


Boyd Research and Education Center, Room 328
Melanie Matchett Wood
William Caspar Graustein Professor of Mathematics
Harvard University
The 2024 edition of our Cantrell Lecture Series will take place TuWedTh March 1214th 2024
The speaker is:
Melanie Matchett Wood
William Caspar Graustein Professor of Mathematics
Harvard University
Tuesday March 12
Title: Central Limit Theorems for matrices and modular arithmetic
Abstract: The Central Limit Theorem is an example of the ubiquitous, yet still surprising, phenomenon in probability that many independent random inputs often combine to give a universal output, insensitive to the input distributions. It is the mathematical reason that the bell curve, or normal distribution, appears so often. We will explore how this phenomenon plays out when investigating the behavior of matrices, especially in modular arithmetic. We will explain how analogs of the bell curve appear for the behavior of matrices, as universal distributions involving the Riemann zeta function. (This talk includes joint work with Hoi Nguyen.)
Wednesday March 13
Title: Finite quotients of 3manifold groups
Abstract: It is wellknown that for any finite group G, there exists a closed 3manifold M with G as a quotient of the fundamental group of M. However, we can ask more detailed questions about the possible finite quotients of 3manifold groups, e.g. for G and H_1,...,H_n finite groups, does there exist a 3manifold group with G as a quotient but no H_i as a quotient? We answer all such questions. To prove nonexistence, we prove new parity properties of the fundamental groups of 3manifolds. To prove existence of 3manifolds with certain finite quotients but not others, we use a probabilistic method, by first proving a formula for the distribution of the fundamental group of a random 3manifold, in the sense of DunfieldThurston. This is joint work with Will Sawin.
Abstract: It is wellknown that for any finite group G, there exists a closed 3manifold M with G as a quotient of the fundamental group of M. However, we can ask more detailed questions about the possible finite quotients of 3manifold groups, e.g. for G and H_1,...,H_n finite groups, does there exist a 3manifold group with G as a quotient but no H_i as a quotient? We answer all such questions. To prove nonexistence, we prove new parity properties of the fundamental groups of 3manifolds. To prove existence of 3manifolds with certain finite quotients but not others, we use a probabilistic method, by first proving a formula for the distribution of the fundamental group of a random 3manifold, in the sense of DunfieldThurston. This is joint work with Will Sawin.
Thursday March 14
Title: Conjectures for distributions of class groups when there are roots of unity in the base field
Abstract: Cohen, Lenstra, and Martinet have given highly influential conjectures on the distribution of class groups of number fields, the finite abelian groups that control the factorization in number fields. Malle, using tabulation of class groups of number fields, found that the CohenLenstraMartinet heuristics for the distributions of class groups of extensions of a number field seemed incorrect when the base field contains roots of unity. We describe a new conjecture for the distribution of class groups (at primes not dividing the order of the Galois group) that corrects for this issue. We explain how large q limit function field results, along with new results on the moment problem for random groups, lead to proofs of new reflection principles over number fields and our conjecture. This talk is on joint work with Will Sawin.
Abstract: Cohen, Lenstra, and Martinet have given highly influential conjectures on the distribution of class groups of number fields, the finite abelian groups that control the factorization in number fields. Malle, using tabulation of class groups of number fields, found that the CohenLenstraMartinet heuristics for the distributions of class groups of extensions of a number field seemed incorrect when the base field contains roots of unity. We describe a new conjecture for the distribution of class groups (at primes not dividing the order of the Galois group) that corrects for this issue. We explain how large q limit function field results, along with new results on the moment problem for random groups, lead to proofs of new reflection principles over number fields and our conjecture. This talk is on joint work with Will Sawin.