A central goal in the study of quantum chaos is being able to make universal statements about the dynamics of generic Hamiltonian systems. Under time evolution, an initially local operator progressively explores the Hilbert space of a system becoming increasingly non-local in the process. We will see that this idea lends itself to a natural notion of operator complexity measured (in the Hilbert space of operators) by the overlap of a time-evolving operator with a basis naturally adapted to time evolution and stratified by the growth in the operator's support. The information contained in this so-called Krylov basis is encoded in a sequence called the Lanczos coefficients which quantify the rate at which an operator is "pushed" along the Krylov basis to successively more complex elements. The universal operator growth hypothesis is then the conjecture that the Lanczos coefficients grow asymptotically linearly in any quantum chaotic system. In this talk, I will present an overview of these ideas and see how they manifest in the example of the well-studied SYK model. This talk is primarily based on 1812.08657.

TBA

TBA

I will review how the large N expansion can be used in the context of the renormalisation group to probe some strongly coupled regimes. In particular, I will discuss a work by Gawedzki and Kupiainen where the authors study the three-dimensional non-Gaussian infrared fixed point of Phi^4 in the case of a hierarchical model of rank-one covariance, and explain how their approach could generalise to more realistic models. 

This is a joint work with Ajay Chandra.  

Alday & Maldacena conjectured an equivalence between string amplitudes in AdS5 ×S5 and null polygonal Wilson loops and 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 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. 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, we find that the nontrivial part of these amplitudes at strong coupling, the remainder function,  is 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 go beyond the semiflat examples with a nontrivial TBA.  These new structures underpinning the N=4 SYM amplitudes  will be important beyond strong coupling.

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.