Applied Mathematical Sciences Colloquium

Time: 16:00-17:00, 5 September 2025 (Friday)

Venue: Room 206, Building 12, Yagami Campus

Speaker: Mahmoud Khabou (Imperial College London)

Title: Markov approximation for controlled Hawkes Jump-Diffusions with general kernels

Abstract: We present a Markov approximation for jump-diffusions whose jump part consists in a Hawkes process with intensity driven by a general (possibly non-monotone) kernel. Under minimal integrability conditions, the kernel can be approximated by a linear combination of exponential functions. This implies that Hawkes jump-diffusions can be approximated with Markov jump-diffusions. We illustrate the usefulness of this approximation by applying it to a class of stochastic control problems.

* This event is co-organized by the Emerging Research Center of Future Climate at Keio University.

Time: 16:30-18:30, 26 December 2024 (Thursday)

Venue: Room 206, Building 12, Yagami Campus

Speaker 1: Ruiao Hu (Imperial College London)

Title: Structure-preserving stochastic modelling of submesoscale ocean dynamics

Abstract: In this talk, we consider two distinct frameworks for introducing structure-preserving stochastic perturbations to fluid dynamics from stochastic variational principles. They are Stochastic Advection by Lie Transport (SALT), and Stochastic Forcing by Lie Transport (SFLT). Out of the three fundamental properties of energy conservation, Kelvin circulation dynamics and conserved properties resulting from invariance of the Lagrangian, SALT preserves the integral conserved quantities; SFLT preserves the energy conservation; both frameworks, also possess their modified Kelvin circulation dynamics. We implement the SALT and SFLT framework for the family of shallow water equations and calibrate the noise to represent the difference between the coarse-scale numerical simulation and some given truth. Then, we consider ensemble simulations of the two stochastically perturbed models and compare their statistics.

Speaker 2: Da-jun Zhang (Shanghai University)

Title: Solitons, integrable systems and discretization

Abstract: Integrable systems involves the study of physically relevant nonlinear equations, which includes many families of well-known, highly important partial and ordinary differential equations, such as the Korteweg-de Vries and nonlinear Schrödinger equation and the Painlevé transcendents. In contrast to the well-established theory of differential equations, the theory of difference equations has until recently been quite undeveloped. Through most recent advances in the theory of discrete integrable systems, the study of difference equations has undergone a true revolution.
This talk aims to give a simple introduction to Solitons, Integrable Systems and their Discretizations from some special aspects. The talk consists of three parts. In the first part, I will recall the story of “solitons”, including the solitary wave found by John Scott Russell in 1834, the KdV equation derived in 1895 by Korteweg and de Vries to model Russell’s wave, later in 1950s the FPU (Fermi-Pasta-Ulam) problem, the Toda lattice and then “solitons” found by Zabusky and Kruskal 1965. The modern theory of integrable systems started from the establishment of Inverse Scattering Transform and the observation of Peter Lax on the connection between integrability and compatibility. In the second part, I will recall several integrable equations found in the early period and their physics backgrounds. There are also many integrable equations that have geometry interpretation. In the final part, we will see how compatibility is understood in discrete case, namely, multidimensional consistency, which has played a central role in the study of discrete integrable systems in the past two decades. 

Time: 17:00-18:30, 15 May 2024 (Wednesday)

Venue: Online (Zoom)

Speaker: Darryl D. Holm (Imperial College London)

Title: Peakons and other Emergent Singular Solutions (ESS) of the b-Equation for Nonlinear Waves

Abstract: We discuss the b-equation and illustrate lots of examples of emergent singular solutions (ESS) in nonlinear fluid wave PDEs.

(1) Integrable ESS (Peakons) in 1D geodesic flows for b=2 Camassa-H [1993], b=3 Degasperis-H-Hone [2002] Then b-equations in N dimensions, H-Staley [2003]

(2) Q: Why is b=2 special?

The ESS is a momentum map on Young measures, H & Marsden [2005]

(3) Many other geodesic equations with b=2 in 1D? Fringer & H [2001]

(4) Stochastic b=2 CH eqn, Crisan & H [2019], Bendall-Cotter-H [2022]

(5) Cotter-H-Pryer [2023] r-CH equation for W^{1,r} norm

(6) Higher dimensional 2D and 3D emergent singular solutions H-Staley [2004]

Time: 16:30-17:30, 21 February 2024 (Wednesday)

Venue: Room 217 (Discussion Room 7), Building 14, Yagami Campus

Speaker: Simone Fiori (Università Politecnica delle Marche)

Title: Automated reorientation of a spacecraft

Abstract: In rational mechanics, the starting point to develop the equations of motion of an object is the definition of its Lagrangian function. A Lagrangian is a combination of the kinetic energy of the object and of a potential energy function. This approach stays valid when one deals with translational motion as well as when one needs to describe the rotational component of motion. The role of the potential energy in determining the translational component of motion is mostly clear, as it essentially deals with gravitational attraction. But what does a potential function represent in rotational motion? The aim of the present lecture is to illustrate an example of application of potential energy in rotational motion in solving the problem of spacecraft reorientation. The application dealt with is the control of attitude of an orbiting spacecraft to a given target while avoiding unwanted directions.

