L3

In this talk, we introduce Newton-MR variants for solving nonconvex optimization problems. Unlike the overwhelming majority of Newton-type methods, which rely on conjugate gradient method as the primary workhorse for their respective sub-problems, Newton-MR employs minimum residual (MINRES) method. With certain useful monotonicity properties of MINRES as well as its inherent ability to detect non-positive curvature directions as soon as they arise, we show that our algorithms come with desirable properties including the optimal first and second-order worst-case complexities. Numerical examples demonstrate the performance of our proposed algorithms.

6d N=(2,0) superconformal field theories have natural surface operators similar in many ways to Wilson lines in gauge theories. In this talk, I will discuss how they can be studied using conformal bootstrap techniques, including connection to W-algebras and the so-called inversion formula, focusing on the limit of large central charge.

TBC

Supersymmetric localization allows one to reduce the computation of the partition function of a supersymmetric theory to a finite-dimensional integral, but the space over which one integrates is often singular. In this talk I will explain how one can use shifted symplectic geometry to get rigorous definitions of partition functions and state spaces in theories with extended supersymmetry. For instance, this gives a field-theoretic origin of DT invariants of CY4 manifolds. This is a report on joint work with Brian Williams.

I will discuss a formulation of an Abelian Chern-Simons theory on the lattice employing the modified Villain formalism. The theory suffers from a well-known problem of having extra zero modes in the Gaussian operator. I will argue that these zero modes are associated with a kind of subsystem symmetry which projects out almost all naive Wilson loops. The operators which survive are framed Wilson loops. These turn out to be topological charges of the associated one-form symmetry, and it has the correct topological spin and correlation functions.

The usual approach to study 2d CFT relies on the Virasoro algebra and its representation theory. Moving away from the criticality, this infinite dimensional symmetry is lost so it is useful to have a look at 2d CFTs from the point of view of more general framework of quantum integrability. Every 2d conformal field theory has a natural infinite dimensional family of commuting higher spin conserved quantities that can be constructed out of Virasoro generators. Perhaps surprisingly two different sets of Bethe ansatz equations are known that diagonalise these. The first one is of Gaudin/Calogero type and was discovered by Bazhanov–Lukyanov–Zamolodchikov in the context of ODE/IM correspondence. The second set is a very natural generalisation of the Bethe ansatz for the Heisenberg XXX spin chain and was found more recently by Litvinov. I will discuss these constructions as well as their relation to W-algebras and the affine Yangian.

 Wide Neural Networks are well known for their Gaussian Process behaviour. Based upon this fact, an initialisation scheme for the weights and biases of a network preserving some geometrical properties of the input data is presented — The edge-of-chaos. This talk will introduce such a scheme before briefly mentioning a recent contribution related to the edge-of-chaos dynamics of wide randomly initialized low-rank feedforward networks. Formulae for the optimal weight and bias variances are extended from the full-rank to low-rank setting and are shown to follow from multiplicative scaling. The principle second order effect, the variance of the input-output Jacobian, is derived and shown to increase as the rank to width ratio decreases. These results inform practitioners how to randomly initialize feedforward networks with a reduced number of learnable parameters while in the same ambient dimension, allowing reductions in the computational cost and memory constraints of the associated network.

The study of the fundamental group of an affine manifold has a long history that goes back to Hilbert’s 18th problem. It was asked if the fundamental group of a compact Euclidian affine manifold has a subgroup of a finite index such that every element of this subgroup is translation. The motivation was the study of the symmetry groups of crys- talline structures which are of fundamental importance in the science of crystallography. A natural way to generalize the classical problem is to broaden the class of allowed mo- tions and consider groups of affine transformations. In 1964, L. Auslander in his paper ”The structure of complete locally affine manifolds” stated the following conjecture, now known as the Auslander conjecture: The fundamental group of a compact complete locally flat affine manifold is virtually solvable.

In 1977, in his famous paper ”On fundamental groups of complete affinely flat manifolds”, J.Milnor asked if a free group can be the fundamental group of complete affine flat mani- fold.
The purpose of the talk is to recall the old and to talk about new results, methods and conjectures which are important in the light of these questions .

The talk is aimed at a wide audience and all notions will be explained 1

After a brief introduction to subject of spherical representations of hyperbolic groups, I will present a new construction motivated by a spectral formulation of the so-called Shalom conjecture.This a joint work with Dr Jan Spakula.

A parabolic representation of the free group  is one in which the images of both generators are parabolic elements of $PSL(2,\IC)$. The Riley slice is a closed subset ${\cal R}\subset \IC$ which is a model for the moduli space of parabolic, discrete and faithful representations. The complement of the Riley slice is a bounded Jordan domain within which there are isolated points, accumulating only at the boundary, corresponding to parabolic discrete and faithful representations of rigid subgroups of $PSL(2,\IC)$. Recent work of Aimi, Akiyoshi, Lee, Oshika, Parker, Lee, Sakai, Sakuma \& Yoshida, have topologically identified all these groups. Here we give the first  substantive properties of the nondiscrete representations using ergodic properties of the action of a polynomial semigroup and identifying the Riley slice as the ``Julia set’’ of this dynamical system. We prove a supergroup density theorem: given any irreducible parabolic representation of $F_2$ whatsoever, {\em any}  non-discrete parabolic representation has an arbitrarily small perturbation which contains that group as a conjugate.  Using these ideas we then show that there are nondiscrete parabolic representations with an arbitrarily large number of distinct Nielsen classes of parabolic generators.