On the group completion of the Burau representation

Abstract: On fundamental groups, the discriminant \prod_{i \neq k} (z_i - z_k) \in \C^\times of a finite collection of points of the plane defines the abelianization homomorphism sending a braid to its number of overcrossings less undercrossings or writhe. In terms of diffeomorphisms of the punctured plane, it defines a kind of `invertible topological quantum field theory' associated to the Burau representation, and in the classical physics of point particles the real part of its logarithmic derivative is the potential energy of a collection of Coulomb charges, while its imaginary part is essentially the Nambu-Goto area of a loop of string in the three-sphere. Its higher homotopy theory defines a very interesting a double-loop map \Z \times \Omega^2 S^3 \to \Pic(S^0) to the category of lines over the stable homotopy ring-spectrum, related to Hopkins and Mahowald's exotic (E_2) multiplication on classical integral homology, perhaps related to the `anyons' of nonclassical physics.

(based on joint work with D Rolfsen)

commutative algebraalgebraic geometryanalysis of PDEsalgebraic topologydifferential geometrygeneral topologygeometric topologymetric geometryoperator algebrasquantum algebrarings and algebrassymplectic geometry

Audience: researchers in the topic

( video )

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Algebra Seminar (presented by SMRI)

Series comments: Algebra Seminar:

'Homological comparison of resolution and smoothing'

Will Donovan (Tsinghua University)

Friday Sep 23, 12:00-1:00PM

Online via Zoom

Register here: uni-sydney.zoom.us/meeting/register/tZEpd-isqD0iGdLoqtmNEEQKxmg0xlakSdCq

Abstract: A singular space often comes equipped with (1) a resolution, given by a morphism from a smooth space, and (2) a smoothing, namely a deformation with smooth generic fibre. I will discuss work in progress on how these may be related homologically, starting with the threefold ordinary double point as a key example.

Biography: Will Donovan is currently an Associate professor at Yau MSC, Tsinghua University, Beijing. He is also a member of the adjunct faculty at BIMSA, Yanqi Lake, Huairou, Beijing and a visiting associate scientist at Kavli IPMU, University of Tokyo. He received his PhD in Mathematics in 2011 from Imperial College London. His interests are algebraic geometry, noncommutative geometry, representation theory, string theory and symplectic geometry.

www.maths.usyd.edu.au/u/AlgebraSeminar/

Note: These seminars will be recorded, including participant questions (participants only when asking questions), and uploaded to the SMRI YouTube Channel www.youtube.com/c/SydneyMathematicalResearchInstituteSMRI

Other upcoming SMRI events can be found here: mathematical-research-institute.sydney.edu.au/news-events/

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