Time: 16:30-17:30, 19 September 2023 (Tuesday)

Venue: Room 216 (Discussion Room 6), Building 14, Yagami Campus

Speaker: Jing Wang (Shanghai University & Waseda University)

Title: Connection between symmetric discrete AKP system and bilinear ABS lattice equations

Abstract: In this talk, we show that all the bilinear Adler-Bobenko-Suries equations (except for Q2 and Q4) can be obtained from symmetric discrete AKP system by taking proper reductions and continuum limits. A new 8-point 3-dimensional lattice equation and a new 8-point 4-dimensional lattice equation are found. Both of them can be considered as extensions of the symmetric discrete AKP equation.

Time: 16:30-18:00, 1 September 2023 (Friday)

Venue: Room 216 (Discussion Room 6), Building 14, Yagami Campus

Speaker: Michael Kraus (Max Planck Institute for Plasma Physics)

Title: Geometric Numerical Integration and Scientific Machine Learning of Hamiltonian Systems

Abstract: Many dynamical systems in physics and other fields possess a Hamiltonian structure and satisfy certain conservation laws such as momentum or energy conservation. Numerical algorithms which preserve these structures usually show greatly reduced errors compared to algorithms that do not preserve these structures, as well as much better long-time stability. In this lecture, important mathematical structures of Hamiltonian systems will be reviewed and consequences of their non-preservation in numerical simulations highlighted. Some basic structure-preserving algorithms for canonical Hamiltonian systems will be introduced and compared with their non-structure-preserving counterparts. Finally, these ideas will be brought forward to the realms of scientific machine learning, where neural networks are used to solve differential equations.

Time: 16:30-18:00, 23 May 2023 (Tuesday)

Venue: Room 217 (Discussion Room 7), Building 14, Yagami Campus

Speaker: Gianluca Frasca Caccia (Università degli Studi di Salerno)

Title: On the numerical preservation of local conservation laws

Abstract: In this talk we present a recently introduced strategy to develop bespoke finite difference methods that preserve conservation laws. Numerical methods that preserve conservation laws are more robust and reliable than standard numerical methods, as their solutions satisfy constraints at a local level that are inherited from the continuous problem. The schemes obtained by this strategy feature certain free parameters that can be arbitrarily chosen. Convenient choices of the parameters yield accurate approximations, but their optimal values are not available a priori and depend on the problem to be solved. We present a new procedure for identifying these optimal values, and we test the effectiveness and efficiency of the proposed strategy. 

Finally, we show how the technique above can be adapted to efficiently solve highly oscillatory problems.

Time: 13:00-14:30, 21 February 2023 (Tuesday)

Venue: Room 219 (Discussion Room 9), Building 14, Yagami Campus

Speaker: Yaqing Liu (Beijing Information Science and Technology University)

Title: Exotic wave patterns in Riemann problem of the high-order Jaulent–Miodek equation: Whitham modulation theory

Abstract: The Riemann problem of the high-order Jaulent–Miodek (JM) equation with initial data of step discontinuity is explored by Whitham modulation theory, which is a modified version of the well-known finite-gap integration method. Based on the reparameterization of the solution with the use of algebraic resolvent of the polynomial defining the solution, the periodic wave solutions of the high-order JM equation are described by the elliptic function along with the Whitham modulation equations. Complete classification of possible wave structures of the high-order JM equation is given for all possible jump conditions at the discontinuity initial value. The analytic results proposed in this work are confirmed by direct numerical simulations.

Time: 16:30-18:00, 20 January 2023 (Friday)

Venue: Room 218 (Discussion Room 8), Building 14, Yagami Campus

Speaker: Sehun Chun (Yonsei University)

Title: Geometry of space and time for neural spike propagation in the brain

Abstract: The brain’s cognitive function has been known to be achieved by the complex wire-like network of neural fibers. Roughly 10^11 neurons make a unique local connection among the neighboring neurons in the grey matter and a relatively universal long-distance connection through the white matter, which make the total connection up to 10^15. This connection is believed to be different for every human being and likely to change over time, mainly depending on experience, learning, and age. Numerous research has been performed to reveal the secret and powerful cognitive function by brain connectivity, but only a handful was revealed for certainty. The most critical mysteries of the brain which could not be explained by the wire-like connection structure are concerned with (1) conceptualization (ability to create an abstract concept from the data), (2) consciousness (unlearned universal network structure), and (3) stability toward adversarial attack (delicate balance between stability and plasticity of learning). One common solver for the mentioned functionality could be related to the field of the brain, i.e., space and time for neurons and surrounding media, called extracellular medium. The space and time of the brain are not just a stack room for neurons, but can play crucial roles in the efficiency and accuracy of information deliverance. In this talk, I will explain the meaning of brain’s space and its role in neural information propagation, particularly when the multidimensional space’s role emerges by the extracellular potential. This will lead to the motivation to use differential geometric tools for signal processing in the brain. An ambiguous concept is concerned with ‘time’ in the brain. Alternatively, we introduce a new notion of time for the analysis of time-dependent neural spike propagation, possibly contributing to the new dynamic connectivity mechanism for the unsolved mysterious function of the brain.

