Project description
Rethinking evolution equations through the lens of optimal transport
Understanding how complex systems evolve over time is a central challenge in mathematics and physics. The ERC-funded OPTiMiSE project seeks to tackle this by combining powerful ideas from the theory of optimal transport with variational methods that describe evolution problems like gradient flows and rate-independent processes. By working in Kantorovich-Wasserstein spaces (where probability measures replace traditional variables), researchers aim to reveal deeper structural insights into how these systems behave. OPTiMiSE will explore new metric approaches to address unresolved questions and find new ways to model the geometry, stability and dynamics of evolving systems.
Objective
Several evolution problems, such as gradient flows or rate-independent processes, are governed by variational principles which are extremely useful for studying the existence, stability, and structural properties of solutions by simple and general constructive approximation methods.
Deep and beautiful ideas from the theory of Optimal Transport have contributed new insights and additional challenging questions to this scenario and have motivated flourishing and original developments. On the one hand, the applications to gradient flows in the Kantorovich-Wasserstein spaces of probability measures reveal the importance, the power, and the flexibility of the metric viewpoint. On the other hand, the interplay with evolutionary problems has in turn brought new ideas and perspectives to Optimal Transport, inspiring a powerful set of techniques for its applications, especially to the analysis and geometry in metric-measure spaces.
In recent years, the PI and his collaborators have given relevant contributions to the general theory of gradient flows, in particular in Kantorovich-Wasserstein spaces, and they have obtained ground-breaking results for metric-measure spaces and Unbalanced Optimal Transport between positive measures with finite mass.
The goal of the project is a wide-ranging analysis which aims to combine and broaden the above themes and perspectives, to address crucial and challenging open problems, and to open up novel research directions:
- new generation results and metric-variational principles for evolution equations,
- the interplay between curvature bounds and convergence of variational approximation schemes,
- a new metric approach to dissipative evolution and saddle-point flows,
- new methods and results for paradigmatic highly nonlinear and non-convex partial differential equations for probability measures,
- the foundation of a mean-field theory for the rate-independent evolution of critical points.
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
This project's classification has been human-validated.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
This project's classification has been human-validated.
- natural sciences mathematics applied mathematics mathematical physics
- natural sciences mathematics pure mathematics geometry
- natural sciences mathematics applied mathematics statistics and probability
- natural sciences mathematics pure mathematics mathematical analysis differential equations partial differential equations
Keywords
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
- Gradient Flows
- Dissipative Evolution
- Rate-Independent Processes
- Optimal Transport
- Wasserstein Metric
- Probability Vector Fields
- Variational Methods
- Metric-Measure Spaces
- Unbalanced Optimal Transport
- Hellinger-Kantorovich Metric
- Minimizing Movements
- JKO Scheme
- Evolution Variational Inequalities
- Energetic and BV Solutions
Programme(s)
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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HORIZON.1.1 - European Research Council (ERC)
MAIN PROGRAMME
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Topic(s)
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Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Funding Scheme
Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
HORIZON-ERC - HORIZON ERC Grants
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Call for proposal
Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
(opens in new window) ERC-2024-ADG
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Net EU financial contribution. The sum of money that the participant receives, deducted by the EU contribution to its linked third party. It considers the distribution of the EU financial contribution between direct beneficiaries of the project and other types of participants, like third-party participants.
20136 Milano
Italy
The total costs incurred by this organisation to participate in the project, including direct and indirect costs. This amount is a subset of the overall project budget.