable language models become less reliable”, Nature, 61-68
- Speaker: Jose Hernandez-Orallo
- Thursday 05 December 2024, 15:00-16:00
- Venue: Centre for Mathematical Sciences, MR14.
- Series: Applied and Computational Analysis; organiser: Matthew Colbrook.
Thu 07 Nov 15:00: Domain-theoretic Semantics for Dynamical Systems: From Analog Computers to Neural Networks
Despite great empirical success, we are still lacking a theory of modern artificial intelligence. In particular, we are missing an interpretation of the ‘sub-symbolic’ computation performed by neural networks. For digital computation, this problem was solved by semantics: the mathematical description of the meaning of program code. In this paper, we work toward an analogous semantics for neural networks and other forms of ‘non-symbolic’ computation like analog computers—which all can be regarded as dynamical systems. To do so, we first summarize the three semantics for digital computation (operational, denotational, logical), and then develop their counterparts for non-symbolic computation (dynamical systems, domains, and modal logic). The key idea is to represent the dynamics of non-symbolic computation as a limit of finite symbolic approximations, which are given by interpretable observations. In an implementation, we thus illustrate the training dynamics of a neural network in a standard machine learning task.
- Speaker: Levin Hornischer
- Thursday 07 November 2024, 15:00-16:00
- Venue: Centre for Mathematical Sciences, MR14.
- Series: Applied and Computational Analysis; organiser: Matthew Colbrook.