About Us - mSpace

Project Overview

mSPACE is a COST Action (CA24122) that seeks to establish a rigorous mathematical foundation for understanding and analyzing multiscale systems, where intricate dynamics arise from the interplay of interactions across various scales that are pervasive in nature and society.

The primary aim is to construct a unified mathematical framework to study multiscale aspects of geometrically structured spaces using analytic, combinatorial, and stochastic approaches. The project focuses on developing theoretical foundations while tailoring this framework to real-world applications.

Key Activities

Short-Term Scientific Missions (STSMs): Twice-yearly calls supporting international collaboration for doctoral students and early-career researchers
Focused Innovative Sessions (FISs): Regular gatherings of 4-5 scientists from different working groups addressing strategic problems
Training Schools: Cross-working group scientific exchange and career development discussions
Annual Meetings: Gatherings of all Action members

Research Focus Areas

The project explores systems across three critical dimensions

Spatial Resolution

From microscopic to coarse-grained to macroscopic scales:
mSPACE studies systems from nanoscale battery materials to continent-wide networks, bridging atomic interactions with large-scale behaviors

Temporal Scale

From fine time dynamics to long-time/low-temperature behavior:
mSPACE analyzes rapid molecular processes alongside slow evolutionary changes, connecting short-term fluctuations to stable long-term patterns.

Stochastic Features

From deterministic to random systems:
mSPACE explores predictable mathematical models versus probabilistic systems, incorporating uncertainty and randomness into complex system analysis.

Working Group Structure

WG1: Analysis of Discrete and Metric Complex Systems

Bridges gaps between discrete and metric graph theory
Extends to hypergraphs, graphons, and open books
Studies functional inequalities and Laplace-type operators
Forms the core of mSPACE, interconnecting with all other working groups

WG2: Spectral Geometry of Continuous Spaces

Focuses on spectral problems for continuous spaces like smooth manifolds
Investigates spectral optimizers and geometric stability properties
Analyzes nodal domains and eigenfunctions
Studies pseudospectrum of non-self adjoint operators

WG3: Dynamics on Structured Spaces

Studies dynamics within structured spaces including networks and hybrid spaces
Focuses on diffusive and dissipative systems through gradient flows
Addresses homogenization properties and discrete-to-continuum limits

WG4: Probabilistic Methods and Mathematical Foundations of Materials Science

Analyzes Poissonian systems and simplicial complexes
Compares stochastic geometry models with machine learning approaches
Explores spatial structures governed by fractional differential equations
Focuses on morphological descriptors and homogenization theories

Implementation Timeline and Activities

Four-Year Structure (2025-2029)

Year 1

Establishment of communication infrastructure, first workshops, and Science Communication Plan development

Years 2-3

Training Schools for students and young researchers, intensive workshop periods with two workshops each year.

Years 4

Final conference, comprehensive scientific report publication, and European Congress of Mathematics session organization

Mathematical Framework and Tools

mSPACE employs six powerful mathematical tools that collectively address multiscale phenomena

Spectral Geometry & Shape Optimization

Analyzes systems through their spectral “fingerprint,” deriving properties of spectrum and eigenfunctions to understand connectivity, robustness, and information flow across systems.

Gradient Flows

Investigates system stability by defining appropriate energy functionals and gradient notions, enabling optimization tasks like designing communication networks with minimal congestion.

Stochastic Geometry & Percolation Theory

Incorporates randomness to create realistic porous medium models, capturing inherent variability and uncertainty of real-world systems.

Complex Networks

Provides tools tailored to highly intertwined and dynamic interaction structures in branched systems, whether natural or laboratory-designed.

Discrete Analysis & Space Homogenization

Extends classical harmonic analysis to discrete point sets while approximating large sparse network evolution through partial differential equations

Optimal Transport

Operates at the interface of functional analysis, stochastics, and geometry, offering theoretical frameworks to transfer properties between different spaces.