Alday & Maldacena conjectured an equivalence between string amplitudes in AdS5 ×S5 and null polygonal Wilson loops together with a duality with amplitudes for planar N = 4 super-Yang-Mills (SYM).  At strong coupling this identifies SYM amplitudes with (regularized) areas of minimal surfaces in AdS.  They reformulated the minimal surface problem as a Hitchin system and in collaboration with Gaiotto, Sever & Vieira they introduced a Y-system and a thermodynamic Bethe ansatze (TBA) expressing the complete integrability that could in principle be used to solve for the amplitude at strong coupling. This lecture will review the parts of this material that we need and use them to identify new geometric structures on the spaces of kinematics for super Yang-Mills amplitudes/null polygonal Wilson loops.   In AdS3, the kinematic space is the cluster variety  M_{0.n} X M_{0,n}, where M_{0,n} is the moduli space of n points on the Riemann sphere moduli Mobius transformations.   The nontrivial part of these amplitudes at strong coupling, the remainder function,  turns out to be the (pseudo-)K ̈ahler scalar for a (pseudo-)hyper-Kaher geometry. It satisfies an integrable system and we give its its Lax form. The result follows from a new perspective on Y-systems more generally as defining the natural twistor space associated to the hyperkahler geometry of the Hitchin moduli space for these minimal surfaces. These connections in particular allows us to prove that  the amplitude at strong coupling satisfies the Plebanski equations for a hyperKahler scalar for  these pseudo-hyperk ̈ahler and related geometries. These hyperkahler geometries are nontrivial, (not semiflat) with a nontrivial TBA that encodes the mutations of the cluster structure.  These new structures underpinning the N=4 SYM amplitudes  will be important beyond strong coupling.  This is based on joint work with Hadleight Frost and Omer Gurdogan, https://arxiv.org/abs/2306.17044.

Any two 1 by 1 real matrices commute.  This is in general not the case for 2 by 2 real matrices.  However, if A, B, C, and D are any 2 by 2 real matrices, then ABCD - ABDC - ACBD + ACDB + ADBC - ADCB - BACD + BADC + BCAD - BCDA - BDAC + BDCA + CABD - CADB - CBAD + CBDA + CDAB - CDBA - DABC + DACB + DBAC - DBCA - DCAB + DCBA = 0.  This identity is the first instance of a general result of Amitsur and Levitski; I will explain a simple graph-theoretic proof due to Swan.

TBA

This is a joint OxPDE and Numerical Analysis seminar.