L4

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.

We establish general conditions for stochastic evolution equations with locally monotone drift and degenerate additive Lévy noise in variational formulation resulting in the existence of a unique invariant probability measure for the associated ergodic Markovian Feller semigroup. We prove improved moment estimates for the solutions and the e-property of the semigroup. Examples include the stochastic incompressible 2D Navier-Stokes equations, shear thickening stochastic power-law fluid equations, the stochastic heat equation, as well as, stochastic semilinear equations such as the 1D stochastic Burgers equation.

Joint work with Gerardo Barrera (IST Lisboa), https://arxiv.org/abs/2412.01381

The normal covering number $\gamma(G)$ of a finite group $G$ is the minimal size of a collection of proper subgroups whose conjugates cover the group. This definition is motivated by number theory and related to the concept of intersective polynomials. For the symmetric and alternating groups we will see how these numbers are closely connected to some elementary (as in "relating to basic concepts", not "easy") problems in additive combinatorics, and we will use this connection to better understand the asymptotics of $\gamma(S_n)$ and $\gamma(A_n)$ as $n$ tends to infinity.

When it comes to the nonlinear heat equation u_t - \Delta u = u^p, a sharp condition for the stability of positive radial steady states was derived in the classical paper by Gui, Ni and Wang.  In this talk, I will present some recent joint work with Daniel Devine that focuses on a more general system of reaction-diffusion equations (which is also also known as the parabolic Henon-Lane-Emden system).  We obtain a sharp condition that determines the stability of positive radial steady states, and we also study the separation property of these solutions along with their asymptotic behaviour at infinity.