L4

In this talk, we present a mathematical model for tumour growth that incorporates viscoelastic effects. Starting from a basic system of PDEs, we gradually introduce the relevant biological and physical mechanisms and explain how they are integrated into the model. The resulting system features a Cahn--Hilliard type equation for the tumour cells coupled to a convection-reaction-diffusion equation for a nutrient species, and a viscoelastic subsystem for an internal velocity.
Key biological processes such as active transport, apoptosis, and proliferation are modeled via source and sink terms as well as cross-diffusion effects. The viscoelastic behaviour is described using the Oldroyd-B model, which is based on a multiplicative decomposition of the deformation gradient to account for elasticity alongside growth and relaxation effects.
We will highlight several of these effects through numerical simulations.
Moreover, we discuss the main analytical and numerical challenges. Particular focus will be given to the treatment of source and cross-diffusion terms, the elastic energy density, and the difficulties arising from the viscoelastic subsystem. The main analytical result is the global-in-time existence of weak solutions in two spatial dimensions, under the assumption of additional viscoelastic diffusion in the Oldroyd-B equation.
This work is based on joint work with Harald Garcke (University of Regensburg, Germany) and Balázs Kovács (University of Paderborn, Germany).

Systems of high-contrast resonators can be used to control and manipulate wave-matter interactions at scales that are much smaller than the operating wavelengths. The aim of this talk is to review recent studies of ordered and disordered systems of subwavelength resonators and to explain some of their topologically protected localization properties. Both reciprocal and non-reciprocal systems will be considered.
 

In the first half of the talk I will review the theory of nuclear magnetic resonance (NMR), leading to the Bloch-Torrey PDE. I will then describe the pulsed-gradient spin-echo method for measuring the Fourier transform of the voxel-averaged propagator of the Bloch-Torrey equation.  This technique permits one to compute the diffusion coefficient in a voxel. For complex biological tissue, as in the brain, the standard model represents spin-echo as a multiexponential signal, whose exponents and coefficients describe the diffusion coefficients and volume fractions of isolated tissue compartments, respectively. The question of identifying these parameters from experimental measurements leads us to investigate the degree of well-posedness of this problem that I will discuss in the second half of the talk. We show that the parameter reconstruction problem exhibits power law transition to ill-posedness, and derive the explicit formula for the exponent by reformulating the problem in terms of the integral equation that can be solved explicitly. This is a joint work with my Ph.D. student Henry J. Brown.

Dey and Xin (J. Appl.Comput.Top. 2022) describe an algorithm to decompose finitely presented multiparameter persistence modules using a matrix reduction algorithm. Their algorithm only works for modules whose generators and relations are distinctly graded. We extend their approach to work on all finitely presented modules and introduce several improvements that lead to significant speed-ups in practice.

Our algorithm is FPT with respect to the maximal number of relations with the same degree and with further optimisation we obtain an O(n³) algorithm for interval-decomposable modules. As a by-product to the proofs of correctness we develop a theory of parameter restriction for persistence modules. Our algorithm is implemented as a software library aida which is the first to enable the decomposition of large inputs.

This is joint work with Tamal Dey and Michael Kerber.

In this talk, we consider Dirichlet forms related to stochastic quantisation for the exp(φ)_{2}-model on the torus. We show strong uniqueness of the corresponding Dirichlet operators by applying an idea of (singular) SPDEs. This talk is based on ongoing joint work with Hirotatsu Nagoji (Kyoto University).

In this talk, I will present the generalization of scaling limit results for stochastic transport equations on torus by Flandoli, Galeati and Luo, to compact manifolds. We consider the stochastic transport equations driven by colored space-time noise(smooth in space, white in time) on a compact Riemannian manifold without boundary. Then we study the scaling limits of stochastic transport equations, tuning the noise in such a way that the space covariance of the noise on the diagonal goes to identity matrix but the covariance operator itself goes to zero, which includes the large scale analysis regime with diffusive scaling.

We obtain different scaling limits depending on the initial data. With space white noise as initial data, the solutions converge in distribution to the solution of a stochastic heat equation with additive noise. With square integrable initial data, the solutions of transport equation converge to the solution of the deterministic heat equation, and we give quantitative estimates on the convergence rate.

We present an existence and uniqueness result for nonlinear Fokker--Planck equations driven by rough paths. These equations describe the evolution of the probability distributions associated with McKean--Vlasov stochastic dynamics under (rough) common noise.  A key motivation comes from the study of interacting particle systems with common noise, where the empirical measure converges to a solution of such a nonlinear equation. 
Our approach combines rough path theory and the stochastic sewing techniques with Lions' differential calculus on Wasserstein spaces.

This is joint work with Peter K. Friz and Wilhelm Stannat.