Rabbit: a small furry mammal. 'Rabbit, rabbit, rabbit: a superstition for day one of the month to bring luck for the rest of it. 'Rabbit and pork: Cockney rhyming slang for talk, as in too much of. Rabbit: a maths puzzle?
You can watch Robin's full 15-minute talk on the Golden Ratio here.
Neural systems in general - and the human brain in particular - are organised as networks of interconnected components. Across a range of spatial scales from single cells to macroscopic areas, biological neural networks are neither perfectly ordered nor perfectly random. Their heterogeneous organisation supports - and simultaneously constrains - complex patterns of activity.
How does the network constraint affect the cost of a specific brain's pattern? In this talk, Dani will use the formalism of network control theory to define a notion of network economy and will demonstrate how the principle of network economy can inform our study of neural system function in health and disease and provide a useful lens on neural computation.
Dani Bassett is the J. Peter Skirkanich Professor at the University of Pennsylvania. In 2016, Dani was named one of the ten most brilliant scientists of the year by Popular Science magazine and in 2018 received the Erdős–Rényi Prize for fundamental contributions to our understanding of the network architecture of the human brain.
Please email @email to register to attend in person.
The lecture will be broadcast on the Oxford Mathematics YouTube Channel on Wednesday 11 February at 5-6 pm and any time after (no need to register for the online version).
The Oxford Mathematics Public Lectures are generously supported by XTX Markets.
A fundamental phenomenon in the representation theory of finite and compact groups is that irreducible characters tend to take smaller values on elements that are far from central. Character estimates of exponential type (that is, bounds of the form |chi(g)|<chi(1)^(1-epsilon)) are particularly useful for probabilistic applications, such as bounding the mixing time of random walks supported on conjugacy classes.
In 1981, Diaconis and Shahshahani established sharp estimates for irreducible characters of the symmetric group S_n, evaluated at a transposition t = (i j). As an application, they proved that roughly n*log(n) random transpositions are required to mix a deck of n playing cards. This was extended in 2007 by Muller--Schlage-Puchta to to arbitrary permutations in S_n. Exponential character bounds for finite simple groups were subsequently developed through a series of works by Bezrukavnikov, Liebeck, Shalev, Larsen, Guralnick, Tiep, and others.
In this talk, Itay Glazer (Technion) will present recent progress on exponential character estimates for compact Lie groups.
This is based on joint work in progress with Nir Avni, Peter Keevash, and Noam Lifshitz.