Time: 16:00-17:30, 2 December 2022 (Friday)

Venue: Room 202C, Building 12, Yagami Campus

Speaker: Ken-ichi Maruno (Waseda University)

Title: Integrable discretizations of integrable nonlinear differential equations with hodograph transformations

Abstract: Solution structure preserving discretizations of integrable systems, i.e., integrable discretizations, have been actively studied in recent years. We have obtained self-adaptive moving mesh schemes of several soliton equations involving hodograph transformations such that the Camassa-Holm equation and the short pulse equation in which mesh intervals are automatically adjusted. In this talk, I will talk about the construction of self-adaptive moving lattice schemes for the  short pulse type equations under  general boundary conditions and its application to numerical computations, as well as integrable discretizations of the SIR model.

Time: 11:00-12:00, 14 June 2022 (Tuesday)

Venue: Online (Zoom)

Speaker: Didong Li (Princeton University & University of California, Los Angeles)

Title: Multi-group Gaussian Processes

Abstract: Gaussian processes (GPs) are pervasive in functional data analysis, machine learning, and spatial statistics for modeling complex dependencies. Modern scientific data sets are typically heterogeneous and often contain multiple known discrete subgroups of samples. For example, in genomics applications samples may be grouped according to tissue type or drug exposure. In the modeling process, it is desirable to leverage the similarity among groups while accounting for differences between them. While a substantial literature exists for GPs over Euclidean domains, GPs on domains suitable for multi-group data remain less explored. In this talk, I'll introduce multi-group Gaussian processes (MGGPs) define on the product of a Euclidean space and a categorical set, where the categorical set represents the group label. General methods to construct valid (positive definite) covariance functions on this domain are provided, together with algorithms for inference, estimation, and prediction. The application to gene expression data illustrates the behavior and advantages of the MGGP in the joint modeling of continuous and categorical variables.

Time: 16:30-17:30, 20 May 2022 (Friday)

Venue: Online (Zoom)

Speaker: Aminreza Karamoozian (Tarbiat Modares University)

Title: Mechanics of Kirigami-based Reconfigurable Structures: Programmable active kirigami metasheets

Abstract: The mechanical behavior of kirigami-based reconfigurable two-dimensional (2D) and three-dimensional (3D) structures is of great interest in fundamental research. Kirigami, an ancient Japanese paper cutting craft, has involved the ability to develop new scientific research and engineering innovations ranging from mechanical metamaterials, stretchable gadgets, and solar tracking to self-assembled 3D meso-structures. However, it lacks a basic understanding of how cut-structures influence the macroscopic mechanical response of kirigami structures. This presentation will discuss the theoretical framework for connecting and integrating macroscopic mechanical behavior and cuts-based microstructures in a novel class of kirigami-based structures that can be reconfigured in 2D and 3D.

Time: 16:30-17:30, 15 April 2022 (Friday)

Venue: Online (Zoom)

Speaker: Xin Zhang (Tongji University)

Title: Elementary introduction to mathematical theory in fluid dynamics 2: Algebra structure and weak solution of N-S equations

Abstract: In this short lecture, we first introduce the modelling of the classical equations in the fluid mechanics, such as Navier-Stokes equations or Euler equations. Then we list some basic properties of the incompressible Navier-Stokes equations (or Euler equations). At last, we introduce the weak solution of the Navier-Stokes equations. 

Time: 16:30-17:30, 21 December 2021 (Tuesday)

Venue: Online (Zoom)

Speaker: Hiroaki Yoshimura (Waseda University)

Title: Dirac dynamical formulation of nonholonomic systems and its applications    

Abstract:  In this presentation, I will mainly talk about the Dirac dynamical formulation of thermodynamics together with some review of Dirac structures in mechanics.

Time: 16:30-17:30, 19 October 2021 (Tuesday)

Venue: Online (Zoom)

Speaker: Xin Zhang (Tongji University)

Title: Elementary introduction to mathematical theory in fluid dynamics 1: Modelling

Abstract: In this short lecture, we first introduce the modelling of the classical equations in the fluid mechanics, such as Navier-Stokes equations or Euler equations. Then we list some basic properties of the incompressible Navier-Stokes equations (or Euler equations). At last, we introduce the weak solution of the Navier-Stokes equations.