On the effectiveness of algebraic and geometric structures in the natural sciences

26.11.2025, 14:00 - 16:00  –  Campus Golm, Building 9, Room 2.22 and via Zoom
Institutskolloquium

Nichol Furey (HU Berlin), Jorge Zanelli (CECs, Chile)

14:00  Nichol Furey (HU Berlin):  The unreasonable effectiveness of division algebras in the natural sciences.
14:45  Tea and Coffee Break
15:15  Jorge Zanelli (CECs, Chile):  The unreasonable effectiveness of Chern-Simons forms in the natural sciences.

Nichol Furey (HU Berlin):  The unreasonable effectiveness of division algebras in the natural sciences.

Abstract:     WIt is demonstrated how a set of particle representations, familiar from the Standard Model, collectively form a superalgebra. Those representations mirroring the behaviour of the Standard Model's gauge bosons, and three generations of fermions, are each included in this algebra - in an unconventional way. This superalgebra is isomorphic to the Euclidean Jordan algebra of \(16\times 16\) hermitian matrices, \(\mathcal{H}_{16}(\mathbb{C})\) and is generated by division algebras. The division algebraic substructure

1. enables a natural factorization between internal and spacetime symmetries, and
2. ) allows for the definition of a \(\mathbb{Z}_2^5\) grading on the algebra.

Those internal symmetries respecting this substructure are found to be \(\mathfrak{su}(3)_C \oplus \mathfrak{su}(2)_L \oplus \mathfrak{u}(1)_Y,\) in addition to four iterations of \(\mathfrak{u}(1)\) For spatial symmetries, one finds multiple copies of \(\mathfrak{so}(3)\). Even though this model involves a superalgebra, it is not supersymmetric.

Very little background knowledge will be assumed in this talk. Students are especially welcome.
    

Jorge Zanelli (CECs, Chile):  The unreasonable effectiveness of Chern-Simons forms in the natural sciences.

Abstract:     The Chern-Simons (CS) forms made their début in mathematics in a classical paper by Shiing-Shen Chern and James Simons in 1974. For the authors the CS forms arose as some unwelcome “boundary term” obstructing their construction of a combinatorial formula for the Pontryagin invariant of a four manifold. This annoying boundary term was also found in physics in connection with anomalies in quantum field theories, which is natural because they also involve the Pontryagin invariant. By the end of the 70s, CS forms were also used as Lagrangians for rather exotic theoretical systems such as supergravity in 11 space-time dimensions and electrodynamics in 3 spacetime dimensions. During the next decade, CS forms were found to be useful to describe the quantum Hall effect and high temperature superconductivity in condensed matter systems. Gradually, CS forms were starting to pop up in some rather obscure exceptional physical systems. But in retrospect, however, one can see that the simplest CS forms were already present in electrodynamics and even in classical mechanics, in use since the mid-nineteenth century. In this talk, I will review some of these examples and argue that CS forms are also at the core of quantum mechanics and can be useful to describe fluid dynamical systems like the earth’s atmosphere.
    

Wenn Sie digital an den Vorträgen teilnehmen möchten, wenden Sie sich bitte an Christian Molle molle @ uni-potsdam.de, um die Zugangsdaten zu erhalten.