Abstract for the talk on 18.05.2022 (14:00 h)

Seminar on Nonlinear Algebra

Sameera Vemulapalli (Princeton University)
Counting Number Fields – Conjectures, Results, and Methods
18.05.2022, 14:00 h, MPI für Mathematik in den Naturwissenschaften Leipzig, G3 10 (Hörsaal)

For a fixed integer d, how many number fields are there of fixed degree d of absolute discriminant less than X? Denote this number N(d, X). This talk will be a friendly foray through the known results, proofs, and main conjectures regarding this question. In particular, it is known that if d = 2,3,4,5 then N(d, X) = O(X). We will discuss the key components of the proofs, which involve the parametrizations of number fields and geometry of numbers. For n >= 6, we will discuss Malle’s conjecture, which predicts the asymptotics of N(d, X) for general degree d. We will also discuss classical upper bounds on N(d, X).

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11.05.2022, 00:08