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Explore advanced differential geometry through this bi-weekly seminar series focused on the SCREAM collaboration project, examining Cartan and parabolic geometries including conformal, projective, CR, and ODE geometry. Delve into sophisticated mathematical structures such as (2,3,5)-distributions and parabolic contact structures while investigating their applications to mechanical systems, geometric robotics, integrable systems theory, and Penrose's Conformal Cyclic Cosmology programme. Study conformal transformations and their role in understanding the universe's beginning, learn classification methods for homogeneous geometric structures, and examine normal forms and symmetries for various distribution types. Investigate dispersionless integrable systems, geometry of rank 2 distributions via abnormal extremals, and Cauchy-Riemann structures on hypersurfaces. Master moving frames and invariants for submanifolds in parabolic homogeneous spaces, gain introduction to 11-dimensional supergravity, and explore Frobenius integrability in Cartan geometries. Analyze symmetries with power series, symmetric trilinear forms and Einstein-like equations, and their connections from affine spheres to Griess algebras. Learn about symmetries, conservation laws and variational principles, study the Cartan-Karlhede algorithm and Cartan invariants for spacetimes, and discover applications of tractor calculus in general relativity. Examine ODEs satisfied by modular forms, dispersionless integrable equations, spacetime G-structures, and the geometry of quantum correlations, providing comprehensive coverage of cutting-edge research in differential geometry and its interdisciplinary applications.
