Universitas Gadjah Mada, Yogyakarta, Indonesia
July 15, 2025
Lecturing Activities
The first lecture was delivered by Prof. Valerio Talamanca on Algebraic Number Theory. He continued from the previous session by introducing the basic notions of Noetherian rings, integrally closed domains, and Krull dimension. He then defined Krull domains and Dedekind domains, discussing their important properties, and provided a remark that the ring of integers đť’Ş_F in a number field F is a Dedekind domain. The lecture concluded with an introduction to fractional ideals and a brief discussion on ramified extensions.
The next lecture was delivered by Prof. Fabien Pazuki on the Geometry of Numbers. He began by explaining the concept of the covolume of a lattice and presented a theorem stating that for a bounded, convex, symmetric subset B in a vector space V, under certain conditions involving the volume of B and the covolume of a lattice L, the intersection of B and L is non-empty. He concluded the lecture by highlighting the important result that there are no unramified extensions of â„š except â„š itself.
The final lecture of the day was given by Dr. Richard Griffon on Projective Geometry. He began with the definition and key properties of the projective plane as an extension of the affine plane. He then defined elliptic curves within the projective plane and explained how the group law can be established on these curves in the projective setting. Additionally, he introduced algebraic maps and isogeny between elliptic curves, concluding with a beautiful result on the isomorphism of elliptic curves characterized by the j-invariant condition.
Exercise Session
The day concluded with an exercise session where participants worked in groups to practice the concepts learned about algebraic number theory and elliptic curves over the past two days. The session took place in a productive and collaborative atmosphere, with active support and guidance provided by the speakers.