Date/Time : 30 Sep 2026 16:30
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Melvin Leok, University of California San Diego

Adjoint systems are widely used to inform control, optimization, and design in systems described by ODEs or DAEs. At the level of PDEs, advection equations and mean field games also have the structure of adjoint systems, and all adjoint systems have a Hamiltonian structure.

We utilize symplectic and presymplectic geometry to investigate the properties of adjoint systems associated with ODEs and DAEs, respectively. We show that the adjoint-variational quadratic conservation laws, which are critical to adjoint sensitivity analysis, arise from (pre)symplecticity of such adjoint systems.

We develop structure-preserving numerical methods for such systems using Galerkin Hamiltonian variational integrators which admit discrete analogues of these quadratic conservation laws. This can be used to train neural networks arising from time-discretizations of neural ODEs, even ones based on implicit-in-time discretization.

Finally, we investigate the geometric structure of adjoint systems associated with evolutionary partial differential equations at the fully continuous, semi-discrete, and fully discrete levels and the relations between these levels. We show that the adjoint system associated with an evolutionary PDE has an infinite-dimensional Hamiltonian structure, which is useful for connecting the fully continuous, semi-discrete, and fully discrete levels.

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Date/Time : 30 Sep 2026 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Jack Thorne, University of Cambridge

Although computing the set of solutions to a given Diophantine equation is hard (!), the solution sets of families of equations often display striking statistical regularity – this is the case, for example, for the family of Weierstrass elliptic curves over Q given by the equations y^2 = x^3+ Ax + B. For a randomly chosen curve in this family, one expects to see either no solutions or infinitely many solutions, each case occurring with probability 1/2. Striking results in this direction have been proved for these curves and for the hyperelliptic curves y2 = x^{2n+1} + … + c_{2n+1} in the Fields medal-winning work of Bhargava (with co-authors including Shankar and Gross).

One of the main ingredients in these works is a bridge between rational points and the space of lattices (i.e. discrete, cocompact subgroups in Euclidean space) — a Riemannian manifold of finite volume that plays a recurring role in number theory, and is familiar (in the rank 2 case) as the quotient of the upper half plane by the group SL(2, Z). I will survey this picture and describe recent work with Jef Laga, which shows that we can use this formalism to understand not only the number of rational points but their physical properties (e.g. their heights).

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Date/Time : 29 Sep 2026 11:00
Venue : S16 #05-18
Speaker : Weihang Zheng, Monash University

In the theory of general relativity, an important heuristic provided by Belinskii-Khalatnikov-Lifshitz (BKL) is that cosmological singularities should be either chaotic and oscillatory or quiescent. While the BKL conjecture has been studied for decades, the behaviour near such singularities remains incompletely understood. There is literature dating back to the last century suggesting some ideas about the Big Bang formation. For the quiescent case, the Kasner family of exact solutions to the Einstein-scalar field equations provides a key model. Recently, a remarkable result was given by Fournodavlos-Rodnianski-Speck where they proved the past nonlinear stability of the Kasner-scalar field solutions of the entire sub-critical regime. In this talk, I will discuss a further result, the localized stability, i.e. the stability on a truncated cone domain. This is joint work in part with Florian Beyer and Todd Oliynyk.

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Date/Time : 29 Sep 2026 16:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Bruno Despres, Sorbonne University, France

Machine learning is the new revolution in applied sciences. For example, a recent major breakthrough is meteorology modelling with hybrid models. However, a hard nut is auto-differentiability of discrete models (required for numerical optimization). I will show that auto-diff is an interesting mathematical problem per se. I will review some recent mathematical and numerical advances which establish a functional setting for auto-differentiability of piecewise C1 functions. Two applications will be detailed: a) new singular functions show up; b) the minimal regularity of activation functions for Deep Ritz/PINNs methods is equal to the number of derivatives of the problems + 1.

