BMST Talks 1/2

Students from the 4EU+ alliance (https://4euplus.eu/4EU-578.html) will give talks on topics related to p-adic numbers, as part of the online course BMST 2026 (https://www.math.ku.dk/english/programmes/bmst-2026/).

Schedule:

• Thursday 9/04, 17:00-18:00. Talk 1: Jonathan Sebastian Carvajal Ocampo (Milan), Introduction to p-adic numbers.

Abstract: If the geometric series satisfies 1+q+q^2+q^3+...=1/(1-q), why not set q=2 ? This gives 1+2+2^2+2^3+...=1/(1-2)=-1. Absurd? Or are we perhaps considering the wrong norm?
This project introduces the p-adic absolute value, the field Q_p of p-adic numbers, and the ring of p-adic integers Z_p. The final goal is Hensel’s lemma and its application to solving polynomial equations over the p-adic numbers.

• Friday 10/04, 9:30-10:30. Talk 2: Piotr Marszalik (Warsaw), Monsky's theorem and equidissections of hypercubes.

Abstract: Can you cut a square into three triangles of equal area? While it sounds like a trivial school geometry puzzle, this question remained open until Paul Monsky’s proof in 1970. Monsky demonstrated that such a dissection is impossible for any odd number of triangles, utilizing techniques from both combinatorial topology and the theory of p-adic valuations.
In 1979, David Mead extended this result to higher dimensions. For instance, he showed that a three-dimensional cube can be divided into m tetrahedra of equal volume if and only if m is a multiple of 6 (3!).
This talk will outline the key steps of these proofs, focusing on the p-adic labeling of vertices and the application of Sperner’s Lemma to identify specific geometric configurations. Finally, we will see how the p-adic valuation of coordinates restricts the possible values of m in any dimension.

• Friday 10/04, 11-12. Talk 3: Mariama Charpentier-Sow (Paris Sorbonne), Monsky's theorem and equidissections of regular polygons.

Abstract: In 1965, F. Richman wondered if it was possible to cut a square into an odd number of non-overlapping triangles of the same area. P. Monsky proved that it was not in 1970. In 1985, the question was asked for other regular polygons and E. Kasimatis proved that for n>=5, a regular n-gon can be dissected into m non-overlapping triangles of equal area if and only if n divides m. In this talk, we will focus on the latter result.
Both of those results regard basic geometrical objects, known and studied for centuries, but their proofs surprisingly use modern tools such as Sperner’s Lemma from combinatorial theory and p-adic valuations and their extensions to R and C.

• Friday 10/04, 13:30-14:30. Talk 4: Sofia Alexovičova (Prague), Tate algebras are the polynomial algebras of rigid geometry, part 1.

Abstract: This talk provides an introduction to Tate algebras, the rings of strictly convergent power series in several variables over complete non-Archimedean fields. We develop Weierstrass theory for Tate algebras, a key tool used to show many of the properties that the Tate algebras have in common with polynomial rings. In particular, we establish Weierstrass Division and the Weierstrass Preparation Theorem, which are analogous to the corresponding results for analytic functions in complex variables. As a consequence, we obtain Strassman’s theorem on the number of zeros of convergent power series in the closed unit ball.

• Friday 10/04, 15:00-16:00. Talk 5: German Mata Gutiez (Paris Sorbonne), Tate algebras are the polynomial algebras of rigid geometry, part 2.

Abstract: In this talk, we will develop properties of Tate Algebras that allow to use them as building blocks for doing geometry in the Non-Archimedean setting. Theorems from the algebraic setting, such as the Noether Normalisation Lemma and the Nullstellensatz, can be proven in this setting. We will also see that Tate Algebras have many familiar properties. Finally, affinoid algebras, the corresponding construction to affine varieties, will be presented, and we will explain how they can be endowed with a natural topology.