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.
Mon 03 Mar 18:00: Protein self-assembly – understanding and controlling the machinery of life
Proteins are the active molecules of life. However, most proteins do not work on their own in health or disease; a key challenge, therefore, is understanding how these molecules interact with each other to give rise to function or malfunction. This talk will outline our efforts to discover, understand and use the basic principles that drive protein assembly into larger scale structures and phases. I will discuss how controlling transitions between such phases can help us ameliorate biological malfunction when it occurs in disease, and well as develop new classes of functional materials.
- Speaker: Professor Tuomas Knowles, Yusuf Hamied Department of Chemistry
- Monday 03 March 2025, 18:00-19:00
- Venue: Bristol-Myers Squibb Lecture Theatre, Department of Chemistry.
- Series: Cambridge Philosophical Society; organiser: Beverley Larner.
Mon 03 Mar 18:00: Protein self-assembly – understanding and controlling the machinery of life
Proteins are the active molecules of life. However, most proteins do not work on their own in health or disease; a key challenge, therefore, is understanding how these molecules interact with each other to give rise to function or malfunction. This talk will outline our efforts to discover, understand and use the basic principles that drive protein assembly into larger scale structures and phases. I will discuss how controlling transitions between such phases can help us ameliorate biological malfunction when it occurs in disease, and well as develop new classes of functional materials.
- Speaker: Professor Tuomas Knowles, Yusuf Hamied Department of Chemistry
- Monday 03 March 2025, 18:00-19:00
- Venue: Bristol-Myers Squibb Lecture Theatre, Department of Chemistry.
- Series: Cambridge Philosophical Society; organiser: Beverley Larner.
Wed 12 Mar 18:00: Towards a Net Zero World: Developing and applying new tools to understand how materials for Li and “beyond-Li” battery technologies function
This talk will start by describing existing battery technologies, what some of the current and more long-term challenges are, and touch on strategies to address some of the issues. I will then focus on my own work – together with my research group and collaborators – to develop new characterisation (NMR, MRI , and X-ray diffraction and optical) methods that allow batteries to be studied while they are operating (i.e., operando). These techniques allow transformations of the various cell components to be followed under realistic conditions without having to disassemble and take apart the cell. We can detect key side reactions involving the various battery materials, in order to determine the processes that are responsible ultimately for battery failure. We can watch ions diffusing in, and moving in and out of, the active “electrode” materials that store the (lithium) ions and the electrons, to understand how the batteries function. Finally, I will discuss the challenges in designing batteries that can be rapidly charged and discharged.
- Speaker: Professor Clare P. Grey
- Wednesday 12 March 2025, 18:00-19:00
- Venue: Bristol-Myers Squibb Lecture Theatre, Department of Chemistry.
- Series: Cambridge Philosophical Society; organiser: Beverley Larner.
Wed 12 Mar 18:00: Towards a Net Zero World: Developing and applying new tools to understand how materials for Li and “beyond-Li” battery technologies function
This talk will start by describing existing battery technologies, what some of the current and more long-term challenges are, and touch on strategies to address some of the issues. I will then focus on my own work – together with my research group and collaborators – to develop new characterisation (NMR, MRI , and X-ray diffraction and optical) methods that allow batteries to be studied while they are operating (i.e., operando). These techniques allow transformations of the various cell components to be followed under realistic conditions without having to disassemble and take apart the cell. We can detect key side reactions involving the various battery materials, in order to determine the processes that are responsible ultimately for battery failure. We can watch ions diffusing in, and moving in and out of, the active “electrode” materials that store the (lithium) ions and the electrons, to understand how the batteries function. Finally, I will discuss the challenges in designing batteries that can be rapidly charged and discharged.
- Speaker: Professor Clare P. Grey
- Wednesday 12 March 2025, 18:00-19:00
- Venue: Bristol-Myers Squibb Lecture Theatre, Department of Chemistry.
- Series: Cambridge Philosophical Society; organiser: Beverley Larner.
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.
Mon 17 Mar 18:00: Acoustics of musical instruments - why is a saxophone like a violin?
Musical instruments like the clarinet and saxophone do not obviously have anything in common with a bowed violin string. This talk will explore the physics behind how these instruments work, and it will reveal some unexpectedly strong parallels between them. This is all the more surprising because all of them rely on strongly nonlinear phenomena, and nonlinear systems are notoriously tricky: significant commonalities between disparate systems are rare. For all the instruments, computer simulations will be used to give some insight into questions a musician may ask: What variables must a player control, and how? Why are some instruments “easier to play” than others?
- Speaker: Professor Jim Woodhouse ( CUED)
- Monday 17 March 2025, 18:00-19:00
- Venue: Bristol-Myers Squibb Lecture Theatre, Department of Chemistry.
- Series: Cambridge Philosophical Society; organiser: Beverley Larner.
Mon 17 Feb 18:00: Why there’s no such thing as “the” scientific advice
During the Covid-19 pandemic, U.K. policy-makers claimed to be “following the science”. Many commentators objected that the government did not live up to this aim. Others worried that policy-makers ought not blindly “follow” science, because this involves an abdication of responsibility. In this talk, I consider a third, even more fundamental concern: that there is no such thing as “the” science. Drawing on the case of adolescent vaccination against Covid-19, I argue that the best that any scientific advisory group can do is to offer a partial perspective on reality. In turn, this has important implications for how we think about science and politics.