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Date/Time : 28 Sep 2026 15:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Jack Thorne, University of Cambridge

The Langlands correspondence predicts the existence of Galois representations associated to certain automorphic representations of GL(n, AF). These are known to exist in the case that F is a CM field and the automorphic representation in question is regular algebraic (this includes, for example, the automorphic representations that should correspond to elliptic curves over imaginary quadratic fields). For arithmetic applications, it is essential to know that this automorphic to Galois dictionary satisfies compatibility with the local Langlands correspondence: in particular, when considering the restriction of the p-adic Galois representation to a p-adic decomposition group, the representation should be potentially semistable, in the sense of Fontaine, and its (semisimple) Weil—Deligne representation should be the one predicted by the local Langlands correspondence. This local-global compatibility has been proved in many cases when the p-adic Galois representation satisfies a global hypothesis (such as being `decomposed generic’). I will discuss a proof of the general case, and describe an application to the Bloch—Kato conjecture. This is joint work with A’Campo, Hevesi, and Whitmore.

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Date/Time : 24 Sep 2026 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : David Lee, Australian Bureau of Meteorology

Energy conserving variational discretisations that preserve the skew symmetric structure of the non-canonical Hamiltonian form of the moist compressible Euler equations are extended to conserve the quadratic variance of thermodynamic tracers. This is important for the simulation of thermal processes since tracer variance represents a convex invariant of the equations of motion, so its conservation is strongly related to the stable simulation of those processes.

Moist processes are incorporated into the energy and tracer variance stable formulation such that the intrinsic thermodynamic variables (pressure, temperature and chemical potentials of the different moisture species) are derived via the derivatives of a thermodynamic potential (the internal energy) with respect to the prognostic model variables (density, entropy and mass fractions of the moisture species). This ensures thermodynamic consistency for the representation of non-equilibrium moist phase exchanges.

A neural network is then trained to learn the phase exchanges within the thermodynamically consistent framework, via kilometer scale output of a regional weather model. The resulting energy and tracer variance conserving, thermodynamically consistent model of the moist compressible Euler equations is then used to simulate three phase convection at sub-kilometer scale.

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Date/Time : 23 Sep 2026 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Frank den Hollander, Leiden University

In this talk I present an overview of what is known and is not known about random processes on dynamic random graphs in co-evolution, i.e., with mutual feedback. The literature offers plenty of heuristics, simulations and conjectures, but so far mathematical results are extremely scarce. I describe some of these results, with a focus on opinion dynamics. In particular, I consider random graphs in which vertices can have one of two possible opinions. Pairs of vertices connected by an edge share their opinions according to a rate that may depend on the size of the graph. Each edge turns on or off according to a rate that depends on whether the vertices at its two endpoints have the same opinion or not. I exhibit joint evolution equations and describe the occurrence of phase transitions between consensus and polarisation.

Based on joint work with Siva Athreya (Bangalore) and Adrian Rollin (Singapore).

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Date/Time : 23 Sep 2026 17:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Péter Komjáth, Eötvös Loránd University

To be announced.

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Date/Time : 22 Sep 2026 10:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Ng Kah Loon and Jonathon Teo Yi Han, National University of Singapore

Prof. Ng Kah Loon and Dr. Jonathon Teo will be conducting a sharing session during recess week on IMMS setup, transferring marks, Canvas grades, and related matters, followed by a session on the use of SoftMark (marking e-copies of a pen-and-paper exam). The aim is to help reduce the likelihood of errors in CA and exam marks entering/transfer.

Kah Loon will start off the session on IMMS/Canvas and related matters for about 30-40 minutes. After a short break, Jonathon will introduce SoftMark and some of its features/set up work required. The entire session should be less than 2 hours. Please bring along your laptop as you will be guided through hands-on sessions.

Marks Transfer and Exam Marking Sharing Session – Fill out form
https://forms.cloud.microsoft/Pages/ResponsePage.aspx?id=Xu-lWwkxd06Fvc_rDTR-gizvpniXA2FDsY7mBxdqnkVUNVo4M0ZVMzEzV1lBTTZaOFA4NlBXNVhHSC4u

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