GSI'25

Keynote speakers

Prof. Nina MIOLANE

Assistant Professor, AI, UC Santa Barbara. Co-Director, AI Center, Bowers Women's Brain Health Initiative. Affiliate, Stanford SLAC

Topological Deep Learning: Unlocking the Structure of Relational Systems

The natural world is full of complex systems characterized by intricate relations between their components: from social interactions between individuals in a social network to electrostatic interactions between atoms in a protein. Topological Deep Learning (TDL) provides a framework to process and extract knowledge from data associated with these systems, such as predicting the social community to which an individual belongs or predicting whether a protein can be a reasonable target for drug development. By extending beyond traditional graph-based methods, TDL incorporates higher-order relational structures, providing a new lens to tackle challenges in applied sciences and beyond. This talk will introduce the core principles of TDL and provide a comprehensive review of its rapidly growing literature, with a particular focus on neural network architectures and their performance across various domains. I will present open-source implementations that make TDL methods more accessible and practical for real-world applications. All in all, this talk will showcase how TDL models can effectively capture and reason about the complexity of real-world systems, while highlighting the remaining challenges and exciting opportunities for future advancements in the field.

Bibliography

  1. Hajij, M., Papillon, M., Frantzen, F., Agerberg, J., AlJabea, I., Ballester, R., … & Miolane, N. (2024). TopoX: a suite of Python packages for machine learning on topological domains. Journal of Machine Learning Research, 25(374), 1-8.
  2. Papamarkou, T., Birdal, T., Bronstein, M. M., Carlsson, G. E., Curry, J., Gao, Y., … & Zamzmi, G. (2024). Position: Topological Deep Learning is the New Frontier for Relational Learning. In Forty-first International Conference on Machine Learning.
  3. Papillon, M., Bernárdez, G., Battiloro, C., & Miolane, N. (2024). TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks. arXiv preprint arXiv:2410.06530.
  4. Papillon, M., Sanborn, S., Hajij, M., & Miolane, N. (2023). Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networks. arXiv preprint arXiv:2304.10031.
  5. Hajij, M., Zamzmi, G., Papamarkou, T., Miolane, N., Guzmán-Sáenz, A., Ramamurthy, K. N., … & Schaub, M. T. (2022). Topological deep learning: Going beyond graph data. arXiv preprint arXiv:2206.00606.

Prof. José Figueroa-O'Farrill

School of Mathematics, University of Edinburgh

Cohomological approach to symplectic reduction and applications in string theory: some old and new results

Coisotropic reduction in symplectic geometry can be phrased cohomologically and goes under the name of BRST cohomology in Physics.  It provides a quantisation procedure for gauge theories which is equivariant under global symmetries.  It was first discovered in the context of gauge theories in the mid 1970s, but it plays a very important role in the quantisation of string theories, where it usually appears in th guise of semi-infinite cohomology, a cohomology theory for certain infinite-dimensional Lie algebras which sits in between homology and cohomology.  I will summarise some of the history of the subject and mention a recent application in the context of so-called non-relativistic strings.

Bibliography:

  1. Kostant and S. Sternberg, Symplectic reduction, BRS cohomology and infinite-dimensional Clifford algebras, Ann. of Physics 176 (1987) 49
  2. B. Frenkel, H. Garland and G.J. Zuckerman, Semi-infinite cohomology and string theory, Proc. Natl. Acad. Sci. USA 83 (1986) 8442
  3. M.Figueroa-O’Farrill and G. S. Vishwa, The BRST quantisation of chiral BMS-like field theories, arXiv:2407.12778 [hep-th]
  4. M.Figueroa-O’Farrill, E. Have and N. Obers, Quantum carrollian strings, work in progress

 

Philip J. Morrison

The University of Texas at Austin , Physics Department

Metriplectic Dynamics: A Geometrical Framework for Thermodynamically Consistent Dynamical Systems

Classical descriptions of matter present many fluid mechanical and kinetic theory dynamical systems. These include, e.g., the Navier-Stokes-Fourier system, the Cahn-Hilliard-Navier-Stokes system for multiphase fluid flow, and various types of collisional kinetic theories for gaseous and plasma modeling.  A desirable feature of such modeling is thermodynamic consistency, i.e., conservation of energy and production of entropy, in agreement with the first and second laws of thermodynamics. Metriplectic dynamics is a kind of dynamical system (finite or infinite) that encapsulates in a geometrical formalism such thermodynamic consistency.  An algorithmic procedure for building such theories is based on the metriplectic 4-bracket, a bracket akin to the Poisson bracket that maps phase space functions to another.  However,  the 4-bracket maps 4 such functions and has algebraic curvature symmetries.  Metriplectic 4-brackets can be constructed using the Kulkarni-Nomizu product or via a pure Lie algebraic formalism based on the Koszul connection.  The formalism algorithmically produces many known and new dynamical systems, and it provides a pathway for constructing structure preserving numerical algorithms.  

Bibliography:

  1. P. J. Morrison and M. Updike, « Inclusive Curvature-Like Framework for Describing Dissipation: Metriplectic 4-Bracket Dynamics,” Physical Review E 109, 045202 (22pp) (2024). 
  2. A. Zaidni, P. J. Morrison, and S. Benjelloun,, « Thermodynamically Consistent Cahn-Hilliard-Navier-Stokes Equations using the Metriplectic Dynamics Formalism,” Physica D 468, 134303 (11pp) (2024).
  3. N. Sato and P. J. Morrison, « A Collision Operator for Describing Dissipation in Noncanonical Phase Space,”Fundamental Plasma Physics 10, 100054 (18pp) (2024)
  4. W.  Barham, P.  J. Morrison, A.  Zaidni, « A thermodynamically consistent discretization of 1D thermal-fluid models using their metriplectic 4-bracket structure, » arXiv:2410.11045v2 [physics.comp-ph] 19 Oct 2024.
  5. A. Zaidni, P. J. Morrison, « Metriplectic 4-bracket algorithm for constructing thermodynamically consistent dynamical systems, » arXiv:2501.00159v1 [physics.flu-dyn] 30 Dec 2024.
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