- Speaker: Professor Stephen John
- Monday 17 February 2025, 18:00-19:00
- Venue: Bristol-Myers Squibb Lecture Theatre, Department of Chemistry.
- Series: Cambridge Philosophical Society; organiser: Beverley Larner.
Mon 03 Feb 18:00: To Bend or to Break? — new views on the hardening of metals
Kipling’s “Iron‒Cold Iron‒is master of them all” captures the familiar importance of metals as structural materials. Yet common metals are not necessarily hard; they can become so when deformed. This phenomenon, strain hardening, was first explained by G. I. Taylor in 1934. Ninety years on from this pioneering work on dislocation theory, we explore the deformation of metals when dislocations do not exist, that is when the metals are non-crystalline. These amorphous metals have record-breaking combinations of properties. They behave very differently from the metals that Taylor studied, but we do find phenomena for which his work (in a dramatically different context) is directly relevant.
- Speaker: Professor Lindsay Greer ( Department of Materials Science and Metallurgy)
- Monday 03 February 2025, 18:00-19:00
- Venue: Bristol-Myers Squibb Lecture Theatre, Department of Chemistry.
- Series: Cambridge Philosophical Society; organiser: Beverley Larner.
Mon 17 Mar 18:00: Acoustics of musical instruments - why is a saxophone like a violin?
Musical instruments like the clarinet and saxophone do not obviously have anything in common with a bowed violin string. This talk will explore the physics behind how these instruments work, and it will reveal some unexpectedly strong parallels between them. This is all the more surprising because all of them rely on strongly nonlinear phenomena, and nonlinear systems are notoriously tricky: significant commonalities between disparate systems are rare. For all the instruments, computer simulations will be used to give some insight into questions a musician may ask: What variables must a player control, and how? Why are some instruments “easier to play” than others?
- Speaker: Professor Jim Woodhouse ( CUED)
- Monday 17 March 2025, 18:00-19:00
- Venue: Bristol-Myers Squibb Lecture Theatre, Department of Chemistry.
- Series: Cambridge Philosophical Society; organiser: Beverley Larner.
Mon 17 Feb 18:00: Why there’s no such thing as “the” scientific advice
During the Covid-19 pandemic, U.K. policy-makers claimed to be “following the science”. Many commentators objected that the government did not live up to this aim. Others worried that policy-makers ought not blindly “follow” science, because this involves an abdication of responsibility. In this talk, I consider a third, even more fundamental concern: that there is no such thing as “the” science. Drawing on the case of adolescent vaccination against Covid-19, I argue that the best that any scientific advisory group can do is to offer a partial perspective on reality. In turn, this has important implications for how we think about science and politics.
- Speaker: Professor Stephen John
- Monday 17 February 2025, 18:00-19:00
- Venue: Bristol-Myers Squibb Lecture Theatre, Department of Chemistry.
- Series: Cambridge Philosophical Society; organiser: Beverley Larner.
Mon 03 Feb 18:00: To Bend or to Break? — new views on the hardening of metals
Kipling’s “Iron‒Cold Iron‒is master of them all” captures the familiar importance of metals as structural materials. Yet common metals are not necessarily hard; they can become so when deformed. This phenomenon, strain hardening, was first explained by G. I. Taylor in 1934. Ninety years on from this pioneering work on dislocation theory, we explore the deformation of metals when dislocations do not exist, that is when the metals are non-crystalline. These amorphous metals have record-breaking combinations of properties. They behave very differently from the metals that Taylor studied, but we do find phenomena for which his work (in a dramatically different context) is directly relevant.
- Speaker: Professor Lindsay Greer ( Department of Materials Science and Metallurgy)
- Monday 03 February 2025, 18:00-19:00
- Venue: Bristol-Myers Squibb Lecture Theatre, Department of Chemistry.
- Series: Cambridge Philosophical Society; organiser: Beverley Larner.
Fri 28 Mar 09:00: The Unknown Maxwell
In the millennium poll, James Clerk Maxwell (1831-1879) was voted the third greatest physicist of all time – behind Newton and Einstein. But Maxwell’s extraordinary range of interests and achievements extended far beyond his well-known equations for electromagnetism and his thermodynamic relations, and this meeting will explore just a few of the fields in which Maxwell did seminal work. The day will begin with an overview of James Clerk Maxwell’s life and achievements. The talks following will highlight some topics in which current research is revealing interesting developments, but also looking back to Maxwell’s insights in laying the foundations for so much of our contemporary science.
There will be a small exhibition of artefacts including some of Maxwell’s models from the Cavendish collection.
James Clerk Maxwell had strong links with the Cambridge Philosophical Society during his time at Cambridge. He studied mathematics as an undergraduate – initially at Peterhouse, but moving to Trinity before the end of his first term. He graduated in 1854, and shortly afterwards presented his first paper On the transformation of surfaces by bending to the Cambridge Philosophical Society. His career took him to Aberdeen, Edinburgh and King’s College London before returning to Cambridge in 1871 to become the first Professor of Experimental Physics. He was President of the Cambridge Philosophical Society 1875-1877.
- Speaker: The Cambridge Philosophical Society
- Friday 28 March 2025, 09:00-17:15
- Venue: Cambridge University Engineering Department. Constance Tipper Lecture Theatre.
- Series: Cambridge Philosophical Society; organiser: Beverley Larner.
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.
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