Special day and time: Friday, October 20, 2023, 11:00 AM-12:00 PM, 210 Mudd, APAM Conference Room

Speaker: José A. Carrillo, Oxford University

Title: Aggregation-Diffusion Equations for Collective Behaviour in the Sciences

Abstract: Many phenomena in the life sciences, ranging from the microscopic to macroscopic level, exhibit surprisingly similar structures. Behaviour at the microscopic level, including ion channel transport, chemotaxis, and angiogenesis, and behaviour at the macroscopic level, including herding of animal populations, motion of human crowds, and bacteria orientation, are both largely driven by long-range attractive forces, due to electrical, chemical or social interactions, and short-range repulsion, due to dissipation or finite size effects.

Various modelling approaches at the agent-based level, from cellular automata to Brownian particles, have been used to describe these phenomena. An alternative way to pass from microscopic models to continuum descriptions requires the analysis of the mean-field limit, as the number of agents becomes large. All these approaches lead to a continuum kinematic equation for the evolution of the density of individuals known as the aggregation-diffusion equation. This equation models the evolution of the density of individuals of a population, that move driven by the balances of forces: on one hand, the diffusive term models diffusion of the population, where individuals escape high concentration of individuals, and on the other hand, the aggregation forces due to the drifts modelling attraction/repulsion at a distance.

The aggregation-diffusion equation can also be understood as the steepest-descent curve (gradient flow) of free energies coming from statistical physics. Significant effort has been devoted to the subtle mechanism of balance between aggregation and diffusion. In some extreme cases, the minimisation of the free energy leads to partial concentration of the mass.

Aggregation-diffusion equations are present in a wealth of applications across science and engineering. Of particular relevance is mathematical biology, with an emphasis on cell population models. The aggregation terms, either in scalar or in system form, is often used to model the motion of cells as they concentrate or separate from a target or interact through chemical cues. The diffusion effects described above are consistent with population pressure effects, whereby groups of cells naturally spread away from areas of high concentration. This talk will give an overview of the state of the art in the understanding of aggregation-diffusion equations, and their applications in mathematical biology.

This talk will be offered in a hybrid format. If you wish to participate remotely, please send an email to [email protected].
 

---

 

José A. Carrillo is Professor of Nonlinear Partial Differential Equations at the Mathematical Institute of the University of Oxford. He is also a Tutorial Fellow in Applied Mathematics at The Queen´s College.

He was previously Chair in Applied and Numerical Analysis at Imperial College London from October 2012 till March 2020 and formerly ICREA Research Professor at the Universitat Autònoma de Barcelona during the period 2003-2012. He was a lecturer at the University of Texas at Austin 1998-2000. He held assistant and associate professor positions at the Universidad de Granada 1992-1998 and 2000-2003, where he also did his PhD.

His research field is Partial Differential Equations (PDE). They constitute the basic language in which most of the laws in physics or engineering can be written and one of the most important mathematical tools for modelling in life and socio-economical sciences. The modelling based on PDEs, their mathematical analysis, the numerical schemes, and their simulation in applications are my general topics of research.

His expertise comprises long-time asymptotics, qualitative properties and numerical schemes for nonlinear diffusion, hydrodynamic, and kinetic equations in the modelling of collective behaviour of many-body systems such as gas molecules in rarefied gases, sand beads in granular media, charge particle transport in semiconductors, synchronization of neurons in computational neuroscience or cell movement by chemotaxis or adhesion forces.

He served as chair of the Applied Mathematics Committee of the European Mathematical Society 2014-2017. He was the chair of the 2018 Year of Mathematical Biology. He is currently the Program Director of the SIAM activity group in Analysis of PDE and member of the Board of the European Society for Mathematical and Theoretical Biology 2021-. He has organized a large number of scientific events and summer schools at BIRS, ICMS, and MFO; research thematic programs: WPI Vienna 2007, IPAM-UCLA 2008, INI-Cambridge 2010, Institut Mittag-Leffler 2016. JAC is co-organizing a thematic program during the first semester of 2022 at the Isaac Newton Institute entitled ``Frontiers in Kinetic Theory: Connecting Microscopic to Macroscopic Scales''.

He has been elected as a member of the European Academy of Sciences, Section Mathematics, in 2018 and SIAM Fellow Class 2019. He is currently the head of the Division of the European Academy of Sciences, Section Mathematics. He has served on ERC panels in Mathematics for Starting and Consolidator Grants.

He has been visiting professor in top universities worldwide, for instance, he held an IBM Visiting Professorship 2017 (Brown-USA) and a Changjiang Visiting Scholarship in China at SWUFE-Chengdu 2018--2020, being one of the few non-Chinese born researchers to have been awarded, such fellowship. He has given more than 400 seminars in leading universities worldwide. He is now Visiting Professor at Shanghai Jiatong University 2021-2014. He has been regularly invited as a plenary speaker at international major conferences in the area: Mathematics and its Applications (Joint French-Italian Math Societies Meeting, 2006), 5th European Congress of Mathematicians (2008), Canadian Mathematical Society Summer Meeting (2013), XV International Conference on Hyperbolic Problems: Theory, Numerics and Applications (2014), and more recently the joint British (Applied) Mathematical Colloquium 2021, the SIAM-CAIMS Annual Meeting (Toronto 2020) and the ENUMATH (Lisbon, 2021) for instance.

He was recognized with the SEMA prize (2003) and the GAMM Richard Von-Mises prize (2006) for young researchers. He was a recipient of a Wolfson Research Merit Award by the Royal Society 2012-2017.

He has an extensive mentoring experience of postdocs and PhD students. He was awarded the 2016 SACA award for best PhD supervision at Imperial College London. He has been Highly Cited Researcher 2015, 2016, 2017, 2018, 2019 and 2020 by the Web of S