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The Landau Lecture series


Edmund Landau The Landau lecture series is a series of three lectures by a distinguished mathematician on the subject of his choice. It is usually suggested that one lecture be accessible to the university scientific community at large and the additional two lectures may be more specialized and of interest only to professional mathematicians and specialists.

About Prof. Edmund Landau (1877-1938)
Edmund Landau (1877-1938)
Edmund Landau and the Hebrew University

The Landau lectures were formerly (1991-2010) under the auspices of The Edmund Landau Minerva Center for Research in Mathematical Analysis and Related Areas


2017:      Prof. Jacob Tsimerman    (University of Toronto)

Lecture 1    Colloquium talk.    Counting Number Fields
Thursday, January 5th    14:30      Department Colloquium    Manchester House (Mathematics Building), Lecture Hall 2
Number fields are fields which are finite extensions of Q. They come with a canonical invariant called the discriminant, which can be thought of as the volume of a certain canonically associated lattice. While these objects are central to modern number theory, it turns out that counting them is extremely difficult. More precisely, what is the asymptotic behavior of N (n,X) the number of degree n field extensions of Q with discriminant at most X as X grows, while n remains fixed? It is conjectured by Linnik that N (n,x)∼ c n, and this has been proven for n<=5 by Davenport-Heilbronn(n=3) and Bhargava (n=4,5) using the theory of pre-homogenous vector spaces. Bhargava has conjectured a precise value for c n. We will explain these developments, as well as recent joint work with Arul Shankar that gives another proof in the case n=3, and which yields a strong heuristic reason to believe Linniks conjecture with Bhargava's value for c n . Along the way, we shall also mention the parallel story in function fields, where much more is known thanks to the theory of Etale cohomology.

This lecture is aimed at a general audience.

Lecture 2    Torsion In Class Groups
Sunday, January 8th    12:00      Seminar Talk    Ross 70
(joint with Bhargava,Shankar,Taniguchi,Thorne, and Zhao) Zhangs conjecture asserts that for fixed positive integers m, n, the size of the m-torsion in the class group of a degree n number field is smaller than any power of the discriminant. In all but a handful of cases, the best known result towards this conjecture is the convex bound given by the Brauer-Siegel Theorem. We make progress on this conjecture by giving asubconvex bound on the size of the 2-torsion of the class group of a number field in terms of its discriminant, for any value of n. The proof is surprisingly elementary, and we give several applications stemming from the case of cubic fields, including improved bounds on the number of A4 fields, and on the number of integer points an elliptic curve can have. Along the way, we prove a surprising result on the shape of the lattice of the ring of integers of a number field. Namely, we show that such a lattice is very limited in how skew it can be.

Lecture 3    Cohen-Lenstra in the Presence of Roots of Unity
Monday, January 9th    16:00      Seminar Talk    Ross 70
(joint with Lipnowski, Sawin) The class group is a natural abelian group one can associated to a number field, and it is natural to ask how it varies in families. Cohen and Lenstra famously proposed a model for families of quadratic fields based on random matrices of large rank, and this was later generalized by Cohen-Martinet. However, their model was observed by Malle to have issues when the base field contains roots of unity. We study this in detail in the case of function fields using methods of Ellenberg-Venkatesh-Westerland, and based on this we propose a model in the number field setting. Our conjecture is based on keeping track not only of the underlying group structure, but also certain natural pairings one can define in the presence of roots of unity.

Past lectures in the Landau lecture series

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