Date/Time : 18 Nov 2026 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Kenji Nakanishi, Kyoto University

Since the pioneering work of Kenig and Merle, there have been numerous results on global behavior of large general solutions to nonlinear dispersive equations that vary depending on the initial data, from asymptotically linear or scattering solutions to solitons and blow-up. The main question is to distinguish the types of solutions or to predict them from the initial data, without any asymptotic solution formula such as those provided by the inverse scattering theory in the special case of completely integrable systems. Despite of some spectacular progress such as the soliton resolution without size restriction, the range of applicable equations has been rather limited, such as the nonlinear wave equation or Schrodinger with simple nonlinearity, because certain monotonicity or invariance properties play crucial roles in the analysis without small factor. The Zakharov system is a model of the Langmuir turbulence in plasma, coupling the Schrodinger and the wave equations, and may be regarded also as a mathematical model to extend the methods from the simple scalar equations. It has been quite successful in the last decade for space dimensions 3 and 4, in spite of a few difficulties such as the derivative loss, the lower degree of nonlinear interactions, and the loss of invariance for scaling and for momentum or coordinate velocity. This talk will review the state of art, together with some detail on most recent results and open problems.

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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 : 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 : 16 Sep 2026 17:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Nikolay Bazhenov, Nanyang Technological University

In recent years, there has been renewed interest in the algorithmic aspects of various Stone-type dualities. For instance, Harrison-Trainor, Melnikov, and Ng (2020) proved that a Boolean algebra B has a computable isomorphic copy if and only if its dual Stone space St(B) is homeomorphic to a computable Polish space. In this talk, following the approach of Ershov and Schwidefsky, we work with effective presentations of topological T_0-spaces with base. We establish a computable version of a Stone-type duality due to Schwidefsky (2023), and we discuss its applications in computable topology.

This is joint work with Iskander Kalimullin and Marina Schwidefsky.

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Date/Time : 16 Sep 2026 13:30
Venue : S16 #05-18
Speaker : Teruhisa Koshikawa, Keio University

By the categorical local Langlands conjecture, the semiorthogonal decomposition indexed by the Kottwitz set and Bernstein decompositions in the geometric side are expected to have counterparts in the spectral side. I will discuss my joint work with Bertoloni Meli realizing them directly. It is closely related to the theory of cuspidal coherent sheaves, but also has connections to some recent developments in algebraic geometry.

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Date/Time : 16 Sep 2026 17:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Nikolay Bazhenov, Nanyang Technological University

In recent years, there has been renewed interest in the algorithmic aspects of various Stone-type dualities. For instance, Harrison-Trainor, Melnikov, and Ng (2020) proved that a Boolean algebra B has a computable isomorphic copy if and only if its dual Stone space St(B) is homeomorphic to a computable Polish space. In this talk, following the approach of Ershov and Schwidefsky, we work with effective presentations of topological T_0-spaces with base. We establish a computable version of a Stone-type duality due to Schwidefsky (2023), and we discuss its applications in computable topology.

This is joint work with Iskander Kalimullin and Marina Schwidefsky.

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Date/Time : 16 Sep 2026 16:30
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Jackie Lok, National University of Singapore

Recently, a class of algorithms based on combining classical fixed-point iterations with repeated random sparsification has been successfully applied to very large-scale linear algebra problems, where storing the full solution vector itself may be prohibitive. However, a general mathematical understanding of when this approach can achieve dimension-independent computational and storage requirements remains limited.

In this talk, we study a randomly sparsified power method for computing the leading eigenvector of a column-stochastic matrix. We prove a dimension-independent error bound that depends on the approximate sparsity of the solution vector and the mixing time of the Markov chain associated with the matrix, establishing the possibility for high-quality approximation in very high dimensions. On the other hand, we show that a deterministically sparsified counterpart can fail, demonstrating a provable separation between deterministic and random sparsification.

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