SMS scnews item created by Catherine Meister at Mon 14 Jul 2025 1456
Type: Seminar
Distribution: World
Expiry: 21 Jul 2025
Calendar1: 17 Jul 2025 1300-1400
CalLoc1: SMRI Seminar Room (A12 Room 301)
Auth: cmeister@159-196-153-133.9fc499.syd.nbn.aussiebb.net (cmei0631) in SMS-SAML

Geometry and Topology Seminar: Chern

New geometric theorems about Fluids and Conics



Speaker: Albert Chern

Abstract: This talk consists of two recent work, one about topological analysis on fluid dynamics, and the other about Penrose’s 8-conic theorem. Vorticity formulation is a widespread description in fluid mechanics. However, its applicability has been limited to simply-connected domains. We show that on non-simply-connected domains, fluid’s cohomology component plays an important role and can interact with fluids. This interaction corresponds to a new equation of motion and new conservation laws which can be viewed as Casimir invariants in Hamiltonian formulation of fluid dynamics. The new equation allows us to construct new analytical solutions to Euler’s equation in terms of the Hilbert transforms on complex manifolds. In the special case of point vortices on surfaces, the configuration can be viewed as divisors on a Riemann surface. In this fluid analogy of the theory of algebraic curve, the cohomology equation can be phrased elegantly in terms of the divisor class group realized in the Jacobi variety. The second part of the talk is about an incidence theorem about conics in double contact, first discovered by Sir Roger Penrose in his undergraduate years but remained unpublished until recently revealed in interviews. The theorem includes as special cases many well-known theorems of projective geometry as well as in metric geometries. Through collaboration with a few projective geometry enthusiasts and Penrose himself, the theorem is further developed. I will show four proof sketches.


Actions:
ball Calendar (ICS file) download, for import into your favourite calendar application
ball UNCLUTTER for printing
ball AUTHENTICATE to mark the scnews item as read
School members may try to .