6:30 pm — 9:00 pm
Monday, 8th Mar 2021
18.30-19.30: Charlotte Hoffmann
20.00-21.00: David Kielak
Ms Charlotte Hoffmann(IST Austraia)
Short words of high imprimitivity rank yield hyperbolic one-relator groups
It is a long standing question whether a group of type $F$ that does not contain Baumslag–Solitar subgroups is necessarily hyperbolic. One-relator groups are of type $F$ and Louder and Wilton showed that if the defining relator has imprimitivity rank greater than $2$, they do not contain Baumslag-Solitar subgroups, so they conjecture that such groups are hyperbolic. Cashen and I verified the conjecture computationally for relators of length at most $17$. In this talk I'll introduce hyperbolic groups and the imprimitivity rank of elements in a free group. I’ll also discuss how to verify hyperbolicity using versions of combinatorial curvature on van Kampen diagrams.
A/Prof David Kielak(University of Oxford)
Recognising surface groups
I will address two problems about recognising surface groups. The first one is the classical problem of classifying Poincaré duality groups in dimension two. I will present a new approach to this, joint with Peter Kropholler. The second problem is about recognising surface groups among one-relator groups. Here I will present a new partial result, joint with Giles Gardam and Alan Logan.
Thursday, 11th Mar 2021SR118, SR Building (and online via Zoom)
Prof. Richard Brent(CARMA, The University of Newcastle)
Primes, the Riemann zeta-function, and sums over zeros
First, I will give a brief introduction to the Riemann zeta-function ζ(s) and its connection with prime numbers. In particular, I will mention the famous “explicit formula” that gives an explicit connection between Chebyshev’s prime-counting function ψ(x) and an infinite sum that involves the zeros of ζ(s). Using the explicit formula, many questions about prime numbers can be reduced to questions about these zeros or sums over the zeros.
Motivated by such results, in the second half of the talk I will consider sums of the form ∑φ(γ), where φ is a function satisfying mild smoothness and monotonicity conditions, and γ ranges over the ordinates of nontrivial zeros ρ = β + iγ of ζ(s), with γ restricted to be in a given interval. I will show how the numerical estimation of such sums can be accelerated, and give some numerical examples.